Met Office Record Temperature Claims Are Fraudulent

From NOT A LOT OF PEOPLE KNOW THAT

By Paul Homewood

I was in the middle of writing up the story of the heatwave and the supposed record June temperature set at Class 5 junk station Santon Downham, when lo and behold three days after the event the Met Office “discovered” an even higher temperature in Norfolk on the Friday!

Amazingly Lingwood, near Norwich, was nearly half a degree hotter than even Santon Downham, which should set alarm bells ringing:

Both are junk Class 5 sites, but Lingwood is truly worse than junk, as Ray Sanders has diligently revealed at Tallbloke.

As Ray notes, the weather station is just yards north of the edge of a thick wood, apparently in someone’s back garden:

The winds were from the south on Friday, so that bank of trees acted as a very efficient sun trap, preventing any air circulation. It is also surrounded on the other three sides by a tall hedge.

It really is difficult imagining a worse site for measuring temperatures.

Lingwood really does sum up how bad the Met Office’s temperature network is. In fact, it has gone beyond descriptions such as “bad” or “poor”. It is now corrupt, in the same way as the Soviets manipulated statistics for political purposes.

The Met Office keep these sites going, and even open new ones, because they need the artificially high temperatures they produce for political purposes.

Their excuse that it is hard to find pristine sites is hogwash. I could drive out of Norwich and within ten minutes find plenty of open fields, perfect for meteorological purposes.

I’m away at the mo, so my full analysis will be published in the next day or so. That is, as long as the Met Office don’t miraculously find an even hotter place in the meantime!

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341 Comments
Stephen Wilde
July 1, 2026 2:07 am

Open field sites?
You must be joking.

July 1, 2026 2:23 am

This suffers from all the other “junk site” nonsense.

Those trees didn’t spring up overnight.

The wind didn’t blow from the south for the first time ever in June on that particular day.

Likewise all the other excuses.

Whether or not the site is pristine is irrelevant to the fact that it recorded its warmest ever June temperature on Friday.

The Met Office keep these sites going, and even open new ones, because they need the artificially high temperatures they produce for political purposes.

And other conspiratorial, evidence-free gibberish…

Reply to  TheFinalNail
July 1, 2026 2:53 am

The problem is, they are not supplying a continuous series of temps from the same site, which would indeed maybe show a trend if there is one. (Though I am not sure a trend from a very poor site can prove much). Instead, they seem to pick the highest temp reading for the whole country, from different sites every time.

What is it supposed to prove? What policy decisions is it supposed to justify?

This is indeed an unusually hot summer in the UK (and Europe). But is it any more than the result of a rare but not unprecedented weather pattern which happens at irregular and infrequent intervals, but which is not attributable to the very small amount of recent global warming that records show? What’s the evidence?

The problem is, there are all kinds of reports like this, hottest June or July or whatever temp for instance. But what is totally lacking is any kind of systematic forecast of what is supposed to happen to UK weather over the coming decades, and what if anything to do in response to this forecast.

I guarantee you that building more wind and solar farms, abolishing heated towel rails, moving to EVs and heat pumps, ripping out air conditioners in Camden, none of that will come up for consideration as a sensible response that will improve the lives of the people who, you know, actually live in Britain.

Anthony Banton
Reply to  michel
July 1, 2026 3:24 am

But is it any more than the result of a rare but not unprecedented weather pattern which happens at irregular and infrequent intervals, but which is not attributable to the very small amount of recent global warming that records show? What’s the evidence?

I gave it in this post …….

https://wattsupwiththat.com/2026/06/22/temperatures-to-hit-record-breaking-38c-as-heat-dome-heads-for-britain/#comment-4209692

Reply to  Anthony Banton
July 1, 2026 4:40 am

Another doom spreading announcement last week was that southern trains were being delayed or cancelled because of the threat of rail buckling due to the heat. I don’t recall hearing about any actual incidents of this occurring, just the possibility it might. But derailments due to heat have happened before, the most famous of which was June 5, 1950 – The Flying Scotsman train derailed at Tollerton, Nottinghamshire due to heat-buckled track.

And before that? from google AI (after having to ask it to dispense with the ‘climate crisis’ claptrap and look at the entire historical record)

“Regarding how many times track buckling had caused derailments across the UK rail network prior to 1950, the answer is dozens, if not hundreds of times. While complete network-wide statistics for minor infrastructure shifts do not exist for the Victorian and Edwardian eras, official records confirm that sun-warped rails were a known summer operational hazard. 

Engineers left small gaps between rail joints to allow for heat expansion. If a summer was hotter than anticipated, the steel expanded until the gaps closed completely.

The Resulting Buckle: 

Once the expansion gaps closed, the immense internal pressure had nowhere to go, causing the track to violently kink outwards.

Documented Pre-1950 Heat-Buckle Derailments:

Whenever summer temperatures spiked, the pre-nationalisation railway companies routinely dealt with track distortions. Prominent historical examples of actual derailments caused by the sun include:

Felling, County Durham (1907): A passenger train was fully derailed after hitting a section of track warped by summer temperatures.

Hartley, Cumberland (1909): A freight train hauled by a North Eastern Railway locomotive left the tracks when the rails buckled under the direct heat of the sun.”

And from Network Rail:

“Railway tracks can buckle when rail temperatures reach roughly 46°C to 50°C (115°F to 122°F)”

“Unprecedented”

Only in your head.

Reply to  Right-Handed Shark
July 1, 2026 5:07 am

The late summer and autumn of 1972 were an important, though sometimes overlooked, period in Scotland’s environmental and political history. Unlike the famous drought of 1976, the 1972 episode was characterised by an unusual combination of prolonged dry conditions followed by concerns over water management and agricultural resilience. 1972 mattered because it was unprecedented in modern times. The UK Parliament passed the Local Government (Scotland) Act 1973, which reorganised Scottish local government from 1975. Alongside this came major changes in water administration.
The new regional councils became responsible for:

  • public water supplies,
  • sewerage,
  • strategic infrastructure planning,
  • investment across much larger geographic areas.

This meant engineers could think in terms of whole river catchments rather than individual towns.
Although these reforms had been under discussion before 1972, the drought strengthened the case that water should be managed regionally.
A weather event that caused changes in policy and I think that the water levels and river flows use 1972 as a worst case situation.

Gregg Eshelman
Reply to  Ben Vorlich
July 2, 2026 1:01 am

Last summer the UK had to close down higher levels of the canal system due to lack of water. They claimed “drought”, while about every other video by all the Canalers who regularly post on YouTube were showing rain.

The real problem is the decrepit state of the canals, both the ones owned by the Canal River Trust and private companies like Peel Holdings.

What anyone can see on those YouTube videos are almost all the canal gates leak, some so bad it takes a long time to fill a lock to raise a boat. The water’s leaking out nearly as fast as the valves can let it in. Banks are crumbling in, trees need to be cut back. Many have fallen in and it takes ages for anyone to get out with a chainsaw to clear them. Then there’s the failures of embankments. There was a big one at the end of December 2025 and a really big one Jan 1st 2025, at the same spot it failed about 50 years before. There have been some minor failures in between.

While budgets are tight, most of the funds for maintenance come from user license fees, the CRT did get right onto repair works on the Llangollen breach because that part of the canal is also a major municipal water supply aqueduct. I’d bet part of the repair funding has been coming from the villages that canal supplies with water. Fixing this one is a must do project, especially with all the pounds being spent pumping water around the breach.

In contrast, the privately owned Bridgewater canal has had little repair work done in a year and half. Peel Holdings acquired it as part of a package deal when they bought a major shipping channel. The main cause of this failure was overtopping of the walls during heavy rain, and no spillway down to the river running very close to that part of the canal. The side that washed out was away from the river. A simple concrete spillway to the river would’ve prevented it.

Anthony Banton
Reply to  Right-Handed Shark
July 1, 2026 7:59 am

“Unprecedented”

Only in your head.”

I never said that.
I just showed that previous record June maxima had an inherently cooler airmass associated with them.
This one had an 850mb free-air temperature of 20 to 23C which is 3 to 5C higher than those previous record Junes.

Hence we ended up with a surface max that was 2C higher also.

Just basic Meteorology.
You cannot get a surface maxima divorced from the DALR, which will extend from 850mb to the surface + 2/3C superadiabatic at the surface.
That you deny the above facts is no surprise as it is necessary to maintain the ideologically motivated dissonance of the majority of WUWTians.

Reply to  Anthony Banton
July 1, 2026 9:06 am

‘That you deny the above facts is no surprise as it is necessary to maintain the ideologically motivated dissonance of the majority of WUWTians.’

Perhaps it is you and your fellow alarmists who are in denial. A temporary ‘blocking high’ quickly sends European temperatures soaring tens of degrees (F) over ‘normal’, yet we are constantly ‘informed’ by the phenomenological physics of climate modeling that convective heat transport is a mere sideshow compared to radiative transfer in the troposphere.

But be sure to continue combing through the highly tampered surface temperature data to find that elusive CO2 greenhouse signal – it has to be there somewhere!

Reply to  Anthony Banton
July 1, 2026 6:21 am

Yes, read it then. The argument appears to be that the weather pattern is normal and of natural cause. Seems plausible enough. Nothing exceptional about its occurrence or dynamics. And then it goes on to argue that the difference is the heat of the air that this weather pattern draws up.

It must being argued that the Sahara is hotter than in the past due to global warming. Consequently when its hot air reaches the UK as it does in these episodes it raises the temp of the heat wave.

What’s the evidence that its any warmer in the Sahara now than in the past? What’s the evidence that if it is, its due to global warming?

The null hypothesis would be that these are very variable weather patterns and weather systems. Sometimes a warm episode in the Sahara will coincide with the weather pattern, sometimes not. Sometimes the Sahara will be warmer than others. Why do we need to invoke global warming, and what exactly is the evidence for the attribution?

I still want to know, even supposing you could make a case for Saharan warming to be the cause, and for that to have increased the temps of the latest heat wave, and even if you expect more of these very hot ones, why on earth does that justify current government policies?

taxed
Reply to  michel
July 1, 2026 8:12 am

A big clue to why the air coming up from the south of the UK is so hot currently is because of the Mediterranean Sea been so warm. It’s currently well above it’s average SST and what that suggests is that there has been persistent high pressure weather patterning over this area during the late spring and early summer. So allowing extended amounts of sunshine and light winds to warm up the Mediterranean Sea much quicker then what is usually the case. Along with all this sunshine not only will the sea be warmer but also the surrounding land as well.

The evidence to support this case will be found recorded in the sunshine records of the holiday resorts around the Mediterranean Sea this spring and early summer.

Reply to  michel
July 1, 2026 8:30 am

Very nice response, along with a series of logical questions, to which I look forward to seeing AB’s answers.

Anthony Banton
Reply to  michel
July 1, 2026 8:57 am

“Sometimes a warm episode in the Sahara will coincide with the weather pattern, sometimes not. Sometimes the Sahara will be warmer than others”

It seems that that air is now hotter than the historical past.

(I know it’s 10 years out if date )

https://journals.ametsoc.org/view/journals/clim/28/16/jcli-d-14-00230.1.xml

Abstract
Evaluation of three reanalyses (ERA-Interim, NCEP-2, and MERRA) and two observational datasets [CRU and Global Historical Climatology Network (GHCN)] for 1979–2012 demonstrates that the surface temperature of the Sahara Desert has increased at a rate that is 2–4 times greater than that of the tropical-mean temperature over the 34-yr time period. ”

comment image

“Annual-mean surface temperature anomalies (K) averaged over the Sahara Desert (red), the entire tropics (blue), tropical land (green), and the entire globe (black) for (a) ERAI, (b) NCEP-2, (c) MERRA, and the (d) CRU and (e) GHCN observations.”

Reply to  Anthony Banton
July 1, 2026 10:01 am

Also from the Abstract:

‘While the response to enhanced greenhouse gas forcing over most of the globe involves the full depth of the atmosphere, with increases in long-wave back radiation increasing latent heat fluxes, the dryness of the Sahara surface precludes this response. Changes in the surface heat balance over the Sahara during the analysis period are primarily in the upward and downward long-wave fluxes.’

Hmmm, I’ve always heard that positive water vapor feedback is a really big deal in CAGW, yet these guys maintain that the dry-as-a-bone Sahara has warmed 2-4 times more than the tropics.

Also, please provide some details regarding the ‘upward and downward long-wave fluxes’, including:

Are they ‘energy’ fluxes and, if so, do they occur simultaneously?

From what part of the wild blue yonder does the downward long-wave flux originate?

How are they measured and using what ‘instrument’?

Reply to  Anthony Banton
July 1, 2026 1:10 pm

This chart says all that needs to be says.

A large proportion is JUNK DATA.

uk-temp-stations-2
Anthony Banton
Reply to  bnice2000
July 2, 2026 3:57 am

“The Met Office weather and climate stations, as well as the subsequent data processing, follow international standards. It is important to note that the observing stations numbered siting classification is not an indication of the station quality or suitability for weather and climate monitoring, it is a measure that helps the scientists understand the ability of the station to represent a region. Some stations are deliberately situated close to airports to provide aeronautical information –and these are different in nature from those used to provide weather and climate information.”

https://wmo.int/content/wmo-statement-uk-met-office-observations

Reply to  Anthony Banton
July 2, 2026 4:48 am

It is important to note that the observing stations numbered siting classification is not an indication of the station quality or suitability for weather and climate monitoring,

Did you really read what you wrote. If the siting classification is not a quality indicator, exactly what does the increase in uncertainty actually mean.

Better yet, it is just another reason why averaging stations is meaningless. If stations being averaged do not al meet the same quality specification you really don’t know what you have ended up with.

Reply to  Jim Gorman
July 2, 2026 5:01 am

I went back and reread the WMO classification document. It says the following:

In the following text, the classification is (occasionally) completed with an estimated uncertainty due to siting, which has to be added in the uncertainty budget of the measurement. This estimation is coming from bibliographic studies and/or some comparative tests

This should lead one to recognize that uncertainty is an important concept and that uncertainty should be properly propagated throughout any calculations. Since uncertainties add, it should be obvious that stations with high uncertainty contaminate any conclusions made from averaging.

Reply to  Anthony Banton
July 2, 2026 12:39 am

OK, that is interesting.

Assume the account is correct, that would explain why this heatwave is hotter than usual for UK summers. It would suggest that future heatwaves could be similar.

It wouldn’t suggest that the heatwave mechanism will kick in any more often. And it would not explain why the Sahara is warming more rapidly – the attribution to global warming of this particular phenomenon is still a question.

And of course the big one for anyone unfortunate enough to be living in the UK, it would not explain why Net Zero as currently being implemented is a rational response to the warming Sahara! Because it isn’t!

strativarius
Reply to  TheFinalNail
July 1, 2026 3:21 am

I showed you a graph and you now know that the so called climate crisis is all in your cranium.

Yet you persist in the delusion. Why is that? I think we should be told.

Alan M
Reply to  TheFinalNail
July 1, 2026 3:42 am

Yes, it recorded its warmest ever June temperature. but what does that show when by definition, the site is not fit for purpose?

SxyxS
Reply to  Alan M
July 1, 2026 9:51 am

These sites were built with the purpose to ” deliver new records “.

MarkW
Reply to  SxyxS
July 1, 2026 3:18 pm

80% of the sites, both in the US and England, when surveyed, were determined to be junk sites, unfit for purpose.
You don’t get that bad, unless someone has been working at it.

MarkW
Reply to  Alan M
July 1, 2026 3:17 pm

The claim seems to be that the site wasn’t fit for purpose 20 years ago, therefore any change between then and now means something.

Reply to  TheFinalNail
July 1, 2026 4:27 am

Anyone with a brain knows that politics has influenced the weather for the last 50 years. The reason they have a class system, is to know which stations are most reliable at giving a correct area temperature. As temperature is an intensive property. Reflecting a local state, which may include contaminated heat close to the station. Not based on an equilibrium temperature of a large area. Where no local sources of heat can raise the temperature. It’s the same with hurricanes, satellites show a low resolution intense wind speed, but a NHC plane records a upper air local wind for a minute which is 30% more intense than the satellite record. This local state is politicised by the media and used to describe the whole hurricane top speed. As wind speed is also a intensive property. And politics today will exploit intensive properties for political purposes.

Sparta Nova 4
Reply to  slindsayyulegmailcom
July 1, 2026 5:21 am

We can anticipate the D.C. temperatures during this present heatwave will decline rapidly as Congress has gone into recess. A lot less hot air. 😉

Reply to  TheFinalNail
July 1, 2026 4:51 am

When its uncertainty is greater than 5 degrees in each direction, that record temp is meaningless.

Reply to  TheFinalNail
July 1, 2026 4:56 am

Whether or not the site is pristine is irrelevant to the fact that it recorded its warmest ever June temperature on Friday.

Of course it is irrelevant, if your goal is to simply use non-scientific measurements. This is supposed to be science, you know. Science requires strict adherence to certain protocols when making measurements so that they can be comparable when analyzing them statistically.

Your screed sounds very much like a two year old screaming, “I don’t want to go to bed” or a teenager yelling, “You can’t tell me what to do”. Grow up and learn some responsibility for doing things correctly instead of just yelling “I know better”, post some fact-based reasons for claiming something is incorrect.

Do you know what an outlier is? How does one go about determining what is an outlier?

Sparta Nova 4
Reply to  Jim Gorman
July 1, 2026 5:22 am

The Veruca Salt syndrome.

paul courtney
Reply to  Jim Gorman
July 1, 2026 12:41 pm

Mr. Gorman: Good spot, Mr. Nail did say the secret part out loud- reliability of measurement “irrelevant” to the fact that he relies on it!!
Next he’ll tell us the unreliability is cleaned up when they are (wait for it) averaged.
Then he’ll tell us what Pat frank (and the Gormans!) are missing.
When it’s really hot out, I can sometimes enjoy summer reruns. Not this time, Mr. Nail, please get some new material.

Reply to  TheFinalNail
July 1, 2026 5:10 am

All is not lost.
The Labour government are giving about a billion in subsidy to a carbon capture scheme which could easily reduce the amount of CO2 in the atmosphere by 1 part in a million in under 15,000 years.
They aren’t stupid you know.

Sparta Nova 4
Reply to  TheFinalNail
July 1, 2026 5:16 am

I could put a thermometer in my oven and record the highest temperature ever.

Lighten up Frances.

Reply to  Sparta Nova 4
July 1, 2026 7:01 am

Depends if the oven door is open or closed – h/t bdgwx

/s

claysanborn
Reply to  Frank from NoVA
July 1, 2026 10:02 am

Too hot with oven door open? That’s OK, to counter that, leave the refrigerator door open too.

Mr.
Reply to  claysanborn
July 1, 2026 1:36 pm

Ah, tabling averaged temperature constructs.
Well, that’s it – slam-dunk, game over!!
/sarc

Sparta Nova 4
Reply to  Frank from NoVA
July 1, 2026 12:40 pm

I suppose I could use the broiler instead of the full oven. 😉

Reply to  Sparta Nova 4
July 1, 2026 4:20 pm

Fine, but you’ll have to clear that with bdgwx first.

Reply to  TheFinalNail
July 1, 2026 8:53 am

If I set up a weather station in my garden, which is only 240 square metres in area, then the maximum temperature I obtained would vary depending on where it was located, even behind a Stevenson screen. To be honest I was expecting the highest temperature to have occurred at one of Marham, Lakenheath or Mildenhall in Norfolk and perhaps earlier in the month Lakenheath or Waddington.

MarkW
Reply to  TheFinalNail
July 1, 2026 3:14 pm

That is the thing about sites that are very bad for many reasons. The acolytes will start screaming that since no one flaw is very meaningful, there is no need to fix the site.
On the other hand, I wonder what their view would be if anyone ever found a site flaw that resulted in a too cool reaing?

He admits that the site has gotten worse over time, but then says it doesn’t matter because we just recorded a new high that was several tenths of a degree warmer than the previous high 20 years ago.

And he completely misses the irony.

strativarius
July 1, 2026 3:05 am

They have over 100 “virtual” weather stations with over 3 million years of observations…

Its hotter than evah or so they say, but I know they’re parroting a narrative…

Time for an official Fifa’ hydration break’.

Keitho
Editor
July 1, 2026 3:11 am

Story tip: I see El Niño is performing to prescription.

https://klimata.org/el-ninometer-real-time-enso-index/?i=1

Nick Stokes
July 1, 2026 3:34 am

It’s always something, isn’t it?

Jet engines, solar panels, now trees!!!

As TFN says, why were those trees doing whatever on this day and not before?

strativarius
Reply to  Nick Stokes
July 1, 2026 4:08 am

There’s always something to nitpick.

A lot of knickers being wet over nothing. Also known as climate anxiety…

Reply to  Nick Stokes
July 1, 2026 5:07 am

We do not know what they were or were not doing on previous days. As to why a given junk site shows a given temp? There is no way to know. That is what makes it a junk site.

The rational approach to this is roughly as follows. This is a very hot UK summer. Not unprecedented, but pretty unusual. Whether a given station shows a UK max temp for any particular month is unimportant if you are looking for evidence to guide policy decisions on housing codes, insulation, energy policy. If the high temp on June 27 or on July 3 it has no climate significance. But one will be trumpeted as a record for June, when if it had happened a few days later it would have been nothing in particular for July.

As to policy, as I said in another post, maybe UK summer temps are going to be higher, maybe there are going to be more frequent heat waves. Fine, what are the policy implications?

I guarantee you they are not going to be what the UK governments have been doing for the last 20 years. It is not a rational response to more frequent UK heat waves to try and move everyone to heat pumps and EVs, and to move electricity generation to wind and solar. Its like spitting in the wind. It just makes the problem, if there is one, worse. And ‘because climate’ is not any kind of reason.

Revise building codes, and retrofit offices and homes with insulation and external shutters, improve ventilation. And allow new builds to install air con! And improve power generation so it delivers cheap abundant electricity.

Reply to  Nick Stokes
July 1, 2026 5:12 am

And what do increasing temperatures have to do with Britain’s Net Zero scheme?
Nothing.
The UK government has abandoned talk of our Net Zero scheme having any effect on temperature.

Reply to  Nick Stokes
July 1, 2026 5:25 am

As TFN says, why were those trees doing whatever on this day and not before?

How about growing? How about additional leaves due to CO2 fertilization?

You of all people should understand that measurement uncertainty is a requirement for adequately addressing statistical significance.

A class 5 station can have an uncertainty interval of ±”systematic+5)°C. And yes, that means the temperature could have been 5°C warmer that what was recorded. But, it could have been 5°C cooler also. No one really knows do they?

The statistical significance disappears within the uncertainty interval, doesn’t it?

Reply to  Jim Gorman
July 1, 2026 8:05 am

A class 5 station can have an uncertainty interval of ±”systematic+5)°C. And yes, that means the temperature could have been 5°C warmer that what was recorded. But, it could have been 5°C cooler also. No one really knows do they?”

LOLLL

You’re treating the event as though it rests on a single weather station, but it doesn’t. This was a continent-wide heat wave observed across hundreds of stations in multiple countries.

To dismiss a Europe wide heat wave as a measurement artifact, you’d have to argue that a large number of independent observations all produced errors in the same direction and of a similar magnitude at the same time. 

Reply to  Eldrosion
July 1, 2026 8:33 am

To dismiss a Europe wide heat wave as a measurement artifact

I am not dismissing it at all. I am just pointing out that the absolute value may or may not be a new record. All measurements are estimates, using a mean value to compare measurements is not statistically correct.

Reply to  Jim Gorman
July 1, 2026 9:30 am

https://wattsupwiththat.com/2026/07/01/met-office-record-temperature-claims-are-fraudulent/#comment-4212851

The top values Bellman gave are not marginal differences at the edge of uncertainty. The top entry (32.8C) is 0.9C above the next highest recent day, and 2.3C above the 2019 entry.

The top 4 are all clustered together in the same event window (Jun. 23 – 26), which is exactly what you would expect from a real heat wave.

Measurement uncertainty matters most when differences are small (e.g., 30.1 Cvs. 30.2C ) and you’re trying to declare a single exact winner.

It’s not relevant when multiple consecutive records are broken by a large margin across several days and several locations.

At that point, the uncertainty would need to be systematically biased in the same direction across multiple days and across multiple stations and instruments. That’s not a realistic error structure.

Reply to  Eldrosion
July 1, 2026 11:23 am

The top values Bellman gave are not marginal differences at the edge of uncertainty. The top entry (32.8C) is 0.9C above the next highest recent day, and 2.3C above the 2019 entry.

You really have no idea what uncertainty means do you? “Marginal differences at the edge of uncertainty” is a meaningless word salad.

Did you not understand what I said earlier? Measurement uncertainty defines an interval where the TRUE VALUE may lay. No value is more likely than any other value inside that interval.

32.8C that is ±5C might be 0.9C above the next highest day or it might be 0.9C below the next highest. In fact, the true value may be several degrees below the next highest. YOU HAVE NO WAY TO KNOW because of uncertainty. That is why proper measurements are a necessity in science.

Here is the very first paragraph in the GUM, JCGM 100-2008.

0.1 When reporting the result of a measurement of a physical quantity, it is obligatory that some quantitative indication of the quality of the result be given so that those who use it can assess its reliability. Without such an indication, measurement results cannot be compared, either among themselves or with reference values given in a specification or standard.

Measurements cannot be compared among themselves. Maybe you have a unique understanding about how measurement uncertainty is related to comparing temperatures among themselves. Please share that understanding or alternatively, refute what the GUM says.

Reply to  Jim Gorman
July 1, 2026 11:54 am

The atmosphere itself imposes physical constraints on what the observations can plausibly be.

If every day’s truly value could equally be anywhere within +/-5 C, then the true series could be (as an example):

Monday – 33.4C
Tuesday – 27.3 C
Wednesday – 36.0C
Thursday – 28.1C
Friday – 35.9C

That’s not a physically plausible weather scenario.

High pressure ridges strengthen over several days, remain stationary for a while, and then weaken.

So the temperature evolution is highly correlated from one day to the next.

Sparta Nova 4
Reply to  Eldrosion
July 1, 2026 12:47 pm

We had a measured temperature change of -50 F within a 24 hour interval.

So temperature evolution is not highly correlated from one day to the next.

Reply to  Sparta Nova 4
July 1, 2026 12:52 pm

That’s a different meteorological situation.

Mr.
Reply to  Eldrosion
July 1, 2026 1:44 pm

Every one of them is.
That’s what weather does. one day to the next, all around the world.
Are things different in your world?
Do tell . . .

MarkW
Reply to  Eldrosion
July 1, 2026 3:28 pm

When in doubt, just deny the evidence.

Reply to  MarkW
July 1, 2026 4:39 pm

I’m not denying anything. A 50°F temperature drop in 24 hours can certainly occur behind a strong cold front.

It’s simply a different meteorological situation from the persistent high-pressure pattern associated with the heat wave under discussion.

Do deniers think cold fronts and high pressure are the same thing?

Reply to  Eldrosion
July 1, 2026 7:08 pm

So the temperature evolution is highly correlated from one day to the next.

So you have just figured out what uncertainty means. Uncertainty is ultimately a reliability value of a measurement.

From JCGM 100-2008, the very first paragraph.

0.1 When reporting the result of a measurement of a physical quantity, it is obligatory that some quantitative indication of the quality of the result be given so that those who use it can assess its reliability. Without such an indication, measurement results cannot be compared, either among themselves or with reference values given in a specification or standard. It is therefore necessary that there be a readily implemented, easily understood, and generally accepted procedure for characterizing the quality of a result of a measurement, thatis, for evaluating and expressing its uncertainty.

Measurement results can not be compared. That is because you are not certain about their individual values.

For round numbers lets use relative uncertainty. 5/35 is about 14%. What if we were talking about being paid for gold. Would you accept an uncertainty of 14% when that amounts to about $4000/oz? That could cost you about $550. It could also mean you made an extra $550. The point is that you don’t know what is correct and there is no way to ascertain the correctness. That is uncertainty.

Reply to  Jim Gorman
July 1, 2026 11:09 pm

Each thermometer has its own uncertainty, but the true temperatures are constrained by the same atmospheric circulation.

During a persistent high pressure ridge, temperatures evolve according to the underlying synoptic pattern.

The true temperatures are therefore strongly correlated in both space and time. A physically consistent weather field places much tighter constraints on the plausible true temperatures than treating each day’s uncertainty interval in isolation.

Regarding the JGCM:

#1) Hundreds, if not thousands, of independent temperature measurements from stations across Europe satisfy the JCGM repeatability criterion (see below) because they measure the same measurand: the regional weather

“Repeatability (of results of measurements): closeness of the agreement between the results of successive measurements of the same measurand.”

#2) Even if repeatability is not satisfied, Type B evaluation explicitly allows scientific judgment based on all available information.

The GUM states:

“For an estimate xi of an input quantity Xi that has not been obtained from repeated observations, the associated estimated variance u2(xi ) or the standard uncertainty u(xi ) is evaluated by scientific judgement based on all of the available information on the possible variability of Xi.”

It then provides examples of this “pool of available information,” and making clear that the list is illustrative rather than exhaustive.

I don’t see why a physically consistent body of meteorological evidence (hundreds of weather stations, radiosondes, numerical weather models, knowledge of the evolving synoptic pattern, etc.) wouldn’t qualify as “all available information” when assessing the plausibility of the true value.

Reply to  Eldrosion
July 2, 2026 5:41 am

Each thermometer has its own uncertainty, but the true temperatures are constrained by the same atmospheric circulation.

The uncertainty of each thermometer must be added into the total combined uncertainty of the measurand.

The GUM says what it means, i.e., “For an estimate xi of an input quantity Xi …”. Each Xᵢ = xᵢ, is an estimate of a unique measurand combined into a single measurand.

The GUM defines “input quantity as follow:

4.1.3 The set of input quantities X1, X2, …, XN may be categorized as:

⎯ quantities whose values and uncertainties are directly determined in the current measurement. These values and uncertainties may be obtained from, for example, a single observation, repeated observations, or judgement based on experience, and may involve the determination of corrections to instrument readings and corrections for influence quantities, such as ambient temperature, barometric pressure, and humidity;

⎯ quantities whose values and uncertainties are brought into the measurement from external sources, such as quantities associated with calibrated measurement standards, certified reference materials, and reference data obtained from handbooks.

Each station is a unique and separate “input quantity”. X1 is station 1, X2 is station 2, etc. Each has its own uncertainty u(xᵢ), as shown in Eq. 10 in the GUM.

Put the uncertainty of each input quantity xᵢ into Eq. 10 and see what you end up with for a combined uncertainty. I think you’ll find that stations with an uncertainty of ±5°C quickly increase the total u꜀(y)

Reply to  Jim Gorman
July 2, 2026 4:17 pm

We’re not using weather stations as input quantities to calculate a single derived measurand via Eq. (10).

We’re assessing whether a particular station’s reported temperature is physically consistent with the broader atmospheric state inferred from many independent observations.

Reply to  Eldrosion
July 3, 2026 6:17 pm

We’re assessing whether a particular station’s reported temperature is physically consistent with the broader atmospheric state inferred from many independent observations.”

You are measuring DIFFERENT THINGS. The measurement uncertainties of different things ADD, they don’t subtract.

If the uncertainty interval is wider than the difference you are trying to identify, YOU DON’T KNOW IF THE DIFFERENCE IS REAL. You can’t even tell the slope of a trend!

Look at the temperatures in the attached graphic for temps in NE Kansas at 8pm, 7/3/2026. The mean temp is 88F. The standard deviation is 2F for the stated values alone. This assumes the stations are measuring the *same* thing – which of course they aren’t. The actual measurement uncertainty would be the RSS sum of individual components. With 14 components and an assumed measurement uncertainty of +/- 1F the total measurement uncertainty would be +/- 3.7F.

How can you infer the accuracy of any specific measurement station with such a wide range of temperatures, such a wide standard deviation of the data, and the measurement uncertainty of the total?

wibw_7_3_2026_8pm
Reply to  Tim Gorman
July 3, 2026 7:59 pm

“You are measuring DIFFERENT THINGS. The measurement uncertainties of different things ADD, they don’t subtract.”

No. The measurand is air temperature being measured.

The GUM defines a measurand as:

“B.2.9 Measurand: particular quantity subject to measurement.”

You can’t even tell the slope of a trend!”

Who brought up slopes? I didn’t.

The standard deviation is 2F for the stated values alone. “

Why are you treating the observed spread as though it’s primarily measurement uncertainty?

That spread is exactly what you’d expect because the stations are in different locations.

Do you really expect expect a station on a hill, another in an urban area, another in a valley, and another over farmland to all report identical temperatures?

Most of that spread reflects real spatial variability, not thermometer error. Those are two completely different things.

Reply to  Eldrosion
July 4, 2026 6:59 am

No. The measurand is air temperature being measured.”

Uh huh. Each of the temperatures in the image is the same measurand.

Here you are again, trying to lecture people on things you know nothing about.

GUM:
———-
B.2.2
value (of a quantity)
magnitude of a particular quantity generally expressed as a unit of measurement multiplied by a number
EXAMPLE 1 Length of a rod: 5,34 m or 534 cm.
EXAMPLE 2 Mass of a body: 0,152 kg or 152 g.
EXAMPLE 3 Amount of substance of a sample of water (H2O): 0,012 mol or 12 mmol.
—————— (bolding mine)

It is *NOT* the lengths of multiple rods, the mass of multiple bodies, or the amount of substance in different samples.

———————
3.1.4 In many cases, the result of a measurement is determined on the basis of series of observations obtained under repeatability conditions
—————-

Each of those temperatures reflects the measurement of a different measurand, i.e. the value of a property of a different object.

Who brought up slopes? I didn’t.”

Of course you did! Now you are reduced to saying that a trend line has no slope?

“Why are you treating the observed spread as though it’s primarily measurement uncertainty?”

YOU are the one trying to classify the temperature measurements shown as repeated measurements of the same measurand. That makes the measurement uncertainty into a Type A. In a Type A measurement uncertainty the standard deviation of the observations is the measurement uncertainty.

That spread is exactly what you’d expect because the stations are in different locations.”

Yep. And that means you ADD the individual measurement uncertainties to get the total measurement uncertainty. The very thing you are trying to say is wrong!

When you add those measurement uncertainty values you quickly reach a point where any differences are subsumed by the measurement uncertainty. You simply don’t know if any differences actually exist or not. Unless you have a crystal ball supplied by climate science.

Do you really expect expect a station on a hill, another in an urban area, another in a valley, and another over farmland to all report identical temperatures?”

I’m sure you don’t realize it but you just refuted the entire concept of homogenization of temperatures by replacing missing temperature data from a station with the data from surrounding stations.

Homogenization is just one more garbage climate science meme. All it does is spread measurement uncertainty around!

Most of that spread reflects real spatial variability, not thermometer error. Those are two completely different things.”

Spatial uncertainty is a *sampling* uncertainty, not a measurement uncertainty. Some of the spread is due to MICRO-ENVIRONMENT differences. Different elevations, different geography, different terrain, different pressures, different surroundings such as UHI or evapotranspiration, even the paint condition on each station from UV effects, etc. And, of course, some is due to things like mis-calibration, cloud cover, humidity, etc.

Climate science simply averages all of these temperatures without regard to the accumulation of measurement uncertainty. It’s a result of the garbage climate science meme of “all measurement uncertainty is random, Gaussian, and cancels”.

Reply to  Tim Gorman
July 4, 2026 8:31 am

Uh huh. Each of the temperatures in the image is the same measurand.”

I didn’t say they were the same measurand. I said they were measurements of the same physical quantity: air temperature.

BUT each station’s air temperature is a different measurand because it is a different particular quantity. That’s entirely consistent with the GUM definition.

Your confusion is on display for all to see.

“Of course you did! Now you are reduced to saying that a trend line has no slope?”

HAHAHA. Anyone reading this thread can see I never mentioned slope.

This is becoming a pattern. You keep projecting arguments onto me that I never made.

Another example is your claim that I was treating the temperatures in your figure as measurements of the same measurand. I explicitly said the opposite: they’re measurements of the same physical quantity (air temperature), but each station’s air temperature is a different measurand because it’s a different particular quantity.

YOU are the one trying to classify the temperature measurements shown as repeated measurements of the same measurand.”

No. That’s not what I said.

I did not classify those station temperatures as repeated measurements of the same measurand.

I said the spread among them is largely due to real spatial variability because they’re measuring temperatures at different location, not because the thermometers are inaccurate.

You’re now arguing against a position I never took.

Yep. And that means you ADD the individual measurement uncertainties to get the total measurement uncertainty. The very thing you are trying to say is wrong!”

Again, you’re assuming I’m calculating a single combined measurand. I’ve already explained that I’m not. I’m evaluating the plausibility of a station observation in the context of the surrounding meteorological field. Those are different problems.

“I’m sure you don’t realize it but you just refuted the entire concept of homogenization of temperatures by replacing missing temperature data from a station with the data from surrounding stations.”

So desperate. Now you have to sidetrack the discussion into homogenization.
Let’s just say you’ve demonstrated, time and again, that you don’t understand how temperature anomalies are calculated.

Spatial uncertainty is a *sampling* uncertainty”

What?????

Spatial variability is not sampling variability. Spatial variability is the real physical difference between temperatures at different locations

That’s not the same thing as sampling variability.

Reply to  Eldrosion
July 4, 2026 10:49 am

 I’m evaluating the plausibility of a station observation in the context of the surrounding meteorological field. Those are different problems.

So much hand waving! Your statement means nothing more than “I’m just eyeballing the spread”. If you can’t specify a measurand, a process, a model, then you are just making stuff up and that has nothing to with science. Your criticism certainly won’t stand up to examination as you have so aptly displayed.

Reply to  Jim Gorman
July 4, 2026 11:14 am

No, Jim.

See here:

“Spatial and temporal continuity (or consistency) checks are based on the fact the value or magnitude of nearby observations are usually similar. The process compares observations adjacent in either space or time with each other, and if specified differences are noted, initiates further investigation. For example, if two airports on opposite sides of a town report dew points values 10 degrees apart, an instrument check may be in order. The evaluator in these situations must ensure the observed differences are not due to some mesoscale feature such as a front or dryline to justify the observed values.”

https://training.weather.gov/nwstc/Hydrology/HYDRO/QCModule/QCConc.HTML

Operational meteorology is not hand waving.

Reply to  Eldrosion
July 4, 2026 1:16 pm

Spatial and temporal continuity (or consistency) checks are based on the fact the value or magnitude of nearby observations are usually similar”

I just gave you an image of NE Kansas temperatures that shows this isn’t true. Would comparing Pikes Peak temperature with the temp in Colorado Springs be spatially and temporally of the same value or even magnitude?

In the summer, San Diego and Ramona, CA temps are more than 30F different, yet they are only 30 miles apart. Geography and terrain are totally different. NO SPATIAL SIMALARITY.

If you want, I can get you several examples of temperatures from the east side and west side of a mountain and show you the temporal difference in their profiles. But it would be better if you did it for yourself so you would actually have to learn something on the subject.

I’ve even had climate scientists tell me that homogenization from stations 500 miles away is legitimate because they are in the same time zone!

Values and magnitudes stop being similar at the first hedgerow between stations let alone between a river valley and the associated river plateau.

Take the east/west side of a mountain. Boise vs Fort Collins. Which do you suppose gets the most sun insolation on a diurnal basis? Are their temperature values similar both in value and time?

Reply to  Tim Gorman
July 4, 2026 5:34 pm

You’re illustrating the point, not refuting it.

The NWS didn’t say nearby observations are always similar. It said they are usually similar, and that when they aren’t, the evaluator must determine whether the difference is explained by known meteorological or geographic features.

In other words, they don’t treat every large difference as instrument uncertainty. They first assess whether the observed atmospheric or geographic conditions physically justify the difference. That’s exactly the kind of physical consistency check I’ve been describing.

Reply to  Eldrosion
July 2, 2026 6:08 am

The true temperatures are therefore strongly correlated in both space and time.”

So what? The MEASUREMENT of those temperatures is what has uncertainty. Please note that the uncertainty interval has both a plus and a minus value. The *true temperature” can be anywhere in that interval FOR ANY SPECIFIC INDIVIDUAL MEASURMENT VALUE.

That uncertainty interval includes both RANDOM and SYSTEMATIC uncertainty effects. The random effects do *NOT* cancel to zero leaving only SYSTEMATIC uncertainty effects. Having only systematic effects, ones that can be IDENTIFIED, is a requirement in order to validate measurements over time being highly correlated.

You simply don’t know and can’t tell if the individual measurements, even ones taken on consecutive days, are strongly correlated or not when the measurement uncertainty is +/- 5C.

What you are saying, in essence, is that you can know the Great Unknown. That’s the claim of a carnival fortune teller with a cloudy crystal ball. And that’s a perfect descriptor of climate science today!

#1) Hundreds, if not thousands, of independent temperature measurements from stations across Europe satisfy the JCGM repeatability criterion (see below) because they measure the same measurand: the regional weather”

You apparently don’t even understand what the word “measurand” means in the GUM. It’s a typical example of the knowledge of metrology by climate science.

Here are the requirements for REPEATABILITY:
——————————–
B.2.15
repeatability (of results of measurements)
closeness of the agreement between the results of successive measurements of the same measurand carried
out under the same conditions of measurement
NOTE 1 These conditions are called repeatability conditions.
NOTE 2 Repeatability conditions include:
— the same measurement procedure
— the same observer
— the same measuring instrument, used under the same conditions
— the same location
— repetition over a short period of time.
————————

Note carefully “the same measuring instrument”, “the same location”, and “the same conditions”.

Measurements of “regional weather” does not meet ANY of these requirements.

I don’t see why a physically consistent body of meteorological evidence (hundreds of weather stations, radiosondes, numerical weather models, knowledge of the evolving synoptic pattern, etc.) wouldn’t qualify as “all available information” when assessing the plausibility of the true value”

“I don’t see” – that’s because like your other compatriots in climate science (bdgwx, bellman, TFN, etc) you ABSOLUTELY REFUSE to study metrology for meaning and context. You cherry pick pieces that you think support your position without regard to whether those “pieces” comport with the meaning and context of metrology.

Each of the hundreds of individual measurements have a measurement uncertainty, that measurement uncertainty represents the Variance of a random variable (i.e. the individual measurement). When you add random variables, each with its own variance, the variance of each individual random variable ADDS. Variance_total = Var1 + Var2 + …

That Var_total becomes the measurement uncertainty for the result of combining all those measurements. If the measurement uncertainty of each individual temperature in the dataset is +/- 0.5C then the uncertainty associated with that data set is (n) * +/- 0.5C. If “n” is 1000 the measurement uncertainty becomes +/- 500C. In other words whatever you do with the dataset is physically useless.

How do you and climate science get around this? By using the unstated assumptions of 1. all systematic uncertainty = 0, 2. all measurement uncertainty is random, Gaussian, and cancels, and 3. the SEM thus becomes the measurement uncertainty which can be made arbitrarily small with more and more individual components being included in the dataset. In other words, GARBABE ASSUMPTIONS. This is even ignoring that the SEM is SAMPLING ERROR, not measurement error.

Then you and climate science refuse to characterize the dataset as a single sample of a lot of components or whether each component represents a single sample of size 1. 1. if it is a single sample then, regardless of the number of entries, the SEM has an in-built uncertainty of 30%, 2. if it is a collection of multiple samples with a sample size of 1 the measurement uncertainty is sum of all of the individual measurement uncertainties.

Climate science makes some more garbage assumptions about the temperature measurements: 1. the measurement uncertainty of each component is Gaussian, 2. that the measurements themselves are not multi-modal, and 3. systematic uncertainty = 0.

From a physical science point of view as well as metrology, climate science should move to using enthalpy and 5-number statistical descriptors. Abandon all the garbage assumptions. We should have data available for at least 40 years to do so.

Reply to  Tim Gorman
July 2, 2026 4:42 pm

“You simply don’t know and can’t tell if the individual measurements, even ones taken on consecutive days, are strongly correlated or not when the measurement uncertainty is +/- 5C.”

Then I have no choice but to believe that you think there was no high pressure over Europe at the time of the heat wave. This is desptie the fact that exceptionally large geopotential heights were measured over the UK at the time.

“Note carefully “the same measuring instrument”, “the same location”, and “the same conditions”.”

Suppose you’re measuring the air temperature at Heathrow Airport at 15:00 UTC on July 1. Now imagine the same thermometer measures that temperature thousands of times over the course of a few seconds. All of those conditions are satisified.

You’re not listening to what I am saying.

“If “n” is 1000 the measurement uncertainty becomes +/- 500C.”

LOLLLLLL, you can’t seriously believe that!

Reply to  Eldrosion
July 3, 2026 6:38 pm

Then I have no choice but to believe that you think there was no high pressure over Europe at the time of the heat wave.”

How did you make the leap to this? What you don’t know is the TRUE VALUE of that high pressure. That’s because the measurement uncertainty is a part of the measurement.

YOU SIMPLY CAN’T IGNORE MEASUREMENT UNCERTAINTY.

Suppose you’re measuring the air temperature at Heathrow Airport at 15:00 UTC on July 1. Now imagine the same thermometer measures that temperature thousands of times over the course of a few seconds. All of those conditions are satisified.”

NO! Not all the conditions are satisfied. 1. Temperature measuring stations do *NOT* make thousands of measurements every second. 2. Because of air flow through the measuring station each of those thousand measurements will be of a different measurand. 3. over one second, hysteresis can cause an asymmetric uncertainty even with digital measuring devices. Meaning the measurement value is dependent on the direction of any temperature change, including over just one second. It’s not even clear if a field measuring device exists with the response time to make a thousand measurements per second.

LOLLLLLL, you can’t seriously believe that!”

Of course I believe that. It’s the MATH. The issue is that you don’t understand metrology at all!

Again, one more time:

When the measurement uncertainty is greater than the difference trying to be identified, you don’t know if the difference exists or not!

You exceed the measurement uncertainty trigger for just two stations with +/- 1C if you are trying to identify a difference in the tenths or hundreds digit! You don’t need to go any further than that! Trying to add in more stations doesn’t help!

Measurement uncertainty is *NOT* the SEM. The SEM is sampling uncertainty, not measurement uncertainty. You can minimize sampling uncertainty with larger samples. You cannot minimize measurement uncertainty with larger samples of different things.

If you were designing a bridge would you use the SEM of measurements of 100 beams as the measurement uncertainty for shear strength of the beams? Climate science does just that!

Reply to  Tim Gorman
July 3, 2026 7:46 pm

“What you don’t know is the TRUE VALUE of that high pressure. That’s because the measurement uncertainty is a part of the measurement.
YOU SIMPLY CAN’T IGNORE MEASUREMENT UNCERTAINTY.”

I’m not ignoring measurement uncertainty. I’m rejecting the assumption that every value within a nominal ±5°C interval is equally plausible once you account for the observed synoptic pattern. The atmosphere itself provides additional constraints on the physically plausible values.

2. Because of air flow through the measuring station each of those thousand measurements will be of a different measurand.”

Simply absurd. Your interpretation makes the concept of repeatability practically unusable for any continuously varying quantity, including the temperature examples discussed throughout the GUM.

You’d know that if you read the document as a whole instead of cherry picking isolated passages that you think support your argument. You have a knack for projecting.

The purpose of repeatability is not to require that the physical world would become perfectly frozen. It’s intended to characterize the dispersion of repeated measurements under nominally the same conditions.

Reply to  Eldrosion
July 4, 2026 6:24 am

I’m not ignoring measurement uncertainty. I’m rejecting the assumption that every value within a nominal ±5°C interval is equally plausible once you account for the observed synoptic pattern. “

That only means you don’t grasp the concept of measurement uncertainty.

You are using the conclusion as part of the argument.

I know the synoptic pattern because I can ignore the measurement uncertainty involved with it so then I can ignore the measurement uncertainty of the components making up the synoptic pattern.

If you don’t know the actual data values because of measurement uncertainty, then you can’t know the synoptic pattern without measurement uncertainty either.

Simply absurd. Your interpretation makes the concept of repeatability practically unusable for any continuously varying quantity, including the temperature examples discussed throughout the GUM.”

Talk about absurd. It is perfectly feasible to use multiple sensors to create multiple observations of the same thing at the same time. The issue is the measurement uncertainty associated with each of those sensors contributes to the total measurement uncertainty of their average.

You simply can’t get away from the concept of measurement uncertainty. It’s just plainly obvious that you have never been involved in the field of ACCURATE measuring at all. For instance, how do you handle the effects of inserting a temperature sensor into the flow of a hot gas or liquid that will then cause turbulence in the flow thus affecting the temperature distribution? Inserting multiple sensors will only increase the turbulence in the flow.

What does the measurement uncertainty budget look like for such a situation?

You’d know that if you read the document as a whole instead of cherry picking isolated passages that you think support your argument. You have a knack for projecting.”

The only people cherry picking around here are you and your compatriots trying to justify how climate science garbage assumptions are “true science”.

You had absolutely *NO* idea of what “repeatability” means when you said temperature data from different measurands using different instruments result in meeting “repeatability” requirements.

Now you are just whining that someone caught you out. The only fix for that is for *YOU* to learn something on the subject before trying to lecture others on how they are wrong.

Reply to  Tim Gorman
July 4, 2026 7:05 am

The only people cherry picking around here are you and your compatriots trying to justify how climate science garbage assumptions are “true science”.

+14242

Pure projection.

You had absolutely *NO* idea of what “repeatability” means when you said temperature data from different measurands using different instruments result in meeting “repeatability” requirements.

Zip, zero, nada.

Now you are just whining that someone caught you out. The only fix for that is for *YOU* to learn something on the subject before trying to lecture others on how they are wrong.

You mean like when bellman calls you stupid or a liar?

Reply to  Tim Gorman
July 4, 2026 9:01 am

“I know the synoptic pattern because I can ignore the measurement uncertainty involved with it so then I can ignore the measurement uncertainty of the components making up the synoptic pattern.”

I never said I ignore measurement uncertainty.

“If you don’t know the actual data values because of measurement uncertainty, then you can’t know the synoptic pattern without measurement uncertainty either.”

The synoptic pattern isn’t inferred from temperature measurements alone. That’s the point I’ve been trying to make.

It’s independently supported by geopotential height analyses, radiosondes, surface pressure observations, satellite observations, and many other datasets.

Those independent observations constrain the physically plausible atmospheric state even though each individual measurement has uncertainty.

Talk about absurd. It is perfectly feasible to use multiple sensors to create multiple observations of the same thing at the same time.”

Agreed. Yet you seem to make an exception for temperature, even though the GUM itself uses temperature in several examples.

Reply to  Eldrosion
July 4, 2026 12:59 pm

The synoptic pattern isn’t inferred from temperature measurements alone. That’s the point I’ve been trying to make.”

Each and every factor in your pattern that is a measurement has measurement uncertainty. *ALL* of those measurement uncertainties add when forming a whole based on the factors.

If you are saying that you are forming a synoptic pattern that is *NOT* based on measurements, then you are basically saying that you are guessing – which has a measurement uncertainty all of its own!

It’s independently supported by geopotential height analyses, radiosondes, surface pressure observations, satellite observations, and many other datasets.”

Each and every one of those factors involve measurements that have uncertainty. They all add to the total uncertainty. You are actually making your case WORSE for yourself instead of better!

“Those independent observations constrain the physically plausible atmospheric state even though each individual measurement has uncertainty.”

Being bounded doesn’t lessen their variance and it is their variance that determines the uncertainty, not the bounds!

Does no one supporting climate science actually read the GUM for meaning and context?

GUM, Sec 2.2.3
—————–
NOTE 3 It is understood that the result of the measurement is the best estimate of the value of the measurand, and that all components of uncertainty, including those arising from systematic effects, such as components associated with corrections and reference standards, contribute to the dispersion.”
——————-

The actual value of the GAT is bounded by the fixed insolation from the sun and the variable of T^4 for earth. That does *NOT* mean that the GAT doesn’t have lots of measurement uncertainty since it is not yet at the bound.

Agreed. Yet you seem to make an exception for temperature, even though the GUM itself uses temperature in several examples.”

More bullshite! Have you been talking with bdgwx? Each of the examples are either associated with a single object or have assumptions allowing different things to be considered as the same thing in order to provide a teaching example. ALL OF THEM!

The GUM simply doesn’t say anywhere that intensive property values for different objects can be averaged. It’s not in there!

Climate science and its supporters like bdgwx always confuse the mathematical ability to perform a calculation and its applicability to the real world. Yes, you can calculate the average value of ten single observations of temperature from ten different waterbaths but that value has no real world significance. It stems from the statistician/mathematician meme of “number is just numbers”. Anything you can do with this set of numbers you can do with any set of numbers.

Reply to  Tim Gorman
July 4, 2026 7:53 pm

*ALL* of those measurement uncertainties add when forming a whole based on the factors.”

Point already addressed.

If you are saying that you are forming a synoptic pattern that is *NOT* based on measurements, then you are basically saying that you are guessing – which has a measurement uncertainty all of its own!”

Anthony Banton posted a radiosonde sounding for Cornwall at 12Z on June 25:

https://wattsupwiththat.com/2026/06/24/some-revealing-screen-dumps-from-the-uk-met-office-website/#comment-4210711

Of course the radiosonde has its own measurement uncertainty. But it provides an independent line of evidence, and it too indicates an record-breaking warmth (global warming is so scary).

Anyway, that’s a pretty good way to show that the uncertainty in the true value is much smaller than the +/-5°C you’re claiming, certainly not enough to move the event out of record-breaking territory.

“More bullshite! Have you been talking with bdgwx? Each of the examples are either associated with a single object or have assumptions allowing different things to be considered as the same thing in order to provide a teaching example. ALL OF THEM!”

That’s not true. Section 4.1.3 explicitly discusses environmental quantities as input quantities:

The set of input quantities X1, X2, …, XN may be categorized as: ⎯ quantities whose values and uncertainties are directly determined in the current measurement. These values and uncertainties may be obtained from, for example, a single observation, repeated observations, or judgement based on experience, and may involve the determination of corrections to instrument readings and corrections for influence quantities, such as ambient temperature, barometric pressure, and humidity;”

Note the word ambient. The GUM is clearly discussing environmental measurements, not just isolated laboratory objects.

Reply to  Eldrosion
July 4, 2026 11:10 am

I’m not ignoring measurement uncertainty. I’m rejecting the assumption that every value within a nominal ±5°C interval is equally plausible once you account for the observed synoptic pattern. The atmosphere itself provides additional constraints on the physically plausible values.

More hand waving pretending to be science. Show us some math that allows you to make any specific conclusion about temperatures based on a “observed synoptic pattern”.

The definition of synoptic is forming a general summary or synopsis, or a broad overview. If that is your idea of calculating a measurand, good luck with that in a scientific discussion.

I’ve done the work. I have seen many monthly standard deviations in round numbers of ±4°F (±2°C). That is a spread of 8°F (4°C). Your synoptic overview isn’t any more accurate than that.

Reply to  Jim Gorman
July 4, 2026 11:56 am

“More hand waving pretending to be science. Show us some math that allows you to make any specific conclusion about temperatures based on a “observed synoptic pattern”.”

The dry adiabatic lapse rate is about 9.8°C/km. A 5°C temperature difference is therefore equivalent to roughly 500 meters of vertical displacement in dry adiabatic terms.

So treating +/-5°C as the effective uncertainty of the true temperature during a coherent, persistent high pressure event is just nonsensical.

Reply to  Eldrosion
July 4, 2026 1:33 pm

The dry adiabatic lapse rate is *NOT* a major uncertainty factor. The condition of the paint on the measuring station after years of UV *is*. The color of the ground underneath the station *is*. The calibration drift of both the sensor and the digitization device from extended heating *is*. The nearest hedgerow blocking wind flow *is*. How many mud dauber wasp nests sit in the air intake of the station *is*. The distance to nearest heat exhaust from air conditioning *is*. UHI *is*.

I could go on but it would probably be wasted on you. It’s obvious that you have never once had to prepare an uncertainty budget for *anything*. My guess is that you’ve never even looked at the ISO documents on uncertainty budgets.

Have you ever, even once, been outside with the wind in your face while the clouds are moving the exact opposite direction? What do you suppose that does to the temperature difference between the ground and the troposphere?

Reply to  Tim Gorman
July 4, 2026 2:19 pm

“The dry adiabatic lapse rate is *NOT* a major uncertainty factor.”

Talk about strawman.

I’ll reiterate my point again:

A +/- 5°C uncertainty for the true temperature is inconsistent with this specific synoptic event because the atmospheric state constrains what temperatures are physically plausible.

Clearly, that doesn’t fit your narrative

Reply to  Eldrosion
July 4, 2026 2:35 pm

In other words you don’t have a single refutation to offer as to why troposphere temperatures can differ significantly from surface temperatures regardless of lapse rate.

Didn’t think you would.

true temperature”

And what, exactly, is “true temperature”. What you think it *should* be?

Reply to  Tim Gorman
July 4, 2026 5:53 pm

“In other words you don’t have a single refutation to offer as to why troposphere temperatures can differ significantly from surface temperatures regardless of lapse rate.”

During a persistent high pressure ridge, the atmospheric column evolves in a physically consistent way. Air subsides from aloft, is compressed as pressure increases, and warms approximately at the dry adiabatic lapse rate.

You’re correct that temperatures aloft and at the surface can differ substantially but they are not evolving independently. Their evolution is linked by the same atmospheric dynamics.

“And what, exactly, is “true temperature”. What you think it *should* be?”

That is one thing the GUM is actually quite clear about.

The true value isn’t what you think it should be. It’s the actual value of the measurand. Although it generally cannot be known exactly, measurement uncertainty characterizes the interval within which it is believed to lie.

That is one thing the GUM is actually quite clear about.

The true value isn’t what you think it should be. It’s the actual value of the measurand. Although it can’t be known exactly, measurement uncertainty characterizes the interval within which it is believed to lie with a stated level of confidence.

—————————————————–

B.2.3

true value (of a quantity)
value consistent with the definition of a given particular quantity
NOTE 1 This is a value that would be obtained by a perfect measurement.
NOTE 2 True values are by nature indeterminate.
NOTE 3 The indefinite article “a”, rather than the definite article “the”, is used in conjunction with “true value” because there may be many values consistent with the definition of a given particular quantity. 

—————————————————–

Again, please read the GUM for comprehension rather than cherry picking isolated passages.

Reply to  Jim Gorman
July 1, 2026 2:08 pm

Measurements cannot be compared among themselves.”

Independent measurements absolutely can be compared. 

Reply to  Eldrosion
July 4, 2026 1:36 pm

Only if they meet repeatability requirements! An independent measurement of the temperature in Mexico City at noon today vs an independent measurement of the temperature in Kansas City at noon will tell you almost nothing of use in the physical world.

Reply to  Tim Gorman
July 4, 2026 2:22 pm

Who else is saying otherwise? It seems like only you are making that claim.

MarkW
Reply to  Jim Gorman
July 1, 2026 3:27 pm

I have yet to meet any alarmist who knew the first thing about statistics.

Reply to  MarkW
July 1, 2026 4:45 pm

hahaha

We’re discussing metrology, not statistics. That distinction matters, especially when dealing with someone like Jim Gorman, who is so particular about not conflating the two.

Tell him, for example, that the uncertainty of an average decreases as 1/√N under the usual assumptions for independent measurements, just as the standard error of the mean does. He won’t like it.

Reply to  Eldrosion
July 2, 2026 11:53 am

Tell him, for example, that the uncertainty of an average decreases as 1/√N under the usual assumptions for independent measurements, just as the standard error of the mean does. He won’t like it.

An average reduces the uncertainty! ROTFLMAO.

Each measurement in a temperature average has a sample size of 1. Regardless of how many measurements/stations there are, each data point has a sample size of 1.

Basically, what you are describing is the Standard Error Measure (SEM) or according to the GUM, the standard uncertainty of the mean. The SEM is not measurement uncertainty. Let’s look at some definitions from the GUM.

B.2.17 experimental standard deviation
for a series of n measurements of the same measurand, the quantity s(qk) characterizing the dispersion of the results

4.2.2 … This estimate of variance and its positive square root s(qk), termed the experimental standard deviation (B.2.17), characterize the variability of the observed values qk , or more specifically, their dispersion about their mean q.

2.2.3 The formal definition of the term “uncertainty of measurement” developed for use in this Guide and in the VIM [6] (VIM:1993, definition 3.9) is as follows:

uncertainty (of measurement)

parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand

C.2.20

variance

a measure of dispersion, which is the sum of the squared deviations of observations from their average divided by one less than the number of observations

C.3.3 Standard deviation

The standard deviation is the positive square root of the variance. Whereas a Type A standard uncertainty is obtained by taking the square root of the statistically evaluated variance, it is often more convenient when determining a Type B standard uncertainty to evaluate a nonstatistical equivalent standard deviation first and then to obtain the equivalent variance by squaring the standard deviation.

I have many other references if you need them. They all repeat the same thing. The measurement uncertainty is the standard deviation of the random variable containing the observations.

There is only one instance where the standard uncertainty of the mean (SEM) is useful. That is when multiple samples of the same measurand are made each of which has “n” observations under repeatable conditions. One can then say that for that one, single, unique measurand that the mean ±SEM is the estimated value with a small uncertainty.

If you want to know how to determine the measurement uncertainty of a heterogeneous material, read Sections F.1.1.2 and H.6. Those cover the measurement uncertainty involved with regional and global temperatures.

Reply to  Jim Gorman
July 3, 2026 6:27 pm

Seeing no response from Eldrosion, I assume I hit the nail on the head.

Hopefully, the response about uncertainty decreasing with 1/√n goes away in the future.

Reply to  Eldrosion
July 3, 2026 6:45 pm

Tell him, for example, that the uncertainty of an average decreases as 1/√N under the usual assumptions for independent measurements, “

You are as bad as bellman and bdgwx. They absolutely refuse to identify whether they are speaking of the SEM or the measurement uncertainty when discussing measurements. They just use the word “uncertainty” hoping to be able to use the Equivocation fallacy when called on their assertions being incorrect.

You are doing the same. The “uncertainty of an average” from using 1/sqrt(N) IS SAMPLING UNCERTAINTY. It is *NOT* measurement uncertainty.

My guess is that you will also continue to not specify whether you are talking about sampling uncertainty or measurement uncertainty – just like bellman and bdgwx.

It all stems from the garbage meme of: “all measurement uncertainty is random, Gaussian, and cancels” that is continually found in climate science.

Reply to  Eldrosion
July 1, 2026 3:17 pm

The claim of Britain’s highest temperature on record IS a single event.

Reply to  Retired_Engineer_Jim
July 1, 2026 4:54 pm

The UK wasn’t the only place experiencing record or near record heat.

https://wmo.int/media/news/record-breaking-heat-spreads-through-europe

Germany broke new temperature records for three consecutive days, with the town of Coschen in eastern Germany near the border with Poland reporting 41.7°C on 28 June. A total 252 weather stations recorded all time temperature records – the highest ever. According to DWD, 46 stations throughout Germany recorded temperatures of above (40°C) -until 27 June. “

Hungary: A new June temperature record of 40.7°C was recorded near the capital Budapest on 28 June, with heat forecast to intensify further. Poland also recorded a new all-time temperature record, provisionally of 40.5°C, as did the Czech Republic.”

Austria set a new June temperature record of 40.0 °C in the Vienna City, with a Red Alert continuing for the capital on 29 June, according to Geosphere Austria. “

“In the Netherlands, the Dutch national meteorological service, KNMI, issued an unprecedented Red Alert for extreme heat for eight provinces for 26 June for 26 June, and reported a number of new station records, and a new national June temperature record of 39.4 °C. “

Global warming is so scary.

MarkW
Reply to  Eldrosion
July 1, 2026 3:25 pm

You are acting as if this was the only defective site.
As recent surveys have shown, most of them are junk.

PS: I love the way you try to shift the terms of the debate.
Nobody has denied that there is a heat wave. It’s the claims that it is a huge record breaker that are being challenged.

Reply to  MarkW
July 1, 2026 5:14 pm

So, to summarize the “issue”:

You acknowledge that Europe experienced an exceptional heat wave.

What’s being contested is whether the UK specifically set a genuine national temperature record, even though it was still an exceptionally hot event.

And never mind that the same synoptic event produced record or near record temperatures across multiple European countries. So, even if the UK record were revised, it wouldn’t change the broader picture.

And to bolster that mountain you’ve made out of a molehill, you resort to the usual conspiratorial blah blah blah about the Met Office and its “junk stations.”

Information originating from Ray Sanders, an incompetent cretin.

MarkW
Reply to  Jim Gorman
July 1, 2026 3:24 pm

There are many ways to make a site hotter than it should be, there are few ways to make it cooler than it should be.

Reply to  Nick Stokes
July 1, 2026 7:28 am

And how does Paul explain the fact that I predicted the eventual outcome in advance?

https://community.netweather.tv/topic/102625-summer-2026-max-temp-heat-watch/page/2/#findComment-5487646

Reply to  Nick Stokes
July 1, 2026 7:47 am

It doesn’t matter what was or wasn’t there before, what matters is what’s there now.

If the ground around the weather station doesn’t comply with WMO standards it’s trash.

Nick Stokes
Reply to  Redge
July 1, 2026 2:03 pm

Clearly it does comply. You can see where this goes…
Level 5? Bottom level. Trash. Get rid of it. Then..
Level 4? Bottom level. Trash. Get rid of it. Then..
Level 3? Bottom level. Trash. Get rid of it. Then..
Level 2? Bottom level. Trash. Get rid of it.Then..
Level 1? Well, maybe

aussiecol
Reply to  Nick Stokes
July 1, 2026 3:31 pm

Just cut the trees down Nick… problem solved.

1saveenergy
July 1, 2026 3:49 am

[“It really is difficult imagining a worse site for measuring temperatures.”]

No, it’s a perfect site for measuring temperatures
If you want to show consistently high results, sheltered, sun-trapped, perfect.
The majority of sheeple will believe the results, especially if they hear it on the BBC (:<((

MrGrimNasty
July 1, 2026 4:23 am

It is not ‘fraudulent’, it is what it is (to use such a ghastly phrase).

It is pub trivia, simply the highest reading recorded on all the official weather stations recording at the time.

There is no supreme level of perfection in all the existing records either.

As I’ve explained before, I have had the same max/min thermometer in the same unchanged position and environs since the early 80s. I recorded an all time high (not just June) by just over 2C. Getting anywhere near 30C used to be exceptional in my coastal location, now it’s very ordinary. 34C is the new 30C.

It clearly was a record breaking heatwave.

Just because a site may look bad and be rated as up to 5C error does not mean it was on this occasion. Arguing the toss over points that are mere conjecture is pretty pointless.

Frankemann
Reply to  MrGrimNasty
July 1, 2026 4:50 am

…may look bad? So you argue it is not bad? Perfect siting for objective reading of temperature representative for the larger area where it is located?

MrGrimNasty
Reply to  Frankemann
July 1, 2026 5:13 am

Yawn.

Reply to  Frankemann
July 1, 2026 1:22 pm

Big mismatch between urban and rural sites in the UK. (see chart)

Met Office likes to use badly sited, highly tainted urban and airport sites, because they have an agenda to keep.

They have made basically zero attempt to maintain a system of sites that are even remotely acceptable for long term measurements.

And have, in fact, deliberately created more sites that are in the class 4,5 bracket.

UK-town-vs-rural
Anthony Banton
Reply to  bnice2000
July 2, 2026 4:10 am

Mr bnice:

How many times have I told you that stations next to the sea have their air temps moderated (controlled) by the SST over which the prevailing wind blows?

Therefore you MUST be aware that the ocean does not have diurnal temp delta as does land.
Do I have to explain the physics ?!

Hence you present Stornoway and Lerwick, both of which have a prevailing wind from the SW ….. from off the Atlantic ocean.

Oh yes, contamination is just fine with bnice so long as it’s a cold one!

Reply to  Anthony Banton
July 2, 2026 4:42 am

How many times have I told you that stations next to the sea have their air temps moderated (controlled) by the SST over which the prevailing wind blows?

One more reason for not averaging temperatures, you are averaging two different things. It is like making a saddle for the average size of a Thoroughbred horse and a Shetland pony. The average is meaningless.

Anthony Banton
Reply to  Jim Gorman
July 3, 2026 4:31 am

One more reason for not averaging temperatures”

Anomalies mr Gorman.
Anomalies.

Reply to  Anthony Banton
July 4, 2026 3:57 am

Anomalies don’t help! This has been pointed out MULTIPLE times here on WUWT. Anomalies inherit the measurement uncertainties of the components used to determine the anomalies.

u_total = u(average) + u(measurement)

regardless of whether you are doing an addition or subtraction of the given stated values.

The anomaly determined by [anomaly = avg – measurement] STILL has the measurement uncertainty of u_total = u(average) + u(meas).

And it’s not just this. If the variances (i.e. the measurement uncertainties) of the average and the measurement are different then some kind of weighting should be done to give the most accurate value (i.e. the smallest measurement uncertainty) more weight.

Climate science does none of this. They don’t properly propagate measurement uncertainty onto the anomaly nor do they weight the components to give more weight to the most accurate value.

ANOMALIES DON’T HELP.

It’s just one more garbage assumption by climate science. If you don’t propagate the measurement uncertainties, then you have no way to judge whether the anomaly actually exists or not or if it is just an artifact of the inaccuracy of the measurements.

Reply to  MrGrimNasty
July 1, 2026 12:21 pm

Error is not uncertainty, and uncertainty is not error.

Reply to  karlomonte
July 3, 2026 11:26 am

Same applies to anomalies and even worse. Variance is not propagated at all. Tell what the variance/standard deviation of a typical anomaly actually is and how it is calculated. Or do you just assume the measurement uncertainty disappears amongst all the averaging?

Reply to  Jim Gorman
July 3, 2026 2:02 pm

The later is the firm position of bigoilbob, and he casts aspersions on anyone who doesn’t buy into this narrative.

July 1, 2026 4:29 am

However, bear in mind that “38 °C” was recorded at RAF Lakenheath and Norwich Airport at around the same time of day.

MrGrimNasty
Reply to  Jim Hunt
July 1, 2026 7:32 am

Within tenths of 40C in Jersey, but that’s B.Isles not UK.

Reply to  MrGrimNasty
July 1, 2026 12:53 pm

More France like than UK like!

Temperature wise at least.

Reply to  Jim Hunt
July 1, 2026 1:24 pm

Airports, OK ! 😉

Reply to  Jim Hunt
July 1, 2026 11:51 pm

Amusingly, after blocking me on Twatter Ray Sanders is now making precisely the same point over on Tallbloke’s Talkshop!

“At 13:26 GMT 26th June 2026 WMO03577 Synoptic station at RAF Lakenheath posted a reading of 38°C. Surely not even the Met Office can deny this.”

I wonder where he got that idea?

Ray-Block-2026-06-29_20-58
July 1, 2026 4:47 am

Note, these records are provisional and have yet to be verified, so I think any claim of fraud is unwise.

Meanwhile, I’d love to know what Homewood’s views are on the accuracy of the previous record, set in 1976 in Southampton, Mayflower.

strativarius
Reply to  Bellman
July 1, 2026 4:59 am

These ‘records’ are based on a nanosecond in geological time. They are laughable at best.

Reply to  strativarius
July 1, 2026 5:05 am

It’s the nanosecond of time I happen to be living in. That means more to me than temperatures during the Mesozoic era.

strativarius
Reply to  Bellman
July 1, 2026 5:09 am

It’s the nanosecond of time I happen to be living in. 

The fact that we do not control the weather and never have done escapes you.
You can come out from under the table, you know… Maybe you can be saved by studying orbital mechanics?

Reply to  strativarius
July 1, 2026 7:16 am

We “humans” do have some control over “the climate” though.

paul courtney
Reply to  Jim Hunt
July 1, 2026 12:59 pm

Mr. Hunt: Well, we talk about it, but nobody does anything! h/t Mark Twain.
Putting humans in scare quotes was startling, then we saw the climate in scare quotes, and we knew that you’re not a serious commenter.

Reply to  paul courtney
July 1, 2026 4:51 pm

“Sneer quotes” in actual fact.

Reply to  Jim Hunt
July 1, 2026 1:25 pm

Only the local urban microclimate.

Global climate.. ZERO.

Reply to  Jim Hunt
July 1, 2026 3:23 pm

No, if climate is the “average” of weather over 30 years, if we have no control over weather, then we have no control over climate. We may, of course, have some control over “Climate”.

SxyxS
Reply to  Bellman
July 1, 2026 10:02 am

It only means something to you because you have been told for decades that it does.

Meanwhile in the real world any aspect, be it the ever rising food production, the non- existing sea level rise, the greening of the planet(the co2 increase in greenhouse gas terms is irrelevant but 40% in terms of plant food)
and everything else are prove that there is no climate crisis(an absolutely absurd term anyway).

You belong to a species that occupies climate territories where the coldest and hottest habited places are 60 degrees apart in terms of average temperatures and more than 100 in terms of peaks.
So stop being such a pussy about a degree less or more.

MarkW
Reply to  Bellman
July 1, 2026 3:31 pm

Translation: I don’t care about science.

Reply to  Bellman
July 1, 2026 5:40 am

Meanwhile, I’d love to know what Homewood’s views are on the accuracy of the previous record, set in 1976 in Southampton, Mayflower.

Put them on a graph and include the uncertainty bars. How much overlap is there? What is the reliance you can put on a temperature measurement of either reading?

Remember that the “mean” is only the center of an interval, and the true value can be anywhere within that interval. A ±5°C interval does not provide much statistical significance to any comparison.

MarkW
Reply to  Bellman
July 1, 2026 3:31 pm

If they aren’t verified, than it is pretty unprofessional of them to be putting out press releases touting them.

Almost as if they never had any intention of being accurate.

July 1, 2026 4:55 am

Meanwhile CET recorded the 4 warmest max June temperatures last month

1 2026-06-26 32.8
2 2026-06-25 31.9
3 2026-06-23 31.5
4 2026-06-24 30.7
5 2019-06-29 30.5
6 1976-06-28 30.3
7 2005-06-19 30.2
8 1976-06-29 30.1
9 1947-06-02 30.0
10 1950-06-06 29.9

Personally though I think “June” record is not very meaningful as there were warmer days in early July. The warmest ever CET maximums

1 2022-07-19 37.3
2 2022-07-18 34.8
3 2019-07-25 34.2
4 1990-08-03 33.4
5 2020-07-31 33.2
6 1976-07-03 33.1
7 2006-07-19 33.0
=8 1990-08-02 32.8
=8 2003-08-09 32.8
=8 2015-07-01 32.8
=8 2026-06-26 32.8

CET also recorded it’s warmest minimum temperature for all months, not just June.

1 2026-06-26 20.1
2 2016-07-20 19.6
3 2022-07-19 18.9
4 1997-08-11 18.8
5 1948-07-29 18.7
6 1975-08-05 18.5
=6 2004-08-09 18.5
=6 2020-08-12 18.5
=9 1949-09-05 18.4
=9 2001-07-29 18.4

Reply to  Bellman
July 1, 2026 5:15 am

Oh please! You and many others are predicting catastrophic climate/weather 100 years into the future based on less that 100 years of mostly questionable data collection. Look at this graphic, study it and the time scales involved and get your mind right. It will make an honest man/women out of you.

Pick two points in this graphic. One on an upward temp trend and one on a downward trend and convince me temperature spikes, hot and cold did not occur with in a 100, 1000 or 10000 year span.

And just because thermometers were not invented doesn’t mean the spikes didn’t occur.

MrGrimNasty
Reply to  Bellman
July 1, 2026 5:19 am

You and your facts, that’ll get you nowhere here with my fellow skeptics.

strativarius
Reply to  MrGrimNasty
July 1, 2026 5:37 am

your facts

His truth….

MrGrimNasty
Reply to  strativarius
July 1, 2026 7:34 am

It’s the best scientific data available, your comment is just silly; especially considering your only reason for rejecting them is that it offends your own beliefs.

Reply to  MrGrimNasty
July 1, 2026 7:47 am

It’s the best scientific data available,

If you include a measurement uncertainty of ±5°C to be scientific, then I suppose your characterization is possible. Using it to gauge statistical probability between two separate measurements is perverse.

Reply to  Jim Gorman
July 1, 2026 1:29 pm

With “warm” measurements from bad sites, you can drop the “minus” off the +/- symbol.

Sparta Nova 4
Reply to  MrGrimNasty
July 1, 2026 12:51 pm

It is not “scientific data” it is measurement data.

Mr.
Reply to  Sparta Nova 4
July 1, 2026 2:00 pm

Not even measurement data.

No numbers are “data” if the original source measurements as recorded are subjected to numerical gymnastics such as “averaging’ or “homogenization” etc.

Such numbers are “constructs”, often where assumptions are baked in.

And “assumptions” are not immutable laws of physics or anything else.

So the probity of temps constructs is rancid.

MarkW
Reply to  MrGrimNasty
July 1, 2026 3:33 pm

If the best available isn’t fit for purpose, charging ahead and using it anyway is a fool’s errand.

Reply to  MrGrimNasty
July 1, 2026 5:48 am

You and your facts, that’ll get you nowhere here with my fellow skeptics.

That is because us skeptics require scientific accuracy for scientific claims. That includes using measurement uncertainty intervals in order to ascertain the statistical significance of any claims.

Occam’s Razor will tell you that if you can draw a straight line that has no change within the uncertainty intervals of the temperatures, then you don’t know what the actual trend is.

MrGrimNasty
Reply to  Jim Gorman
July 1, 2026 7:37 am

It’s the CET. Your ignorance is astounding.
Take a look at the record. If you can’t see the truth there’s no point with further debate.

Reply to  MrGrimNasty
July 1, 2026 7:51 am

If you can’t see the truth there’s no point with further debate.

From a popular movie, “You can’t handle the truth”! Funny how you must use the argumentative fallacy of Appeal to Authority rather than discuss the scientific details of measurements.

Reply to  MrGrimNasty
July 1, 2026 1:30 pm

CET is a mish-mash of data from bad sites.

The methodology and sites were changed since the Met-Office took over.

CETpage5_30046_image002
Reply to  MrGrimNasty
July 1, 2026 1:35 pm

If CET had been curated in the same way Manley did it, the results are very difference.

CET-Manley-Method
MarkW
Reply to  MrGrimNasty
July 1, 2026 3:34 pm

You say that as if it is proof that their numbers are beyond reproach.

Reply to  Bellman
July 1, 2026 5:42 am

Meanwhile CET recorded the 4 warmest max June temperatures last month.

Why do you never show the measurement uncertainty for each of those measurements? Does the statistical significance disappear when you do that?

Reply to  Jim Gorman
July 1, 2026 6:07 am

Agree. Show it for the individual measurements. Show it for the trends. Doh!

I need to spoon feed my point. The “measurement uncertainty” for the individual measurements would tend to diminish with the more of them used to calculate the resulting trends. And ponder before you “wudabout correlation?”. That would tend to further diminish trend uncertainty.

strativarius
Reply to  bigoilbob
July 1, 2026 6:38 am

Spoon feeding: a very middle class concept, Bob. Popular at Eton and Harrow.

Has it ever been hotter before? I wonder…

comment image

Reply to  strativarius
July 1, 2026 6:58 am

Certainly. And?

MarkW
Reply to  bigoilbob
July 1, 2026 3:36 pm

Proves that CO2 has nothing to do with the current temperatures.
Which btw are still below each of the 4 previous warm periods as well as all of the Holocene Optimum.

MrGrimNasty
Reply to  strativarius
July 1, 2026 7:41 am

There you go shifting the goal posts again.
We are talking about the MO records data that exists for the UK, not past reconstructions. No one that is ‘against you’ disputed that so why bring it up?

Sparta Nova 4
Reply to  MrGrimNasty
July 1, 2026 12:53 pm

There you go again, shifting the goal posts.
The article was about one measurement at one site declaring the hottest eva.

Reply to  strativarius
July 1, 2026 11:57 am

A fake graph which the listed authors have said is a misrepresentation of their work!

Reply to  bigoilbob
July 1, 2026 7:11 am

 The “measurement uncertainty” for the individual measurements” s/b ” The “calculated uncertainty” for the trends”

The poster regrets the error…

Reply to  bigoilbob
July 1, 2026 7:38 am

 The “measurement uncertainty” for the individual measurements” s/b ” The “calculated uncertainty” for the trends”

If you are assuming the residuals are the “calculated uncertainty” of the trend, then you are assuming the data points are 100% accurate with no uncertainty of their own determined from the measurement uncertainty inherent in each measurement.

Residuals only tell you the fit of the model, i.e., regression of 100% accurate data points.

The uncertainty in the slope and intercept uses a covariance matrix of the fitted parameters using measurement uncertainty, not the residuals. There are several techniques that can be used to assess the trend uncertainty. One is weighted least squares.

 In addition, these are time series, not x-y related data. They should be analyzed using time series analysis techniques. If CO2 is the main driver, then the x-axis should be CO2 and the y-axis temperature. Do you ever wonder why the supposed main culprit in heating is never plotted along with temperature? Why is that not done?  

Reply to  Jim Gorman
July 1, 2026 7:56 am

“If you are assuming the residuals are the “calculated uncertainty” of the trend, then you are assuming the data points are 100% accurate with no uncertainty of their own determined from the measurement uncertainty inherent in each measurement.”

No need to assume that each data point is “accurate”. I am assuming that what you are aksing for – the total “uncertainty” including ALL sources of accuracy/uncertainty, both correlated and not, is provided. These data distributions – whether constant or changing, throughout, can be used to provide the referenced, distributed, trends, with an uncertainty that takes all into account. As the number of data points increases, and/or the number of different “errors” are introduced, the trend error will tend to decrease. Of course in the cherry picked example of early systemic trend errors jetting low, and later systemic errors jetting high, the overall trend would be thus off (goes both ways). But as the number of those systemic errors increases, the error they would introduce in the trend would tend to decrease.

Reply to  bigoilbob
July 1, 2026 12:51 pm

As the number of data points increases, and/or the number of different “errors” are introduced, the trend error will tend to decrease.

You wish it to be true — it isn’t.

Error is not measurement uncertainty, and measurement uncertainty is not error.

Reply to  karlomonte
July 1, 2026 3:32 pm

You wish it to be true — it isn’t.”

Ever wonder why the Pat Frankian Bizarro World statistical theories get relegated to just about just this subterranean forum? I mean, how many years can you and yours keep telling yourself that long drilled down upon scientific/statistical truths, are Dr. Evil conspiracy theories, promulgated by hundreds of thousands of scientific pro’s? For why? I invite you to ponder…

Reply to  bigoilbob
July 1, 2026 5:32 pm

For why? I invite you to ponder…

I don’t need to ponder. I can give you several resources to start with. I expect you to respond with resources rather than taking the position that I know what is correct. Many scientists have no training in metrology at all. I have discussed this with several that said that is what statisticians are for. Statisticians that have never even seen a measurement.

From, SOP 29 Assignment of Uncertainty

3.2.1.2 Standard deviation from a series of replicate measurements. Measure a stable test object at least seven times, no two measurements of which should be made on the same day. Calculate the standard deviation in the conventional manner to obtain the standard deviation of the process, sp, keeping in mind that it does not fully represent the measurement process under all typically encountered conditions and that additional uncertainty values may need to be addressed more fully. Actual degrees of freedom will be identified from the control chart. Note: Repetitive measurements made on the same day estimate the short-term standard deviation of the process and may underestimate actual measurement variability.

From: Statistical distributions commonly used in measurement uncertainty in laboratory medicine – PMC

The SD of normal distribution is also the standard uncertainty of the measurements.

From: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1255808/

So, if we want to say how widely scattered some measurements are, we use the standard deviation. If we want to indicate the uncertainty around the estimate of the mean measurement, we quote the standard error of the mean.

From: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2959222/#

Unlike SD, SEM is not a descriptive statistics and should not be used as such. However, many authors incorrectly use SEM as a descriptive statistics to summarize the variability in their data because it is less than the SD, implying incorrectly that their measurements are more precise.

From: https://dianacuesta.com/wp-content/uploads/2010/07/20-errores-estadisticos-de-los-articulos.pdf

Error #5: Using the standard error of the mean (SEM) as a descriptive statistic or as a measure of precision for an estimate

The main problem associated with Dr. Frank’s work was attaching a “t⁻¹” to account for the time period that an average covered. Dr. Frank is an analytic chemist. This profession lives and dies with using dimensional analysis for calculating solutions, reactions, etc.

Here is an average, 25°C for a year. When doing dimensional analysis, this turns into:

(25°C/year) × x years = (25°C/year) × x years = 25°C× x

If you don’t do this you end up with, 25°C·years. High school chemistry and physics students learn this.

The other big complaint was what you have claimed occurs. Multiple data points eliminate error. If you are banking on this, you need some backup math. Uncertainty adds every time a new iteration is run using data that has its own uncertainty. No one ever offered any math that refuted that.

Reply to  bigoilbob
July 1, 2026 4:07 pm

As the number of data points increases, and/or the number of different “errors” are introduced, the trend error will tend to decrease.

This is absolutely incorrect. Uncertainty adds, always.

Don’t mix errors with uncertainty. They are different. Errors are determined by “true value + error”. The true value of a measurand is not known, therefore errors are not a reliable method of describing a measurement.

The very first paragraph of the GUM, JCGM 100-2008 says:

0.1 When reporting the result of a measurement of a physical quantity, it is obligatory that some quantitative indication of the quality of the result be given so that those who use it can assess its reliability. Without such an indication, measurement results cannot be compared, either among themselves or with reference values given in a specification or standard. It is therefore necessary that there be a readily implemented, easily understood, and generally accepted procedure for characterizing the quality of a result of a measurement, that is, for evaluating and expressing its uncertainty.

This was agreed to internationally, and the error paradigm was discontinued in favor of uncertainty.

Uncertainty adds, always. Why? Because that is the internationally accepted treatment of uncertainty. Eq. 10 from the GUM is:

u꜀(y)² = ∑ (df/dxᵢ)² u(xᵢ)² (Eq 10)
See those squared terms? That means there are no negative values, they are all positive signs and ADD to obtain the combined uncertainty.

Of course in the cherry picked example of early systemic trend errors jetting low, and later systemic errors jetting high, the overall trend would be thus off (goes both ways).

Again, errors are an old paradigm and should be discarded.

This is from the GUM, and describes what uncertainty is.

D.5.2 Uncertainty of measurement is thus an expression of the fact that, for a given measurand and a given result of measurement of it, there is not one value but an infinite number of values dispersed about the result that are consistent with all of the observations and data and one’s knowledge of the physical world, and that with varying degrees of credibility can be attributed to the measurand.

The uncertainty of an input quantity is never decreased by the uncertainties of other input quantities. They combine as in Eq. 10. This makes a trend absolutely uncertain, no matter how many data points are included.

Reply to  Jim Gorman
July 1, 2026 5:44 pm

See those squared terms?

Futile to keep asking you, but what do you think df/dx is for an average?

Reply to  Bellman
July 1, 2026 7:11 pm

Futile to keep asking you, but what do you think df/dx is for an average?

Do you think  (df/dxᵢ)² is ever going to end up with a negative sign?

Reply to  Jim Gorman
July 2, 2026 2:42 am

It doesn’t have to be negative to make the value smaller, does it? We’ve been over this enough times. What if (df/dxᵢ)² = 1/N²?

Reply to  Bellman
July 2, 2026 5:04 am

It doesn’t have to be negative to make the value smaller, does it? We’ve been over this enough times. What if (df/dxᵢ)² = 1/N²?

You keep ignoring what I said. UNCERTAINTIES ADD, ALWAYS.

Why don’t you address what I said rather than going off on a tangent.

Reply to  Jim Gorman
July 2, 2026 6:17 am

“You keep ignoring what I said. UNCERTAINTIES ADD, ALWAYS. ”

Because for the last 5 years you’ve been claiming that this means the uncertainty of an average, or a linear regression, increases with sample size. If you are now admitting you were wrong and that uncertainties adding can result in smaller uncertainty, then good – we can end this stupid argumen.

Reply to  Bellman
July 2, 2026 1:34 pm

Blame me for your waste of time. Why did I poke this rats nest when I know that these merely are the remaining few Branch Davidians of statistical bad wiring? Not even a handful of cultic holdouts, with (thankfully) the few folks in superterranea who even know they exist doing smiley SMH’s.

I was not even going to, and will not, rebut my 10 responses, as I have shit to do during my waking hours. But you told me why you take the time and I get it.

Reply to  Jim Gorman
July 1, 2026 6:10 pm

I’ve no intention in going through another pointless discussion, so this will be my only comment about this, at this time.

See those squared terms? That means there are no negative values, they are all positive signs and ADD to obtain the combined uncertainty.

  1. You don’t need negative numbers to reduce the uncertainty. The reduction in an average is caused by the partial derivative df/dx in the equation scaling all the values down.
  2. In fact you can have negative terms in the general equation. That happens when the uncertainties are not independent and use the expanded for of the equation. If the correlation is negative, or the function involves subtraction, then the extra term can be negative.

Again, errors are an old paradigm and should be discarded.

You are using the old paradigm the GUM tried to pass. The newer thinking is that this is not a useful distinction.

Von Clarmann et al provide a lucid, thorough analysis of the alleged difference between the error and uncertainty approaches, and conclude that ‘the error concept and the uncertainty concept are the same. Arguments in favor of the contrary have been analyzed and found not to be compelling. Neither was any evidence presented in GUM-2008 that ‘errors’ and ‘uncertainties’ define a different relation between the measured and true values of the variable of interest.’

I have expressed agreement with this conclusion, and now suggest the following allegoric characterization of the relation between measurement errors and measurement uncertainty that justifies the continued and harmonious use of both concepts, emphasizing one or the other, depending on the application and context: measurement errors are the carriers of measurement uncertainty, similarly to how gluons are the carriers of the strong nuclear force.

https://iopscience.iop.org/article/10.1088/1681-7575/adfb80

Reply to  Bellman
July 2, 2026 4:24 am

You don’t need negative numbers to reduce the uncertainty. The reduction in an average is caused by the partial derivative df/dx in the equation scaling all the values down.

That is a whole different issue. The uncertainties still add. And, the (df/dxᵢ) term in many cases is simply equal to 1. Remember, if you divide by a counting number to calculate an average, then you are dividing. Division requires the use of relative uncertainties and the (df/dxᵢ) will simply cancel with the divisor which in essence, makes the sensitivity factor equal to 1. Again, all the uncertainty adds to create the combined uncertainty.

2. In fact you can have negative terms in the general equation. That happens when the uncertainties are not independent and use the expanded for of the equation. If the correlation is negative, or the function involves subtraction, then the extra term can be negative.

Are you saying that temperatures have a correlation because they are not independent? If so, you have introduced an entirely new factor into the process, one that is never used. Congratulations.

Von Clarmann et al provide a lucid, thorough analysis of the alleged difference between the error and uncertainty approaches, and conclude that ‘the error concept and the uncertainty concept are the same.

The document you discuss says this about uncertainty. It is useful to point out that seldom does climate science include “the corresponding coverage probability’ interval. This would normally be a 68% chance of the true value being in the interval when using a standard deviation. To create a 95% coverage interval, NIST and most other national measurement bodies, recommend using a factor of 2 times the standard deviation.

For a scalar measurand, y, the size of the margin of doubt can be expressed by the standard uncertainty, u(y), and the severity of the doubt is the probability that the true value of the measurand lies in the interval . This suggests that u(y) alone achieves only an incomplete characterization of the uncertainty, and should be accompanied by the corresponding coverage probability of this interval (which, obviously, is contingent on the probability distribution that fully characterizes the uncertainty that surrounds y).

The document also says this about the ‘error’ approach.

Since the true value of the measurand typically remains out of reach, then so does the measurement error if it is conceived as the difference between the measured value and the true value. 

I would add that the error and the uncertainty paradigms MAY have identical results, however, error calculation still requires KNOWING the scalar true value of a measurand in order to calculate the error magnitude. It is unlikely that in most measurements that the actual true value is known. Certainly, with temperature measurements, there is no known way to decipher what the true value is.

Reply to  Jim Gorman
July 2, 2026 6:16 am

Remember, if you divide by a counting number to calculate an average, then you are dividing. Division requires the use of relative uncertainties and the (df/dxᵢ) will simply cancel with the divisor which in essence, makes the sensitivity factor equal to 1. Again, all the uncertainty adds to create the combined uncertainty.”

bellman doesn’t believe in relative uncertainty. GUM Equation 12 is just garbage for him.

Reply to  Jim Gorman
July 2, 2026 7:16 am

” Division requires the use of relative uncertainties and the (df/dxᵢ) will simply cancel with the divisor which in essence, makes the sensitivity factor equal to 1.”

Are you competing with Tim to see who can misunderstand this the most?

I’m trying to avoid yet another endless argument, so I will state this just once and let you whine as much as you want.

The equation from the GUM (10) works for any function, whether it involves addition, division, powers, trigonometry or any analytical operations.

In all cases the procedure is the same. Work out the partial derivative for watch term and apply it to the absolute uncertainty of that term. You do not substitute relative uncertainties for absolute ones just because there is a multiplication or division in the function.

If the function consists of nothing but multiplication, divsion, and raising to powers, you can take the result from equation 10 and divide through by the square of the result. When you do that there is a lot of cancellations and you end up with an equivalent equation consisting of adding relative uncertainties.

This also leads to the simplified procedure where you break an equation into parts and use absolute uncertainties for adding and relative uncertainties for multiplication.

What you cannot do is just say because there is a division in the function you can ignore the partial derivatives and just use relative uncertainties. That only works if you have nothing but multiplication and division.

An average does not consist of just division, it consists of adding and division, therefore you can not use the relative uncertainty simplification.

You have two options with an average. Just use the partial derivatives with equation 10. In which case the partial derivative for each term is 1/N. Or you can work out just the uncertainty for the sum and then feed that into another function involvin, dividing by N. In either case you get the same result which is the uncertainty of the sum divided by N. Hence the more things you are a raging the smaller the measurement uncertainty.

Reply to  Bellman
July 2, 2026 8:17 am

I’m trying to avoid yet another endless argument, so I will state this just once and let you whine as much as you want.

Oh my, the expert is lecturing (again). Y’all better listen up — the magic number can to ANYTHING needed to get those milli-Kelvin numbers!

Trendology uber alles.

Reply to  Bellman
July 2, 2026 5:30 pm

What you cannot do is just say because there is a division in the function you can ignore the partial derivatives and just use relative uncertainties.

That is not what I said at all. I said the partial derivative cancels with relative uncertainties. That is using n input quantiles of xᵢ/n. Each input quantity of (xᵢ/n) is a unique quantity and is calculated using division.

An average does not consist of just division, it consists of adding and division, therefore you can not use the relative uncertainty simplification.

You mean you can’t use it with your function definition. Look at the process you have derived for uncertainty.

Let’s look at your function definition.
Y = f(X1, X2, …, Xn, n) = [∑(xᵢ)]/n
So Y = μ = [∑(xᵢ)] / n]

Now what does the standard deviation equal. The normal formula is:

σ = √[ (∑(Xᵢ – μ)²/n]

You have ruled out using individual (xᵢ)/n since that would mean individual input quantities. The only thing left is;

σ = √[ ∑(Xᵢ – μ)²/n] =√ [∑{∑xᵢ/n – μ)²}/n]² = √ [∑{∑xᵢ/n –xᵢ/n]²/n

Well golly gee, we end up with σ = 0/n. Just what you wanted, all uncertainty cancels! /sarc

Or you can work out just the uncertainty for the sum and then feed that into another function involvin, dividing by N. In either case you get the same result which is the uncertainty of the sum divided by N. 

Do you find this anywhere in the GUM? You have just created your own definition for uncertainty, and it has no international agreement on its use or even a mention of “another function”.

And again, if you deny that (xᵢ/n) are a number of unique inputs q in favor of having one single ᵢX that consists of ( ∑(xᵢ)/n, then you have no way to assess it with a Type A statistical analysis. In case you think so, show us how you will calculate a standard deviation and what value you will divide the standard deviation by to obtain a standard uncertainty of the mean.

Reply to  Jim Gorman
July 2, 2026 5:59 pm

What you said is “Division requires the use of relative uncertainties”. That’s where you are wrong, and why you keep coming up with nonsense about cancelling the scaling factors.

I keep having to argue this becasue it is so easy, and the only reason you try to complicate it is so you can keep making mistakes and get the result you want.

All you have to do is work out what your function is, work out the partial derivatives, then put them all in equation 10. If any of your convolutions don’t give you the same answer then you know you have made a mistake.

You mean you can’t use it with your function definition.

If by “it” you mean equation 12, then yes. That’s the point. There’s a very specific assumption about the form of the function that works with (12), and that rules out using it with the mean function.

Now what does the standard deviation equal.

See what I mean about you keep over complicating this. We are not calculating a standard deviation – I’ve no idea why you keep bringing it up.

Do you find this anywhere in the GUM?

No because I’m capable of thinking for myself.

Reply to  Bellman
July 3, 2026 3:16 pm

What you said is “Division requires the use of relative uncertainties”. That’s where you are wrong, and why you keep coming up with nonsense about cancelling the scaling factors.

Since you know more that I, I will entertain you with references that show multiplication and division require the use of relative uncertainties.

+++++++++++++++++++++++++++++++++++++++

Data Reduction and Error Analysis, Bevington and Robinson;

Chapter 3, Multiplication and Division, Page 44

Multiplication – equation -> x = auv where a = constant

σₓ²/x² = σ²/u² + σ²/v² + 2(σᵤ²/uv

Division – equation -> x = au/v

σₓ²/x² = σᵤ²/u² + σᵥ²/v² – 2(σᵤᵥ²/uv
+++++++++++++++++++++++++++++++++++++++

Experimentation and Uncertainty Analysis for Engineers, Coleman and Steele
Chapter 3-3.2, Page 54

When equation is -> r = kX₁ᵃX₂ᵇX₃ᶜ … where k is a constant

{Uᵣ/r}² = a₁{{Uₓ₁/X₁}² + b₁{Uₓ₂/X}² + …
+++++++++++++++++++++++++++++++++++++++

All you have to do is work out what your function is, work out the partial derivatives, then put them all in equation 10. If any of your convolutions don’t give you the same answer then you know you have made a mistake.

The function you are attempting to justify is

Y = f(X1, X2, …, Xn, n) = ∑(Xᵢ)/n

You have two choices for input quanties.

  1. Multiple unique input quantities -> (X1, X2, …, Xn) where X1 = x1/n, X2 = x2/n, …, Xn = xn/n), which you have already said is not allowed, or
  2. one single input quantity. ∑(Xᵢ)/n.

See what I mean about you keep over complicating this. We are not calculating a standard deviation – I’ve no idea why you keep bringing it up.

I am complicating nothing. The GUM Type A statistical analysis requires calculating a mean AND a standard deviation to obtain an estimated value and an uncertainty value. You have declared that there is only one input quantity with a standard deviation of zero.

The GUM and every other resource in metrology requires multiple observations qk which are data points in a random variable in order to calculate a mean and a standard deviation of the probability distribution of the qk observations. With only one observation, ∑(Xᵢ)/n, you can calculate neither.

You are stuck way out in left field here. Your so-called function is not a functional relationship, as I have told you before. It is a statistical calculation to determine a parameter of a distribution.

Show some evaluated math equations if you wish to have your hypothesis considered.

Reply to  Jim Gorman
July 3, 2026 4:47 pm

I will entertain you with references that show multiplication and division require the use of relative uncertainties.

We are talking about equation 10. Everything you entertain me with are the result of equation 10. Really, it’s simple – equation 10 is the general rule that works with any function and uses absolute uncertainties. From this can be derived the simple rules stated in most course work where you break the function up into parts, and use absolute uncertainties for adding and subtracting and relative uncertainties for multiplication and division.

And if the function consists of nothing but multiplication, division and raising to powers, you can use equation 12. What you cannot do is take a function that includes both multiplication and adding, and use equation 12.

You have two choices for input quanties.

No. You only have one. Just the values you are averaging. You can if you want include N as a separate value as you keep doing, but that’s redundant as it’s just a constant specified by the number of input quantities.

“Multiple unique input quantities -> (X1, X2, …, Xn) where X1 = x1/n, X2 = x2/n, …, Xn = xn/n), which you have already said is not allowed, or
one single input quantity. ∑(Xᵢ)/n.”

Both are the same equation, in both the input quantities are X1, X2 … Xn. Adding and dividing by n is what the function does. It is not something you do before imputing the quantities.

And I did not say 1. is not allowed. In fact that’s specifically the form you want to work out the partial derivative for each term.

I am complicating nothing. The GUM Type A statistical analysis requires calculating a mean AND a standard deviation to obtain an estimated value and an uncertainty value.

That’s how you do a type A uncertainty, it is not what is being discussed in the use of equation 10. Each specific input quantity may be the result of averaging multiple measurements, in which case you can use the Standard Deviation of the Mean as an estimate of that inputs uncertainty. Or it can be the result of a single measurement with a Type B estimate of the uncertainty. But either way equation 10, or 12 or any other method, just requires you know what each input value is, and what each individual uncertainty is.

You problem I suspect, is you can’t separate the different averages. Every time we talk about using equation 10 with the mean as the function, we are simply saying you have a number of different things and we want to know what their average is. Not that we are making multiple measurements of the same thing in order to get a better measurement.

You have declared that there is only one input quantity with a standard deviation of zero.

When did I say that? Please supply an exact quote. The number of input quantities is n. This is going to be more than 1. As always you keep going on about standard deviations without ever explaining which one you are talking about. If you mean the standard uncertainty, each individual input will have it’s own uncertainty, which is going to be greater than 0.

If you mean the standard deviation of all your things, then that’s not relevant to this exercise, which is simply to calculate the combined uncertainty of the measurement. Now, if you want a more “real-world” use of a mean, then you would want to treat the various items being averaged as a sample, and the uncertainty of the mean can be described by the SEM, and that will depend on the sample standard deviation.

You are stuck way out in left field here.

Yes, because taking a mean is such a strange concept that has only been around for a few centuries.

Your so-called function is not a functional relationship

By definition it is. But as you still don;t seem to understand what “function relationship” means I understand why you are so confused.

Show some evaluated math equations if you wish to have your hypothesis considered.

What hypothesis? That you can determine the mean by adding all values and dividing by n? That is you apply that function to equation 10 you will get the correct result? That by an extraordinary coincidence that result is exactly the same as you get if you average a set of IID random variables, and is of the same form as you get when you apply it to measuring the same thing multiple times, or taking the values as a sample and obtaining the SEM?

Reply to  Bellman
July 4, 2026 3:09 am

We are talking about equation 10.”

No, Eq 10 is used for MEASUREMENT UNCERTAINTY. Σu(i)/N is the STANDARD ERROR, better defined as the “standard deviation of the sample means”. It is a measure of SAMPLING UNCERTAINTY, i.e. the uncertainty associated with estimating the mean using a small sample size. SAMPLING UNCERTAINTY ≠ MEASUREMENT UNCERTAINTY except for one, single situation which doesn’t apply to averaging multiple measurands like temperature.

GUM:

=================

4.2.1 In most cases, the best available estimate of the expectation or expected value μ_q of a quantity q that varies randomly [a random variable (C.2.2)], and for which n independent observations q_k have been obtained under the same conditions of measurement (see B.2.15), is the arithmetic mean or average q

(C.2.19) of the n observations:
===============(bolding mine, tpg)

===================

4.2.3 The best estimate of σ^2 (q ) = σ^2/n , the variance of the mean, is given by

s^2(q_bar) = s^2(q_k)/n

The experimental variance of the mean s^2(q_bar) and the experimental standard deviation of the mean s(q_bar) (B.2.17, Note 2), equal to the positive square root of s^2(q), quantify how well q_bar estimates the expectation μ_q of q, and either may be used as a measure of the uncertainty of q .

==================

4.2.1 specifies the need for the q_k values to have been obtained under the same conditions of measurement. That simply does not apply to temperature data sets consisting of single measurements of different measurands using different instruments while measuring under different conditions for each measurands.

4.2.3 states “quantify how well q_bar estimates the expectation μ_q of q“. That is how closely the average is to the population average and NOT how accurate the average value is when based on data elements with measurement uncertainty.

You are still caught in the catch22 of whether the temperature data sets represent a “single sample of size n” or represents “multiple samples of size 1″.

1. If you define the data sets as a single sample of size n” then there is no SEM since there is no standard deviation of the sample meanS. Even with two samples the SEM will be 20% too low.
2. If the data sets are multiple samples of size 1 then the SEM is the standard deviation of the sample data because sqrt(1) = 1.

Bottom line? s^2(q_bar) is how close you are to the population average and NOT how close q_bar is to being an accurate measurement, i.e. the measurement uncertainty.

Eq. 10 is for calculating MEASUREMENT UNCERTAINTY. s^2(q_bar) is for calculating how well you have estimated the average because of SAMPLING UNCERTAINTY and is *not* the measurement uncertainty of the average value.

You and your compatriots have NEVER understood what measurement uncertainty actually means. It is how well you estimate the average and is *not* how accurate that estimated average is.

One more time: The average is a STATISTICAL DESCRIPTOR. It is *not* a measurement. All the standard deviation of the sample means can tell you is how well you are at estimating that STATISTICAL DESCRIPTOR. It simply can not tell you how accurate that estimated STATISTICAL DESCRIPTOR is because of the measurement uncertainty portion of the data.

ONE MORE TIME: You need to stop using just the word “uncertainty” and specifically categorize it by using one of the adjectives of “measurement” or “sampling”. The SEM, s^2(q_bar) = s^2(q_k)/n is the SAMPLING uncertainty associated with the statistical descriptor known as the average and is not the measurement uncertainty of the average. Eq 10 is for measurement uncertainty and has no sqrt(n) in it.

Reply to  Tim Gorman
July 4, 2026 4:01 am

“No, Eq 10 is used for MEASUREMENT UNCERTAINTY. ”

I was talking about measurement uncertainty.

“Σu(i)/N is the STANDARD ERROR, better defined as the “standard deviation of the sample means”.”

You are so confused it isn’t worth arguing.

“You are still caught in the catch22 of whether the temperature data sets represent a “single sample of size n” or represents “multiple samples of size 1″”

No confusion . The answer has always been the first one.

“If you define the data sets as a single sample of size n” then there is no SEM since there is no standard deviation of the sample meanS.”

The SEM is standard deviation divided by root N. Hope that clears up your misunderstanding.

Reply to  Bellman
July 4, 2026 7:33 am

I was talking about measurement uncertainty.”

No, you are talking about SAMPLING uncertainty. The SEM is SAMPLING uncertainty. It is sampling uncertainty when you are talking about the standard deviation of the sample means.

The standard deviation of the sample means is generated from the sampling distribution, not from the measurement uncertainty.

“No confusion . The answer has always been the first one.”

If you have one sample then the uncertainty of the SEM is over 20% (and is probably undefined meaning the SEM is useless). You have no idea if the SD of the single sample is representative of the population SD.

“The SEM is standard deviation divided by root N. Hope that clears up your misunderstanding.”

Taylor: “The standard error of the mean is the standard deviation of the distribution of sample means.”

Bevington: “The standard error of the mean is the estimated standard deviation of the mean value.”

GUM: “standard deviation of the mean

NIST: “The standard error of the mean is the standard deviation of the sampling distribution of the mean.”

The formula of SEM = SD/sqrt(N) IS ONLY AN ESTIMATOR of the standard deviation of the sample means. And the SD of the sample means doesn’t exist if there is only one sample!

With just two samples the uncertainty of the SEM is 20% – that is HUGE. With just one sample the SEM is undefined since there is no standard deviation.

Are you *sure* you want to stick with the definition that the temperature data sets only represent a single sample? In that case your “formula” doesn’t work!

Why do all the temperature statistical analyses of climate science always turn out to be such garbage?

And if you are thinking about reversing and saying that the data sets are a collection of samples with a sample size of 1, then the measurement uncertainty of the average just becomes the sum of the sample measurement uncertainties. u_c = Σu(x_i)/1

The entire edifice of a “global average temperature” is built on a foundation of sand from the very beginning to the top of the hierarchy, all based on the garbage meme of “all measurement uncertainty is random, Gaussian, and cancels”.

Reply to  Tim Gorman
July 4, 2026 8:34 am

“No, you are talking about SAMPLING uncertainty.”

Then you need to define exactly what you mean when you use those terms.

Equation 10 is about propagating measurement uncertainties. The assumption for this is we are calculating an exact mean, just the summof the values divided by N. The only uncertainty being the uncertainty of the individual measurements.

By sampling uncertainty I mean treating the individual values as selected at random from a larger population. The average you get is just one possible value, a different sample will give you a different average even if there were no measurement uncertainty.

“It is sampling uncertainty when you are talking about the standard deviation of the sample means.”

Nobody calls it that. It’s the standard error if the mean SEM. Or if you prefer the standard deviation of the mean SDOM. But as usual you just invent your own terms.

But I was not talking about the SEM, I was specifically talking about equation 10, the combined measurement uncertainy.

“If you have one sample then the uncertainty of the SEM is over 20% (and is probably undefined meaning the SEM is useless).”

20% of what?

“Taylor” er al.

Yes that’s what the SEM is. You usually calculate it by taking the sample standard deviation and dividing by √N. I’m surprised you haven’t heard this before, given I’ve explained it to you every few weeks for the past year.

“And the SD of the sample means doesn’t exist if there is only one sample!”

Which is why calling it a standard deviation is misleading, The preferred term is standard error. Regardless though, none of your sources actually think you have to take multiple samples on order to estimate the SEM, they understand that you estimate it more easily by staking the SD of the sample and dividing it by root N.

“Are you *sure* you want to stick with the definition that the temperature data sets only represent a single sample? In that case your “formula” doesn’t work! ”

You can slice the data in as many ways as you want but what you never have are multiple samples of the same population.

And as I keep explaining, the simple SEM equation does not work for global temperatures because they are not a random sample. That’s why none of the uncertainty analysis for global anomalies says to take the average of all stations and divide the standard deviation by √N.

“And if you are thinking about reversing and saying that the data sets are a collection of samples with a sample size of 1”

Stop blaming this nonsense on me. You are the only people who have claimed such a nonsense.

Reply to  Bellman
July 4, 2026 5:26 am

“By definition it is. But as you still don;t seem to understand what “function relationship” means I understand why you are so confused.”

The equation you use doesn’t describe the relationship you keep implying.

The equation u(avg) = Σu(x_i)/sqrt(n) gives the variability of a statistical descriptor and is *NOT* a propagation of measurement uncertainty. It is SAMPLING uncertainty, not measurement uncertainty.

The sampling uncertainty is useless in the real world of metrology. Since the best estimate of the average should have the same decimal places as the measurement uncertainty there is no use in trying to increase the precision of the calculation of the average by decreasing the sampling uncertainty past that point.

Reply to  Tim Gorman
July 4, 2026 5:43 am

“The equation you use doesn’t describe the relationship you keep implying. ”

You are very confused. The function Σx_i/N most definitely does define the mean.

“The equation u(avg) = Σu(x_i)/sqrt(n) gives the variability of a statistical descriptor and is *NOT* a propagation of measurement uncertainty.”

That is not the function I was talking about. Try reading in context for a change.

And you are just plain wrong about the uncertainty here. I’ve no idea we’re you even got that equation, but it is neither the equation for measurement uncertainty derived from equation 10, nor is it the SEM.

If you can’t to keep having this endless argument you really need to learn what it is you ate arguing against.

“The sampling uncertainty is useless in the real world of metrology. ”

Why? You keep insisting there can be no measurement uncertainty because the mean isn’t a mesdurand, then you dismiss treating it as a statistic. Just what do you want, apart from just wanting to make up some meaninglessly large number, just so you can ignore all evidence of global warming?

Reply to  Bellman
July 4, 2026 9:18 am

That is not the function I was talking about. Try reading in context for a change.”

Of course it is your equation!

“And you are just plain wrong about the uncertainty here. I’ve no idea we’re you even got that equation, but it is neither the equation for measurement uncertainty derived from equation 10, nor is it the SEM.”

Put down the bottle!

u(x_i) is the standard deviation. It is the square root of the variance of the data.

u(avg) as you have it defined *IS* the SEM since it is the standard deviation divided by the square root of the sample size.

It is *NOT* the measurement uncertainty!

Why? You keep insisting there can be no measurement uncertainty because the mean isn’t a mesdurand, then you dismiss treating it as a statistic. “

Did you bother to proof read this before hitting the post key? No one ever has said there is not measurement uncertainty because the average isn’t a measurand.

THE MEASUREMENT UNCERTAINTY OF THE AVERAGE IS THE PROPAGATED MEASUREMENT UNCERTAINTY OF THE DATA COMPONENTS. It is *NOT* the SEM.

Your formula is *exactly* what is in the GUM:

s^2(q_bar) = s^2(q_k)/n

This is the SEM estimator!

This is ONLY the measurement uncertainty if the observations meet repeatability requirements AND meet SEM requirements which you always ignore. (SEM requirements like iid, Gaussian distribution, etc).

Just what do you want, apart from just wanting to make up some meaninglessly large number, just so you can ignore all evidence of global warming?”

It is *NOT* a meaningless large number. It is Eq 10! You simply refuse to understand that measurement uncertainty ADDS – willful ignorance.

Adding more measurements will *NOT* decrease the measurement uncertainty. It only moves the best estimate closer to the actual population average. But how close you are to the population average is *NOT* the measurement uncertainty.

Reply to  Tim Gorman
July 4, 2026 9:52 am

“Of course it is your equation!”

Just saying “of course” and repeating the lie is not an argument.

Here is Jim’s comment I was replying to

Your so-called function is not a functional relationship, as I have told you before. It is a statistical calculation to determine a parameter of a distribution.

So yes that’s ambiguous as he doesn’t specify which function he’s talking about. But as he previously said

The function you are attempting to justify is
Y = f(X1, X2, …, Xn, n) = ∑(Xᵢ)/n

He meant that one rather than the equation for the SEM which had not even been mentioned.

Reply to  Jim Gorman
July 6, 2026 7:17 am

“Since you know more that I, I will entertain you with references that show multiplication and division require the use of relative uncertainties.”

OK, let’s try again to explain how this works using equation 10.

Multiplication – equation -> x = auv where a = constant
σₓ²/x² = σᵤ²/u² + σᵥ²/v² + 2(σᵤᵥ²/uv)”

The function is x = auv, for a constant a.

I’ll rename u to w to avoid confusing it with u for uncertainty. So

x = awv

and I’ll assume w and v are independent.

Equation 10 states

u(x)² = (∂x/∂w)² u(w)² + (∂x/∂v)² u(v)²

So now we need the partial derivatives

∂x/∂w = av

∂x/∂v = aw

So,

u(x)² =(av)² u(w)² + (aw)² u(v)²

This makes sense because the measurement of w is being multiplied by both a and v, so the uncertainty also has to be multiplied by the same amount.

But now we can divide both sides by x²

u(x)²/x² =(av)²/x² u(w)² + (aw)²/x² u(v)²

And then we can cancel terms. x = awv, so

(av)²/x² = (av)²/(awv)² = 1/w²

and

(aw)²/x² = (aw)²/(awv)² = 1/v²

So the final equation becomes

u(x)²/x² = u(w)²/w² + u(v)²/v²

The basic equation from equation 10 involves absolute uncertainties and multiplication by the over values. But this simplifies to an equation involving just relative uncertainties.

Now if you have correlation between w and v, you need to use equation (13) from the GUM, which becomes

u(x)² = (∂x/∂w)² u(w)² + (∂x/∂v)² u(v)² + 2(∂x/∂w)(∂x/∂v)u(w,v)

That is,

u(x)²/x² =(av)²/x² u(w)² + (aw)²/x² u(v)² + 2(a² vw)u(w,v)

And dividing by and cancelling terms

u(x)²/x² = u(w)²/w² + u(v)²/v² + 2(u(w,v)/(wv))

Which is the same as the equation from Bevington.

The point being that equation 10, or 13 work as is, there is no changing absolute uncertainties to relative ones. The relative uncertainty rule for multiplication is derived from the correct use of equation 10. But that simplification only works when there is no addition or subtraction in the function.

Reply to  Jim Gorman
July 2, 2026 7:18 am

“Are you saying that temperatures have a correlation because they are not independent?”

I was just giving you an example of where the combined uncertainty might involve subtraction.

Reply to  Jim Gorman
July 2, 2026 7:22 am

“however, error calculation still requires KNOWING the scalar true value of a measurand in order to calculate the error magnitude. ”

You’ve said all this before, and it’s still just as ignorant. The whole point of uncertainty is you do not know the true value or the error. If you did there would be no uncertainty. The piintbis you calculate the expected error using the rules if probability, and it makes no difference if you call this uncertainty or error. It’s a distinction without a difference.

Reply to  Bellman
July 2, 2026 11:06 am

The piintbis you calculate the expected error using the rules if probability, and it makes no difference if you call this uncertainty or error. It’s a distinction without a difference.

Well this is doubling-down on the nonsense: how can you claim to be an expert on the subject when you don’t understand that error is not measurement uncertainty?

And please to explain exactly what a “piintbis” is…

Reply to  karlomonte
July 2, 2026 11:22 am

“Well this is doubling-down on the nonsense”

Yes, it should have said

The point is you calculate the expected error using the rules of probability, and it makes no difference if you call this uncertainty or error. It’s a distinction without a difference.

Sorry about my typos.

“how can you claim to be an expert on the subject ”

I don’t. I try to make it clear I’m the opposite of an expert.

“you don’t understand that error is not measurement uncertainty”

They are not the same, but it makes no difference if you model uncertainty by an uncertainty probability distribution or by an error probability distribution.

“And please to explain exactly what a “piintbis” is…”

It’s a type of covfefe.

Reply to  Bellman
July 2, 2026 12:28 pm

They are not the same, but it makes no difference if you model uncertainty by an uncertainty probability distribution or by an error probability distribution.

The probability distributions are unlikely to be the same between error and uncertainty. I don’t think you were ever trained on the error paradigm. It was primarily instrument related. If I had a voltmeter with an error expectation of 2% of full scale that controlled the final description. If I happened to make several readings that were within 1% of the mean, the 2% controlled the stated value. If I was using a 1000 volt scale at 2%, that meant the range was ±20 volts. If all my readings were within that range, then my process was ok. If not, there had to be something in the experimental setup causing problems. Under the uncertainty paradigm, and using an uncertainty budget, I have a process that anticipates a variety contributions ahead of time and my readings should be compatible with that.

Reply to  Jim Gorman
July 2, 2026 2:32 pm

” I don’t think you were ever trained on the error paradigm.”

It seems not as what you describe doesn’t sound anything like the use of error in uncertainty. I’ll ask again if you have a reference to your “error paradigm”?

Reply to  Bellman
July 3, 2026 5:27 pm

It seems not as what you describe doesn’t sound anything like the use of error in uncertainty.”

Here we go again. If you want to know the future, i.e. the “true value” of something you need to measure, just call bellman. He *knows* the “true value” of what you are going to measure ahead of time. That way you can tell how much error your instrument has. No need to calibrate your measuring instruments, just ask bellman!

Reply to  Tim Gorman
July 3, 2026 6:55 pm

If you want to know the future, i.e. the “true value” of something you need to measure, just call bellman.

These lies are just getting pathetic. I keep telling you that no-one knows the true value of something. I’ve no idea why you think I’m claiming anything different. All I’ve asked is for either of you to give me a link to something describing the “error paradigm”.

You keep claiming that it required knowing the true value before you say what the uncertainty was. I think that’s absurd, and your inability to support your claims suggests you know it’s nonsense as well. But rather than admit it you just keep trolling, in the hope that nobody will notice.

Reply to  Bellman
July 4, 2026 6:05 am

I keep telling you that no-one knows the true value of something. “

Then why do you keep bringing up the “true value +/- error” paradigm?

“You keep claiming that it required knowing the true value before you say what the uncertainty was.”

No, you need to know the true value in order to specify the error! The uncertainty does *NOT* require knowing the true value. You can’t even get this simple concept straight.

“All I’ve asked is for either of you to give me a link to something describing the “error paradigm”.”

How about a definition of the term “paradigm”?

—————–
a philosophical and theoretical framework of a scientific school or discipline within which theories, laws, and generalizations and the experiments performed in support of them are formulated
——————–

Using the adjective “error” with the noun “paradigm” isa perfectly acceptable way to define the concept of “true value +/- error”. The fact that you can’t seem to understand this is just one more indication of your lack of reading comprehension skills.

This is nothing more than the argumentative fallacy known as Literal Evidence Fallacy. Demanding a specific phrase is *NOT* refutation of the concept the phrase describes.

Reply to  Tim Gorman
July 4, 2026 6:57 am

“Then why do you keep bringing up the “true value +/- error” paradigm?”

Because you don’t need to know the true value to know it exists. As far as I can see you and your brother are the only people who claim you need to know it. Most people would understand that “uncertainty” means you do not know the true value.

Did you read what your virtual chum said about the error paradigm? It specifically says you do not normally know the true value.

“No, you need to know the true value in order to specify the error! ”

And as before, you do not need to specify the error.

“The uncertainty does *NOT* require knowing the true value.”

And as keeps happening Tim realises what I keep saying is correct, but then claims I’m the one who doesn’t understand it.

So you accept you do not need to know the true value in order to calculate the uncertainty using the “error paradigm”? So what on earth are you arguing about?

“How about a definition of the term “paradigm”?”

Why not a definition of error paradigm, as I asked. You kept claiming it was wrong because it required knowing the true value, now you accept that’s wrong, but you still won’t provide a reference to this older method.

“Using the adjective “error” with the noun “paradigm” isa perfectly acceptable way to define the concept of “true value +/- error”. ”

Yes, that’s what I:ve been saying. What I’m asking for is a reference to your claim that it only works if you know the true value, or Jim saying it meant there was a different probability distribution.

“Demanding a specific phrase is *NOT* refutation of the concept the phrase describes.”

All I’m doing is asking you or your brother to support your claims.

If it helps here’s what Jim said about the paradigm

The probability distributions are unlikely to be the same between error and uncertainty. I don’t think you were ever trained on the error paradigm. It was primarily instrument related. If I had a voltmeter with an error expectation of 2% of full scale that controlled the final description. If I happened to make several readings that were within 1% of the mean, the 2% controlled the stated value. If I was using a 1000 volt scale at 2%, that meant the range was ±20 volts. If all my readings were within that range, then my process was ok. If not, there had to be something in the experimental setup causing problems. Under the uncertainty paradigm, and using an uncertainty budget, I have a process that anticipates a variety contributions ahead of time and my readings should be compatible with that.

That was the comment I was asking about when you jumped in with your predicting the future nonsense.

Reply to  Bellman
July 4, 2026 11:17 am

Because you don’t need to know the true value to know it exists.”

See my reply further down. If you don’t know the true value then you can’t determine the error term. wi = ui – εi has three terms. You can’t solve for one without knowing the other two. If you don’t know the error term then you don’t know the true value either. If you don’t know the true value then you don’t know the error term either.

The second paragraph in the GUM states:
———————
0.2 The concept of uncertainty as a quantifiable attribute is relatively new in the history of measurement, although error and error analysis have long been a part of the practice of measurement science or metrology. It is now widely recognized that, when all of the known or suspected components of error have been evaluated and the appropriate corrections have been applied, there still remains an uncertainty about the correctness of the stated result, that is, a doubt about how well the result of the measurement represents the
value of the quantity being measured”
———————–

The “true value +/- error” paradigm does *NOT* offer a method to represent the doubt about how well the result of the measurement process represents the value of the quantity being measured. The uncertainty paradigm *does*.

Some day you really should read the GUM from end to end instead of just cherry picking from it.

And as before, you do not need to specify the error.”

If you don’t know either the true value and/or the error then the error paradigm is useless. It provides absolutely no way to judge the accuracy of a measurement!

You are trying to claim that a paradigm that is useless is just as good as one that *is* of use. Typical. There is a *reason* why the uncertainty paradigm came into being. See Section 0.2 above!

Why not a definition of error paradigm, as I asked. You kept claiming it was wrong because it required knowing the true value, now you accept that’s wrong, but you still won’t provide a reference to this older method.”

The error paradigm has three factors. You must know two of them to calculate the third. If you don’t know two of the factors then the paradigm is useless – except to you I guess. The error paradigm simply didn’t offer most of us involved in measurements a good way to judge the accuracy of a measurement – too much doubt remained about the values presented.

I understand that doesn’t matter much to a statistician or mathematician to whom the meme of “numbers is just numbers” is second nature.

What I’m asking for is a reference to your claim that it only works if you know the true value,”

GUM:
—————
Section 2.2.4: “Although these two traditional concepts are valid as ideals, they focus on unknowable quantities: the “error” of the result of a measurement and the “true value” of the measurand (in contrast to its estimated value), respectively.
——————————-(bolding mine, tpg)

Again, some day you need to READ AND COMPREHEND THE GUM. Not just cherry pick from it. In a function of three factors, wi = ui – εi, you need to know either the true value or the error term in order to calculate the third vale. But both the true value and the error term are unknowable quantities (since one depends on the other if you don’t know one then you don’t know the other) and their significance is impossible to convey to others in a usable way. The uncertainty paradigm provides that “usable” way.

All I’m doing is asking you or your brother to support your claims.”

The support is right there in the GUM. You just have to read it instead of remaining willfully ignorant.

That was the comment I was asking about “

jim: “The probability distributions are unlikely to be the same between error and uncertainty”

In fact, since the error term is a scalar value rather than an interval how can it have a probability distribution at all?

It’s why the uncertainty paradigm was developed!

Reply to  Tim Gorman
July 4, 2026 11:32 am

Some day you really should read the GUM from end to end instead of just cherry picking from it.

Unacceptable, this would require him to give up the tiny “error bars” of climatology.

Reply to  karlomonte
July 4, 2026 1:19 pm

That’s the whole issue in a nutshell – for the entirety of climate science as well.

Reply to  Tim Gorman
July 4, 2026 12:18 pm

“If you don’t know the true value then you can’t determine the error term.”

How many times are you going to repeat this before you understand that nobody is suggesting you can know the error term or the true value. Uncertainty is about not knowing these things.

Again, if you think the error paradigm was about finding the true value rather than uncertainty just provide a reference.

In response to me asking for a reference saying you need to know the true value to use the error paradigm, you quote.

Section 2.2.4: “Although these two traditional concepts are valid as ideals, they focus on unknowable quantities: the “error” of the result of a measurement and the “true value” of the measurand (in contrast to its estimated value), respectively.

Which does say what you are claiming. In the contrary, they point out that the true value is unknowable.

“The error paradigm simply didn’t offer most of us involved in measurements a good way to judge the accuracy of a measurement – too much doubt remained about the values presented.”

Huh? Even the GUM admits you get identical results using either paradigm.

“The support is right there in the GUM.”

It isn’t. But even if it was as this whole discussion is about the GUM now being regarded as wrong quoting it isn’t much of an argument.

“In fact, since the error term is a scalar value rather than an interval how can it have a probability distribution at all?”

For the last time, the error term is not the uncertainty. The uncertainty is the standard deviation of all possible errors. It’s a probability distribution just like the uncertainty distribution used in the GUM.

Reply to  Bellman
July 2, 2026 1:02 pm

 but it makes no difference if you model uncertainty by an uncertainty probability distribution or by an error probability distribution.

Another argumentum handwavium — you assert many things that are not true.

Reply to  karlomonte
July 2, 2026 2:43 pm

See for example the GUM

E.5.3 In practice, the difference in point of view does not lead to a difference in the numerical value of the measurement result or of the uncertainty assigned to that result.

Reply to  Bellman
July 2, 2026 4:34 pm

You skipped over lots and lots to get here:

“E.5 A comparison of two views of uncertainty

“E.5.1 The focus of this Guide is on the measurement result and its evaluated uncertainty rather than on the unknowable quantities “true” value and error (see Annex D). By taking the operational views that the result of a measurement is simply the value attributed to the measurand and that the uncertainty of that result is a measure of the dispersion of the values that could reasonably be attributed to the measurand, this Guide in effect uncouples the often confusing connection between uncertainty and the unknowable quantities “true” value and error. ”

and

“E.5.4 While the approach based on “true” value and error yields the same numerical results as the approach taken in this Guide (provided that the assumption of the note of E.5.2 is made), this Guide’s concept of uncertainty eliminates the confusion between error and uncertainty (see Annex D). Indeed, this Guide’s operational approach, wherein the focus is on the observed (or estimated) value of a quantity and the observed (or estimated) variability of that value, makes any mention of error entirely unnecessary. ”

Yet trendology clings like barnacles to error.

Reply to  karlomonte
July 2, 2026 5:03 pm

Yes I decided not to cut and paste the entire GUM and just to extract the part that was relevant. Even they admit that using error instead of uncertainty will still give you the same result.

Reply to  Bellman
July 3, 2026 5:31 pm

Yes I decided not to cut and paste the entire GUM and just to extract the part that was relevant. Even they admit that using error instead of uncertainty will still give you the same result.”

It will give you the same thing IF YOU KNOW THE TRUE VALUE!

GUM: “rather than on the unknowable quantities “true” value and error

Most of us are not omnipotent. We don’t *know* the true value of things we measure. The GUM recognizes this.

You seem to be the only one claiming to know the Great Unknown. You should be able to make a fortune telling people’s futures!

Reply to  Tim Gorman
July 3, 2026 6:58 pm

“It will give you the same thing IF YOU KNOW THE TRUE VALUE!”

And as usual when Tim loses an argument he just repeats his nonsense in bold capitals.

If you know the true value there is no uncertainty. How can this give you the same uncertainty as when you don’t know the true value?

Reply to  Bellman
July 4, 2026 6:07 am

If you know the true value there is no uncertainty. How can this give you the same uncertainty as when you don’t know the true value?”

You were equating “true value +/- error”. True value +/- error is *NOT* uncertainty!



Reply to  Tim Gorman
July 4, 2026 7:10 am

“You were equating “true value +/- error”. True value +/- error is *NOT* uncertainty!”

Of course it is. But I can see why you are confused. The word error can have multiple meanings.

In one sense it can mean a specific error. I measured a 10m rod, but had an error of +1cm, so my measurement was 10.01m. But when used as the basis of uncertainty you are not describing one specific error but the dispersion of all possible errors. Your true value ± error is not describing a single error, but he range of likely error.

Reply to  Bellman
July 3, 2026 5:25 pm

 but it makes no difference if you model uncertainty by an uncertainty probability distribution or by an error probability distribution.”

Bellman is omnipotent. He knows the “true value” of everything he measures. I don’t understand why someone that knows the “true value” ahead of time even bothers with measurements!

Reply to  Tim Gorman
July 3, 2026 6:51 pm

He knows the “true value” of everything he measures.

Wrong.

Reply to  Bellman
July 4, 2026 5:51 am

Wrong”

5.3 In practice, the difference in point of view does not lead to a difference in the numerical value of the measurement result or of the uncertainty assigned to that result.”

This is *NOT* saying that the use of the “true value +/- error” is equivalent to using measurement uncertainty. In one case you cannot determine the error without knowing the “true value”, in the other you don’t need to know the “true value”. “In practice” one is useful, the other is not.

Reply to  Tim Gorman
July 4, 2026 6:34 am

“In one case you cannot determine the error without knowing the “true value””

Stop just saying that and provide a reference.

The GUM says

In this case, µi is viewed as the unknown, unique “true” value of input quantity wi and each wi is assumed to be related to its “true” value µi by wi = µi + εi, where εi is the error in wi.

See the word “unknown”. Do you think that might be a clue to the fact that you do not need to “know” the true value?

Reply to  Bellman
July 4, 2026 9:49 am

See the word “unknown”. Do you think that might be a clue to the fact that you do not need to “know” the true value?”

Unfreakingbelievable

wi = ui + εi

rewrite this as ui = wi – εi

And you think you can calculate ui if you don’t know both components, wi and εi?

You know wi, the measurement value.

ui is the true value. How do you get it if you don’t know εi?

rewrite this as εi = wi – ui

how do you determine the error if you don’t know ui?

In an equation with three variables, you *have* to know two of them in order to calculate the third. An infinite number of combinations for ui and εi exist that give the same wi. So you have to know either ui or εi in order to calculate the other.

If you don’t know ui then you *have* to know εi. But how do you know what the error is if you don’t already know ui?

This was *always* the problem with the error paradigm. Needing to know two of the components when you only *know* one. You can *estimate* εi in order to calculate ui but that *still* leaves uncertainty. How do you know εi(estimated) = εi(actual)?

This was the driving force behind developing the entire uncertainty paradigm. If you have some uncertainty, then just incorporate it in a comprehensive uncertainty interval. You develop a “best estimate” and modulate that with an uncertainty interval encompassing the reasonable values that could be attributed to the measurand. It allows others to make judgements as to whether their subsequent measurements are “reasonable” or not.

Reply to  Tim Gorman
July 4, 2026 10:01 am

“And you think you can calculate ui if you don’t know both components, wi and εi? ”

Stop going round in circles. We are not interested in knowing the error we want to know the uncertainty as defined by the deviation of possible errors. If you want to just say you don’t know the one specific error, then you are correct, but irrelevant. The error paradigm is not about knowing any specific error it’s about using the spread of errors to understand uncertainty.

“This was *always* the problem with the error paradigm. ”

Your inability to come up with a single reference to this error paradigm requiring you to know the true value in order to calculate uncertainty, demonstrates the fact you are just looking for excuses. The reason the GUM tries to avoid using errors and the reason why it’s now considered fine to use errors has nothing to do with needing to know the true value.

Reply to  Bellman
July 4, 2026 11:34 am

We are not interested in knowing the error we want to know the uncertainty as defined by the deviation of possible errors.

Huh?

Reply to  karlomonte
July 4, 2026 11:46 am

What bit didn’t you understand?

Reply to  Bellman
July 4, 2026 12:20 pm

How do you find the standard deviation of the errors if you don’t know their values?

If you use the standard deviation of the stated values for the standard deviation of the error distribution then you are using the uncertainty paradigm, not the error paradigm.

Reply to  Tim Gorman
July 4, 2026 3:48 pm

How do you find the standard deviation of the errors if you don’t know their values?

I’m pretty sure Taylor and Bevington explain this. The obvious answer is the same way as you find the uncertainties using the GUM approach – either by repeated measurements or by prior logic. What the GUM terms Type A or Type B.

If you use the standard deviation of the stated values for the standard deviation of the error distribution then you are using the uncertainty paradigm, not the error paradigm.

Then again, you need to provide a source where the error paradigm is used, and they tell you not to use repeated measurements. If you find such a source, maybe it will answer your question.

As an aside, though – I’m not sure of your logic that using the standard deviation of repeated measurements is the uncertainty paradigm. It’s actually the opposite. You are seeing the standard deviation of measured values, i.e. the spread of errors.

Reply to  Bellman
July 5, 2026 5:08 pm

We are not interested in knowing the error we want to know the uncertainty as defined by the deviation of possible errors.”

“I’m pretty sure Taylor and Bevington explain this. The obvious answer is the same way as you find the uncertainties using the GUM approach – either by repeated measurements or by prior logic.”

You *still* don’t get the error paradigm. Taylor and Bevington do *NOT* explain this for the error paradigm.

The error paradigm is wi = ui + εi. If you don’t know ui then there is no way to get εi! Repeated measurements don’t help if you don’t know ui!

Assuming or estimating a value for ui ADDS uncertainty which the paradigm doesn’t allow to be specified. No amount of “prior logic” can fix this.

Again, this is why the error paradigm was abandoned for the uncertainty paradigm.

Then again, you need to provide a source where the error paradigm is used”

You’ve been given the source OVER AND OVER AND OVER AD INFINITUM!

READ THE GUM! IT’S IN THERE! Start with Annex E! Quit whining that no one has given you this reference!

 and they tell you not to use repeated measurements”

Repeated measurements don’t help if you don’t know ui!

εi = wi – ui

If you don’t know ui how do you know εi?

The GUM tells you that εi inherits the variance of wi BUT IF YOU DON’T KNOW UI YOU DON’T εI. Knowing the variance of wi tells you the variance of εi but NOT IT’S VALUE! And if you just guess at ui all you do is stick in more uncertainty that you have no way to specify! You will have added systematic uncertainty to the statement of the measurement.

One more time, the error paradigm is incomplete. It’s why it was abandoned for the uncertainty paradigm.

I’m not sure of your logic that using the standard deviation of repeated measurements is the uncertainty paradigm. “

You can’t believe how damn frustrating it is to continually have to give you the same quotes from the GUM over and over and over … ad infinitum!

READ THE DAMN GUM!

———————-
3.3.5 The estimated variance u^2 characterizing an uncertainty component obtained from a Type A evaluation is calculated from series of repeated observations and is the familiar statistically estimated variance s^2 (see 4.2).
———————

READ THE DAMN GUM!

You are seeing the standard deviation of measured values, i.e. the spread of errors.”

READ THE DAMN GUM!

This is the variance of the error term εi. That does not help if you don’t know the value of ui! Again, the error paradigm has no way to completely characterize measurements that can be used to judge the reasonableness of subsequent measurements.

One more time, it’s why the error paradigm was abandoned for the uncertainty paradigm.

Reply to  Tim Gorman
July 5, 2026 6:34 pm

You *still* don’t get the error paradigm.

Then give me a reference. You claim that everyone used the paradigm before the GUM came along. People surely wrote it down somewhere. The longer you hide from this the more obvious it becomes that you are just making this up.

You’ve been given the source

Only in your head. All you ever do is quote the GUM which cannot be a reference to the error paradigm seeing as they are trying to replace it. I’m asking for a pre-GUM reference that describes how the error paradigm required you to know the true value in order to calculate the uncertainty. I very much doubt such a source exists becasue it would have been a really useless concept.

One more time, it’s why the error paradigm was abandoned for the uncertainty paradigm.

So why is the uncertainty paradigm being rejected?

Reply to  karlomonte
July 4, 2026 12:18 pm

Don’t need to know the errors. Somehow I guess we are supposed to just know the standard deviation of the errors without actually knowing what the values of the errors are.

Reply to  Bellman
July 4, 2026 12:16 pm

Stop going round in circles. We are not interested in knowing the error we want to know the uncertainty as defined by the deviation of possible errors.”

Again, if you don’t know the true value then you can’t generate a deviation of possible errors. If the true value is unknown then so is the error.

εi = wi – ui. If you don’t know ui then how do you know εi?

In a Type A measurement uncertainty the uncertainty is the standard deviation of the measured values, not of some guess at an εi value.

A Type B measurement uncertainty is similar, just the process of determining the reasonable values is different.

You are really blowing it out your backside now, aren’t you?

 The error paradigm is not about knowing any specific error it’s about using the spread of errors to understand uncertainty.”

unfreakingbelivable!

How do you know the spread of errors if you don’t know the value of the errors? Are you using your cloudy crystal ball again? Do you imbibe something before using the crystal ball?

Your inability to come up with a single reference to this error paradigm”

I’ve given you the reference. It’s right there in the GUM that you never bother reading.

GUM:

-“this Guide in effect uncouples the often confusing connection between uncertainty and the unknowable quantities “true” value and error.
-” this Guide in effect uncouples the often confusing connection between uncertainty and the unknowable quantities “true” value and error.

Here is another one:

————————-
E.5.4 While the approach based on “true” value and error yields the same numerical results as the approach taken in this Guide (provided that the assumption of the note of E.5.2 is made), this Guide’s concept
of uncertainty eliminates the confusion between error and uncertainty (see Annex D). Indeed, this Guide’s operational approach, wherein the focus is on the observed (or estimated) value of a quantity and the observed (or estimated) variability of that value, makes any mention of error entirely unnecessary.
—————————–

Error and uncertainty ARE NOT THE SAME THING. If wi = ui – εi then you need to know at least two of the factors, wi/ui, wi/εi, or ui/εi. Since you don’t know either ui or εi you simply can’t use the formula without making guesses which leaves lingering doubt abut the values given.

READ THE GUM!

Reply to  Tim Gorman
July 4, 2026 1:56 pm

There is one case where the error distribution and the uncertainty distribution are both Gaussian.

Here are just some the error paradigm problems.

  • No adjustments of accuracy, the center value was the true value.
  • No adjustments for experimental components that affect accuracy.
  • No standard procedure for combining error components in functional relationships with multiple inputs.
  • No ability to deal with non-Gaussian probability distributions.
  • No ability to compensate for “systematic” components.
Reply to  Jim Gorman
July 4, 2026 4:10 pm

Here are just some the error paradigm problems

Still waiting for some evidence that the error paradigm ignores those problems.

All the sources I’ve seen talk about systematic errors, combining error components, non-Gaussian distributions etc.

No standard procedure for combining error components in functional relationships with multiple inputs.

It’s called the General Formuia for Error Propagation. It’s functionally identical to equation 10 in the GUM.

Reply to  Bellman
July 6, 2026 6:47 am

It’s called the General Formuia for Error Propagation. It’s functionally identical to equation 10 in the GUM.

Been reading Wikipedia again have we.

  1. Input quantiles each have unique measurements and uncertainty.
  2. In the uncertainty paradigm a single input quantity is evaluated separately using either a Type A statistical analysis or a Type B from some other source.
  3. Once all the uncertainties for each input quantity have been determined — the point has arisen where all the uncertainties must be propagated into a combined uncertainty.

No one has ever said the final step of either error analysis or uncertainty analysis is not the same! Why? Because it has always been necessary to combine all the various errors/uncertainty together with a process that recognizes different dimensions and absolute values. This is normally done with partial derivatives and relative uncertainties regardless of the type of analysis of the individual components. This turns values into dimensionless quantities and weights them according to their value in a functional relationship.

A big difference was the use of the SEM rather than the standard deviation. Why? Because of the prevailing view that a negative error could offset a corresponding positive error. This resulted in a narrowing of the distribution. There was a lot more attention given to identifying “outliers” and removing them in order to obtain a nice Gaussian distribution. If you could show that there were sufficient offsetting errors, the “mean” became the true value with a very small error.

In addition, there was no attention paid to different probability distributions. It was Gaussian or else. Likewise, there was no concept of developing an uncertainty budget. Most errors used manufacture’s stated device error range such as %2 of full scale.

Climate science still holds to this traditional concept. Errors offset and the SEM is the uncertainty. The fact that you never see an uncertainty budget that shows different components is a dead giveaway.

Reply to  Jim Gorman
July 6, 2026 7:36 am

Been reading Wikipedia again have we

Nope, though I’m not sure why you would care. It’s better than getting your AI to tell you what you think.

Input quantiles each have unique measurements and uncertainty.
In the uncertainty paradigm a single input quantity is evaluated separately using either a Type A statistical analysis or a Type B from some other source.
Once all the uncertainties for each input quantity have been determined — the point has arisen where all the uncertainties must be propagated into a combined uncertainty.

Seems a fair summary of what I’m saying.

No one has ever said the final step of either error analysis or uncertainty analysis is not the same!

/then you need to explain exactly what you are saying. Because all I hear is “you couldn’t use the error paradigm if you didn’t know what the true value was.” and “No standard procedure for combining error components in functional relationships with multiple inputs.”

A big difference was the use of the SEM rather than the standard deviation.

You are not making any sense. What used the SEM and what didn’t? Both error and uncertainty paradigms seem to use their equivalent of the SEM for getting the uncertainty of the mean of multiple measurements. Taylor calls it SDOM, the GUM calls it the experimental standard deviation of the mean.

Because of the prevailing view that a negative error could offset a corresponding positive error.

Which is true whichever paradigm you are using.

In addition, there was no attention paid to different probability distributions. It was Gaussian or else.

Again, Taylor and Bevington both describe different distributions.

Likewise, there was no concept of developing an uncertainty budget. Most errors used manufacture’s stated device error range such as %2 of full scale.

You are describing methodology rather than paradigms here.

Climate science still holds to this traditional concept.

Where? You keep making these sweeping statements but you never quote a single paper doing whatever you claim is being done.

Errors offset and the SEM is the uncertainty.

Where and in what context? Are you still talking about measurement uncertainty or are you talking about statistics?

The fact that you never see an uncertainty budget that shows different components is a dead giveaway.

Yes, it suggests you have never looked.

Reply to  Tim Gorman
July 4, 2026 4:01 pm

Again, if you don’t know the true value then you can’t generate a deviation of possible errors. If the true value is unknown then so is the error.

And that’s your big mistake – or at least one of them. If you take repeated measurements the standard deviation of your measurements will be the standard deviation of the errors.

In a Type A measurement uncertainty the uncertainty is the standard deviation of the measured values, not of some guess at an εi value.

No – it’s an estimate of the uncertainty, just as it’s an estimate of the standard deviation of the errors.

unfreakingbelivable!

Thank you. I like to consider myself incredible.

How do you know the spread of errors if you don’t know the value of the errors?

Same answer as above. How do you know the uncertainty if you don’t know which values it’s reasonable to attribute to the measurand? You either guess or you take a sample of measurements.

It’s right there in the GUM that you never bother reading.

That is not describing your error paradigm. This shouldn’t be that difficult. Before the GUM was cobbled together there was a way of using measurement uncertainty you call the error paradigm. You have made a lot of claims about how useless this method was, and how it couldn’t be used without first knowing the true value. All you need to do is produce a document written before the GUM which describes the error paradigm in the way you suggest.

While the approach based on “true” value and error yields the same numerical results as the approach taken in this Guide

And you still won;t acknowledge that this destroys your argument. You are saying it was impossible to determine an uncertainty using the error paradigm, the GUM is saying you could use it to get the same result.

Reply to  Bellman
July 5, 2026 6:39 pm

And that’s your big mistake – or at least one of them. If you take repeated measurements the standard deviation of your measurements will be the standard deviation of the errors.”

WHAT IS THE VALUE OF THE ERROR?

εi +/- σ_wi TELLS YOU WHAT if you know σ_wi but not εi?

If you don’t know ui then you don’t know εi!

The error paradigm is *NOT* the uncertainty paradigm!

The error paradigm
-uses a fixed, unknown “true value”.
-uses an unknown error term for each measurement

The uncertainty paradigm
-uses a “best estimate” for the value
-uses an uncertainty interval for the measurement

These are *NOT* the same!

That is not describing your error paradigm.”

READ THE GUM! Start with Annex E!

Before the GUM was cobbled together there was a way of using measurement uncertainty you call the error paradigm”

NO! There wasn’t. ui and εi ARE NOT DISTRIBUTIONS. They are unknown values. READ THE GUM!!!

 All you need to do is produce a document written before the GUM which describes the error paradigm in the way you suggest.”

Bullshite! *YOU* need to prove my assertions wrong, not the other way around. I’ve given you the internationally accepted GUM that explains all of this and is exactly what I have asserted – I have QUOTED the applicable entries from the GUM!

If you *really* need a reference other than the GUM, look up Eisenhart (1939) “The Meaning of Error in Experimental Measurement”.

“The error of an observation is the difference between the observed value and the true value.”

Eisenhart also says that in the error paradigm probability distributions belong ONLY TO THE ERROR TERM, not to the true value. This is *exactly* what I’ve been trying to tell you and show you with quotes from the GUM. The true value is FIXED AND UNKNOWN. quote: “The true value is never known.”

Eisenhart also uses the exact same expression as the GUM: wi = ui + εi

I know you aren’t going to go look this up. You are just going to continue whining that no one ever gives you any references. It’s one of your go-to excuses when it’s obvious you have no actual understanding of the subject at hand.

And you still won;t acknowledge that this destroys your argument.”

Oh, malarky! You get the same things IF YOU KNOW THE TRUE VALUE!

You seem to be the only person on the Earth that knows the TRUE VALUE of everything. Where did you get your crystal ball? Amazon?

Reply to  Tim Gorman
July 5, 2026 6:51 pm

WHAT IS THE VALUE OF THE ERROR?

You don;t kn ow – that’s why it’s uncertain.

Reply to  Tim Gorman
July 5, 2026 7:33 pm

If you *really* need a reference other than the GUM, look up Eisenhart (1939) “The Meaning of Error in Experimental Measurement”.

That’s a start, but I can;t find an online reference. I’ll look through some of the other Eisenhart sources.

“The error of an observation is the difference between the observed value and the true value.”

Yes, that’s the definition of error.

Eisenhart also says that in the error paradigm probability distributions belong ONLY TO THE ERROR TERM, not to the true value.

Of course. The true value is fixed, it is not a random variable.

This is *exactly* what I’ve been trying to tell you and show you with quotes from the GUM.

Then you would be correct – but all I keep hearing from you is that you need to know the true value in order to calculate the uncertainty.

Eisenhart also uses the exact same expression as the GUM: wi = ui + εi

Are you actually going to get top the part where Eisenhart agrees with you about the need to know the true value?

I know you aren’t going to go look this up.

A link would be a help. Searching so far hasn’t turned up a copy.

You seem to be the only person on the Earth that knows the TRUE VALUE of everything. Where did you get your crystal ball? Amazon?

You know you are losing the argument every time you resort to lying like this. I keep saying you don’t know the true value.

Reply to  Tim Gorman
July 6, 2026 4:30 am

“εi +/- σ_wi TELLS YOU WHAT if you know σ_wi but not εi?”

What is this nonsense? The standard deviation of εi is the same as the standard deviation of wi. That’s all you need to know.

You are just misunderstanding everything for the sake of making a point, or just trolling. You claim to have a lifetime experience in metrology. How did you cope before the GUM arrived?

Reply to  Bellman
July 6, 2026 10:36 am

You are just misunderstanding everything for the sake of making a point

Here is a paper from 1966 about errors. Please note the use of “corrections” throughout the paper. Corrections under error analysis were a tool to limit the distribution surrounding the mean value.

Notes on the use of propagation of error formulas

This discussion on error vs uncertainty is worthless. The world has spoken and uncertainty has won. If you have a problem with that, take it up with your national measurement agency.

Reply to  Jim Gorman
July 6, 2026 2:26 pm

Here is a paper from 1966 about errors.

And what do you think is different than what I’ve been saying? Where does it say you need to know the true value? And how is any of that different to the uncertainty paradigm?

Please note the use of “corrections” throughout the paper.

3 times. The first is correcting for systematic errors, just as is described in the GUM. The other times are demonstrating that the general equation is a reasonable approximation, and the higher order terms can usually be ignored.

This discussion on error vs uncertainty is worthless.

I wouldn’t go that far, but as I said there is little to choose between the two.

The world has spoken and uncertainty has won.

You are still ignoring Possolo’s contribution: “the error concept and the uncertainty concept are the same.”

Reply to  Jim Gorman
July 7, 2026 6:50 am

This discussion on error vs uncertainty is worthless.

Absolutely completely worthless, although this time around bellman let it slip out that he thinks the GUM was just cobbled and tossed together..

The world has spoken and uncertainty has won. If you have a problem with that, take it up with your national measurement agency.

He won’t, but bellman Must — HaveTheVeryLastWord.

Reply to  Bellman
July 5, 2026 9:40 pm

Before the GUM was cobbled together

HAHAHAHAHAHAHAHAHAHA

Your slip is showing.

Reply to  Bellman
July 6, 2026 7:59 am

And you still won;t acknowledge that this destroys your argument. You are saying it was impossible to determine an uncertainty using the error paradigm, the GUM is saying you could use it to get the same result.

The statement “the GUM is saying you could use it to get the same result” should be “you can use the error process, IF DONE PROPERLY, to achieve the same result.

The problem is that if you dig deeply enough, the error paradigm requires the assumption of a Gaussian distribution for all input quantity assessment. That is the only option. GUM D.6.2 graphs may help you.

Reply to  Jim Gorman
July 6, 2026 11:08 am

“The problem is that if you dig deeply enough, the error paradigm requires the assumption of a Gaussian distribution for all input quantity assessment.”

You keep trying to switch the claim. I’m not talking about any issues with the use of statistics. I’m trying to get you to support the claim that the error paradigm required you to know the true value. That’s the claim I find unbelievable.

I’m not sure you are correct about the requirement for Gaussian distributions, though. It depends on exactly what you are doing. I suspect you are talking about an issue to do with expanded uncertainties. The issue, from what I’ve seen, is a lot of early uncertainty texts are sloppy with exactly what an uncertainty interval represents. It might be a standard deviation, or it could be a confidence interval, or anything.

The rules for propagating uncertainty are based around a standard deviation. You can also use them them with a multiple of an SD, that is an expanded uncertainty. The problem is interpreting the interval as a specific confidence level. If you plug a 95% confidence interval into a propagation equation, and your distributions are not normal, the resulting uncertainty might no longer represent an exact 95% confidence interval.

This isn’t really anything to do with error verses uncertainty, it’s just that the GUM is a bit more specific about using standard uncertainties rather than confidence intervals.

Reply to  Jim Gorman
July 2, 2026 10:58 am

Let me clarify something. The mean is a STATISICAL DESCRIPTOR. Nothing more. It is *not* a measurement. It has no measurement uncertainty. The formula the climate science supporters keep coming up with is the SEM, σ/sqrt(n). The clue is that as sample size, “n”, goes up the SEM approaches 0 (zero). Meaning that as the sample size approaches the population size sampling error goes to 0 (zero) and you get a better and better estimate of the population mean. IT TELLS YOU NOTHING ABOUT THE MEASUREMENT UNCERTAINTY THAT GOES WITH THAT MEAN! The mean of even the entire population may be very inaccurate if the component values are very inaccurate. It is the accuracy of the mean that is the measurement uncertainty, not the accuracy of the statistical descriptor known as the mean.

Physically the SEM is pretty much useless. It can’t be used to calibrate a measuring device. It can’t be used for go-nogo tolerance, for system reliability, for load capacity or distribution, failure probability, and anything else that is meaningful in the real, physical world. It’s really only useful when you have multiple measurements of the same thing, using the same JUST CALIBRATED measuring device, under the same environmental conditions. NONE OF THESE APPLY TO MEASUREMENT OF TEMPERATURE.

In other words, it’s pretty much a statistical world value with no point of congruity to the real world most of us live in.

Reply to  Bellman
July 2, 2026 7:42 am

1. You don’t need negative numbers to reduce the uncertainty. The reduction in an average is caused by the partial derivative df/dx in the equation scaling all the values down.

You are incorrect. If xᵢ is divided by “n”, then you must use the division rule to calculate the uncertainty. This rule uses relative uncertainties as follows:

if the input quantity is (xᵢ)/n, then,
the uncertainty of xᵢ is u(xᵢ), and the relative uncertainty is [u(xᵢ)/(xᵢ/n)].

Then (∂f/∂(xᵢ/n) = 1/n and the propagation equation using relative uncertainties is:

[u(y)/y]² = (1/n)² [u(xᵢ)/(xᵢ/n)]² =
(1/n²) [nu(xᵢ)/xᵢ]² =
(1/n²)(n²)[u(xᵢ)/xᵢ]² =
[u(xᵢ)/xᵢ]²

Therefore,
[u(y)/y]² = [u(xᵢ)/xᵢ]², and

u(y)/y = [u(xᵢ)/xᵢ]
u(y) = y [u(xᵢ)/xᵢ], and guess what, “n” disappears!

You can extend this for as many inputs as you want, but the “n” disappears in each input quantity xᵢ. So you end up with an equation of propagation of:

u(Y) = Y√∑u(xᵢ)/xᵢ
This is the same result as if you made the normal assumption that the uncertainty of a constant or counting number is zero.

From the book, Experimentation and Uncertainty Analysis for Engineers, Coleman and Steele.

The function used is Mₛ = (2LaF)/(πR⁴θ)

“Noting that this is in the special for of Eq. (3.18) and assuming as usual that the factors 2 and π have zero uncertainties, …”

The counting number of (1/n) is no different and has no contribution to uncertainty.

Reply to  Jim Gorman
July 2, 2026 9:03 am

“You are incorrect.”

Which bit? That the partial derivative of x/N is 1/N or the way you use this in equation 10?

All your convoluted arguments are wrong if you don’t get the same result as using equation 10, because all other methods are derived from it.

“if the input quantity is (xᵢ)/n, then”

And there’s your first mistake. The input quantity is not xi/n, it’s xi. Dividing by n is the function.

Reply to  Bellman
July 2, 2026 4:08 pm

And there’s your first mistake. The input quantity is not xi/n, it’s xi. Dividing by n is the function.

No, you are incorrect. The function isY = f(X1, X2, …., Xn) = x1/n + x2/n + … + xn/n).

I have searched my engineering handbooks and far and wide on the internet for an equation for temperature that does what you claim an average does.

I know you think your function of f(X1, X2, …, Xn) = (x1 + x2 + …, + xn)/n is a good function for determining temperature, but it does not fit into any GUM process.

The GUM in section 4 shows how to determine the uncertainty of a series of observations -> qₖ. The observations are members of a random variable that has both a mean and standard deviation. Your definition does not use a random variable to determine both of these statistical parameters in a Type A statistical analysis.

This means your definition does not match with the GUM and consequently is outside the normal metrology paradigm.

For example, what is the standard definition deviation for your mean since you claim (xᵢ/n) is not proper? How do you treat a single entity of (∑xᵢ/n) in calculating a standard deviation? Is the standard deviation really zero? In other words (∑xᵢ/n – ∑xᵢ/n)²/(n-1)? That is what you are trying to sell.

Reply to  Jim Gorman
July 2, 2026 5:41 pm

I know you think your function of f(X1, X2, …, Xn) = (x1 + x2 + …, + xn)/n is a good function for determining temperature

It’s a good function for determining the mean. I’m sorry if your engineering handbooks don’t mention that.

For example, what is the standard definition deviation for your mean since you claim (xᵢ/n) is not proper?

I said (xᵢ/n) is not the input. xᵢ is. And it doesn’t have a standard deviation, it’s just a single value. The uncertainty of xᵢ is expressed as a standard deviation, and if you obtained the input by averaging multiple measurements, those measurements have standard deviation.

But that’s all the help I’m giving you. You need to stop mixing up all these different parts of the GUM – just focus on the task of averaging individual measurements of different things, and how you get the measurement uncertainty of that average.

Reply to  Bellman
July 3, 2026 6:16 pm

I said (xᵢ/n) is not the input. xᵢ is.

If xᵢ are the input quantities that determine Y, then Y = f(X, X₂, …, Xₙ).
The estimate of Y denoted as “y” is determined by the estimates of the input quantities denoted as xᵢ. And, y = f(x, x, …, xₙ).

At this point, a measurement model must be defined. You have in the past used the following.

y = f(x, x, …, xₙ, n) = (∑xᵢ)/n

This is a formula for calculating a mean of a collection of individual values. It is not a functional relationship that maps input variables into a unique output.

I asked CoPilot to describe the difference between a mean and functional relationship.

Q: Is an average or mean considered a functional relationship

Short Answer: No — an average (mean) is not a functional relationship.  It is a statistic, not a mapping.

Why a mean is not a functional relationship

A functional relationship requires:

  • A defined input variable 𝑥
  • A defined output variable 𝑦
  • A reproducible mapping 𝑦=𝑓(𝑥)

A mean does none of these. It:

  • Takes a collection of values
  • Collapses them into a single number
  • Does not describe how one variable depends on another
  • Does not predict new values
  • Does not encode a physical law or causal rule

The mean is an operator, like integration or summation:

=(1/n)∑xᵢ

This is a procedure, not a relationship between variables.

Why don’t you see if you can convince an AI otherwise and show the conversation here.

Reply to  Jim Gorman
July 3, 2026 7:12 pm

If xᵢ are the input quantities that determine Y, then Y = f(X₁, X₂, …, Xₙ).

By George, I think he’s got it.

This is a formula for calculating a mean of a collection of individual values.

Yes. That’s the definition of “mean”.

It is not a functional relationship that maps input variables into a unique output.

Oh dear, and you were doing so well.

Calculate the mean of 1,2,3. How many different answers do you get. I can only see one possible answer (1 + 2 + 3) / 3 = 2. That is the unique output. By definition if something is a function it defines a functional relationship.

I asked CoPilot

Why do you keep doing this. You know that any response from an AI is not based on understanding. You know that I’ll just try a different question, get a different answer, which you will pretend doesn’t exist.

Why don’t you see if you can convince an AI otherwise and show the conversation here.

OK – but I kept doing this during the greenhouse effect argument, and kept being ignored. I suspect the same thing will happen here.

—————————————————————————–

My question: “Is the exact mean of a number of values a functional relationship?”

Copilot:

Short answer: Yes — the exact arithmetic mean of a finite set of values is a deterministic functional relationship. The mean is a function because the inputs (the values) uniquely determine the output (the mean).

Why the mean is a functional relationshipA functional (deterministic) relationship is one in which each input leads to exactly one output. In statistics, this is defined as a relationship where the output is precisely determined by the inputs, with no randomness involved.
The arithmetic mean of n numbers
xˉ=1n∑i=1nxi
is exactly determined by the list (x1,x2,…,xn). There is no scatter, no uncertainty, and no multiple possible outputs. It satisfies the definition of a function: each input tuple maps to one and only one output.
So the mapping
(x1,x2,…,xn)↦xˉ
is a deterministic functional relationship.

Why this matters conceptuallyThis distinction is important because:

  • A statistical relationship involves scatter or randomness (e.g., height vs. weight).
  • A deterministic functional relationship is exact (e.g., Celsius → Fahrenheit, or mean → data).

The mean belongs to the deterministic category: it is a mathematical operator, not a statistical trend.

Subtle point you may be probingIf your deeper question is whether the mean is a physically meaningful functional relationship (e.g., whether the mean temperature of two bodies is a meaningful physical quantity), that’s a different issue — and one with rich metrological nuance. But mathematically, the mean is unquestionably a function.

——————————————————————————

Reply to  Bellman
July 3, 2026 7:21 pm

By the way – seeing as we are back to using statistical word generators as authorities, and given you keep failing to provide a reference for your definition of the “error paradigm”, I though I’d ask copilot.

The answer was pretty much how I interpret it – and certainly does not suggest you need to know the true value in order to calculate the uncertainty.

——————————————————————————

In the field of metrology, what is the “error paradigm” of uncertainty?

Answer:

The error paradigm in metrology is the older, pre‑GUM framework in which measurement quality is described in terms of errors—that is, the (unknown) difference between a measured value and the true value. In this paradigm, uncertainty is treated as a property of errors themselves, not of the measurement result.

The error paradigm treats uncertainty as arising from errors—unknown deviations of the measurement result from the true value. It assumes that a true value exists, that errors can be decomposed into identifiable components, and that the goal of analysis is to estimate or bound these errors.

What the error paradigm asserts

  • A true value exists (even if unknown).
  • Error = measured value − true value is the fundamental quantity.
  • Uncertainty is interpreted as the estimated magnitude or distribution of this error.
  • Measurement quality is expressed through:
  • Systematic error (bias)
  • Random error (scatter)
  • Corrections applied to reduce systematic error
  • Residual error bounds after corrections

This approach is historically tied to the theory of errors used in classical physics, geodesy, and engineering.

How this differs from the modern GUM uncertainty paradigmThe GUM (Guide to the Expression of Uncertainty in Measurement) rejects the idea that uncertainty is about errors. Instead, it defines uncertainty as a degree of belief about the range of values that could reasonably be attributed to the measurand.

  • GUM: uncertainty is about knowledge.
  • Error paradigm: uncertainty is about errors.

The GUM explicitly states that “error” and “uncertainty” are conceptually different—though this claim has been challenged in recent literature.

But the debate isn’t settledRecent work argues that the distinction between error and uncertainty is overstated and that both paradigms ultimately describe the same relationship between measured and true values. This critique suggests that the GUM’s dismissal of the error paradigm may be conceptually unnecessary.

SummaryError paradigm = uncertainty as estimated error, grounded in the assumption of a true value and decomposable error sources.
GUM paradigm = uncertainty as state of knowledge, not error.
Both frameworks aim to quantify measurement reliability, but they differ philosophically and mathematically.

—————————————————————————–

Reply to  Bellman
July 4, 2026 1:13 pm

In the field of metrology, what is the “error paradigm” of uncertainty?

The answer was pretty much how I interpret it – and certainly does not suggest you need to know the true value in order to calculate the uncertainty.

Oh brother, when are you going to learn error and uncertainty are two different things.

The error paradigm in metrology is the older, pre‑GUM framework in which measurement quality is described in terms of errors—that is, the (unknown) difference between a measured value and the true value.

From GUM

E.5.1 The focus of this Guide is on the measurement result and its evaluated uncertainty rather than on the unknowable quantities “true” value and error

From your AI: The GUM (Guide to the Expression of Uncertainty in Measurement) rejects the idea that uncertainty is about errors. Instead, it defines uncertainty as a degree of belief about the range of values that could reasonably be attributed to the measurand.

This is exactly what we’ve been telling you. The uncertainty paradigm defines a range of values that can be attributed to the measurand. A Type A statistical analysis uses multiple observations of an input quantity that are collected into a random variable from which a probability distribution is developed and from that a mean and standard deviation. There is no guarantee that the interval contains the true value but by having an accuracy component in the uncertainty budget that is added to the combined uncertainty and correction values determined from a calibration, then expanding the uncertainty to 2σ, one can have an idea that the interval has a good chance of containing the true value.

The error paradigm CAN be the same as an uncertainty but it relies solely on an assumption that the error distribution is random. In other words, a Gaussian distribution where error above the mean, exactly match the errors below. There was no consideration for distributions such as binomial, Poisson, or Lorentzian, and many others. The uncertainty paradigm can handle these.

Reply to  Jim Gorman
July 4, 2026 3:41 pm

Oh brother, when are you going to learn error and uncertainty are two different things.

I am not your brother, and I have not said that they are the same thing.

This is exactly what we’ve been telling you.

It would be great if for once you could state exactly what you think you are telling me.

A Type A statistical analysis uses multiple observations of an input quantity that are collected into a random variable from which a probability distribution is developed and from that a mean and standard deviation.

You do not collect measurements into a random variable, but aside from that, how is any of that different to the error paradigm method of making repeated observations and deriving a probability distribution from the results?

then expanding the uncertainty to 2σ, one can have an idea that the interval has a good chance of containing the true value.

Again, how does that differ from the error paradigm?

The best you can say about the GUM approach is that they are using Bayesian rather than frequentist probability (Hence the degree of belief they talk about). Strictly speaking you shouldn’t talk about a “good chance” of the true value being within an interval using Frequentist probability. But as the GUM refuses to admit to using Bayesian probability, and ignores any suggestion of priors – it’s not a major issue.

The error paradigm CAN be the same as an uncertainty but it relies solely on an assumption that the error distribution is random.

And what do you think your Type A evaluation is doing?

In other words, a Gaussian distribution where error above the mean, exactly match the errors below.

Random does not equal Gaussian. And you are confusing symmetric with Gaussian. Again do you have a source that the error paradigm requires all error distributions to be be Gaussian?

There was no consideration for distributions such as binomial, Poisson, or Lorentzian, and many others.

Huh? Taylor and Bevington both list different distributions.

The uncertainty paradigm can handle these.

This really makes no sense. Both error and uncertainty use probability distributions, then can be any shape you think appropriate. Both use identical equation to handle them.

Reply to  Bellman
July 5, 2026 4:36 pm

You do not collect measurements into a random variable,”

Unfreakingbelievable.

GUM:
————————–
4.2.1 In most cases, the best available estimate of the expectation or expected value μq of a quantity q that varies randomly [a random variable (C.2.2)], and for which n independent observations qk have been obtained under the same conditions of measurement (see B.2.15), is the arithmetic mean or average q (C.2.19) of the n observations:
———————- (bolding mine, tpg)

You JUST CAN’T STOP MAKING GARBAGE ASSERTIONS, CAN YOU?

but aside from that, how is any of that different to the error paradigm method of making repeated observations and deriving a probability distribution from the results?”

Because wi = ui – εi. If ui and εi are unknowable then how do you come up with what wi should be?

The error paradigm does *NOT* collet repeated observations and use them to derive a probability distribution.

The error paradigm is a STATE FUNCTION. If you know the true value, ui, and the error, εi, then that determines wi. Nor can you go the other way. If you don’t know εi then you can’t determine ui.

WAKE UP! If the paradigms were the same then why do you think the international metrology community would all of a sudden just copy the error paradigm into a new document labeled “Uncertainty”?

JUDAS PRIEST! How many times have you been told to read the GUM? It’s right there in the second paragraph of the introduction. Have you not even bothered to read the intro to the document?

—————————
0.2 The concept of uncertainty as a quantifiable attribute is relatively new in the history of measurement, although error and error analysis have long been a part of the practice of measurement science or metrology. It is now widely recognized that, when all of the known or suspected components of error have been evaluated and the appropriate corrections have been applied, there still remains an uncertainty about the correctness of the stated result, that is, a doubt about how well the result of the measurement represents the value of the quantity being measured.
————————-

The error paradigm doesn’t have a way to encompass the uncertainty associated with stated result! There is *NO* uncertainty interval used in the error paradigm! There is no “best estimate” value obtained from averaging.

Again, how does that differ from the error paradigm?”

Do you actually have something against learning? How many times must this be explained to you before it sinks in?

The best you can say about the GUM approach is that they are using Bayesian rather than frequentist probability”

Do you even understand the terms you are using? Using the MEAN of a distribution as the BEST ESTIMATE *IS* a frequentist approach!

Both error and uncertainty use probability distributions, then can be any shape you think appropriate. Both use identical equation to handle them.”

Exactly where in the error paradigm equation do you think probability distributions are used? With the error paradigm wi = ui + εi where wi is the measured value, ui is the true value, and εi is the error. None of these components are assigned a probability distribution based on a measurement model. The only probability that ever enters is if multiple error terms identified during multiple measurements can be described by a distribution. But that distribution is *NOT* used in the wi = ui + εi equation. If you rewrite the equation into εi = wi – ui and ui is a true value (i.e a fixed constant) then the εi term will inherent the variance of the measured values, wi. The problem is that you *still* have to have a “true value” in order to come up with the εi term and there is no way to use a probability distribution for that.

GUM:
———————–
Second, because εi = wi − μi, and because the μi represent unique, fixed values and hence have no uncertainty, the variances and standard deviations of the εi and wi are identical.
———————

The error paradigm is just incomplete. It has no way to completely specify uncertainty in the measurement. It is why it was abandoned in favor of the uncertainty paradigm.

You’ve STILL not bothered to read the GUM at all, have you?

Reply to  Tim Gorman
July 5, 2026 5:31 pm

Unfreakingbelievable.”

Such a convincing argument. But nothing else you say or quote suggests you understand why “collecting measurements into a random variable” is correct in this case. You quote literally says that q is considered to be a random variable – not that the measurements have been collected into the variable.

But as always you get hysterical about a minor correction, in an effort to distract from all your other mistakes, so let’s just leave it. It’s not worth it.

Because wi = ui – εi. If ui and εi are unknowable then how do you come up with what wi should be?

If you are saying wi is the measured value, then I think there’s an obvious way of coming up with the value – by taking a measurement,.

The error paradigm does *NOT* collet repeated observations and use them to derive a probability distribution.

You can keep repeating this as much as you like – it’s not something I’m going to believe from you. Provide a reference to this paradigm.

Here’s a quote from Taylor

Whenever a measurement can be repeated, it should usually be made several times. The resulting spread of values often provides a good indication of the uncertainties, and the average of the values is almost certainly more trustworthy than any one measurement. Chapters 4 and 5 discuss the statistical treatment of multiple measurements.  Here, I emphasize only that if a measurement is repeatable, it should be repeated, both to obtain a more reliable answer (by averaging) and, more important, to get an estimate of the uncertainties.

The error paradigm is a STATE FUNCTION.”

Provide a reference. I’m not interested in whatever nonsense you think the error paradigm is, unless you can back it up with evidence.

The error paradigm doesn’t have a way to encompass the uncertainty associated with stated result!

Seriously, you need to provide a reference to this error paradigm, because it was clearly one of the most pointless ways of defining uncertainty imagined. You are saying metrology could not determine the uncertainty of a measurement until the GUM came along.

How many times must this be explained to you before it sinks in?

Once, if you provide a reference.

Do you even understand the terms you are using?

Yes. In case you hadn’t noticed statistics is a hobby of mine.

From the GUM E.3.5.

In contrast to this frequency-based point of view of probability, an equally valid viewpoint is that probability is a measure of the degree of belief that an event will occur

Recommendation INC-1 (1980) upon which this Guide rests implicitly adopts such a viewpoint of probability since it views expressions such as Equation (E.6) as the appropriate way to calculate the combined standard uncertainty of a result of a measurement.

Exactly where in the error paradigm equation do you think probability distributions are used?

For the last time how can I tell you something about this paradigm, when you won;t provide any reference to it. I have no interest in trying to decide what error prone fantasy you have about the error paradigm.

The error based approach as explained in say Taylor and Bevington uses probability distributions in the adding in quadrature rules, and in the General equation for error propagation. They are used when you calculate the standard error of the mean (or SDOM if you prefer).

If you rewrite the equation into εi = wi – ui and ui is a true value (i.e a fixed constant) then the εi term will inherent the variance of the measured values, wi.

Yes, that’s the point. I’m amazed that you can’t see it even as you write it. You know ui is a fixed value, you know the distribution of wi (to an approximation) therefore you know the distribution of ei (to the same approximation).

The problem is that you *still* have to have a “true value” in order to come up with the εi term and there is no way to use a probability distribution for that.

You say that, then immediately quote the GUM telling you how to do just that.

Reply to  Bellman
July 3, 2026 8:56 pm

But mathematically, the mean is unquestionably a function.

Funny that you left off the “relationship” part of your description.

I checked Grok also. Here is what it said.

Q: I think you are misusing the definition of a simple function. The mean of a probability function is not a mapping to a physical value such as E=mc^2 or V=IR.

You’re right to draw a distinction — I see where the confusion lies. Let’s clarify the terminology more carefully without conflating categories.

In pure mathematics, any rule that assigns exactly one output to each valid input is a function.

The mean absolutely qualifies:

In experimental physics, engineering, and many science textbooks, when people say “functional relationship” between physical quantities, they usually mean:



A relationship (often an equation or law) that expresses how one physical quantity depends on others according to a physical principle or model.

Bottom line

You are correct:

calling the mean a “functional relationship” in the same sense as E=mc² or V=IR is misleading or imprecise in a physics context.

In physics:

Functional relationships → physical laws or models linking quantities (like the examples you gave).

Mean / Expected value → statistical tool / data reduction procedure.

The mean is extremely useful in physics (for handling measurement error, random variables, etc.), but it belongs in the statistical toolbox rather than the “physical functional relationships” category.

As I’ve told you before, the GUM is based upon multiple observations of an input quantity being collected into a random variable that has a mean AND a standard deviation.

Your expressing a formula that is happens to be a mathematical function as a functional relationship that maps physical phenomena to each other is mischaracterizing what a mean describes.

Reply to  Jim Gorman
July 4, 2026 3:45 am

“Funny that you left off the “relationship” part of your description.”

Huh? I was quoting your chatbot.

“Q: I think you are misusing the definition of a simple function. The mean of a probability function is not a mapping to a physical value such as E=mc^2 or V=IR.”

Takk about asking a leading question. It’s not even a question, just a statement of what you want the chatbot to tell you.

“As I’ve told you before, the GUM is based upon multiple observations of an input quantity being collected into a random variable that has a mean AND a standard deviation.”

You keep confusing terms You are just describing a Type A uncertainty of a single measurement. This is not the same as a combined uncertainty, obtained using equation 10.

But you keep ignoring type B uncertainties. They are also treated as random variables with a mean and a standard deviation. I.e. a standard uncertainty. As far as calculating the combined uncertainty, it doesn’t matter.

“Your expressing a formula that is happens to be a mathematical function as a functional relationship that maps physical phenomena to each other is mischaracterizing what a mean describes.”

You need to make your mind up. Do you want the measurement uncertainty of a mean, or do you want to say it doesn’t exist. You seem quite happy to torture the statistics if you end up with a large uncertainty. But when your errors are pointed out, you go back to saying you can’t calculate the measurement uncertainty if a mean because a mean isn’t a “physical phenomenon”.

Reply to  Bellman
July 4, 2026 1:44 pm

Takk about asking a leading question. It’s not even a question, just a statement of what you want the chatbot to tell you.

You can see from the answer that Grok first responded to the question as to whether an average is a mathematical function. It is.

It is not however, a functional relationship that maps the connections between various physical quantities. An average is a statistical parameter describing the central value of a probability distribution. One that is used in Type A evaluation of the observations of an input quantity.

Reply to  Jim Gorman
July 4, 2026 6:11 am

The answer was pretty much how I interpret it – and certainly does not suggest you need to know the true value in order to calculate the uncertainty.”

That is *NOT* what anyone suggested. You were trying to equate “error” with “uncertainty”. You do *NOT* use the “true value” to calculate uncertainty, you use it to calculate error.

Error is *NOT* uncertainty. A concept you *still*, even after years of having it explained to you, do not grasp.

Reply to  Jim Gorman
July 4, 2026 7:03 am

It’s all just one more garbage climate science meme: “numbers is just numbers”. The real world be damned.

Reply to  Bellman
July 4, 2026 5:46 am

My question: “Is the exact mean of a number of values a functional relationship?””

Unfreakingbelivable!

The issue is whether the SEM is a functional relationship! The argument isn’t how to calculate the mean.

The mean is a deterministic function of the data.
The SEM is a characteristic of the sampling model.

The use of the SEM has requirements that you do not meet .

-iid samples
-a well-defined sampling distribution
-the sampling SD is the same as the population SD

Sampling uncertainty is *NOT* measurement uncertainty!

Reply to  Tim Gorman
July 4, 2026 6:06 am

“The issue is whether the SEM is a functional relationship!”

Just calm down and read what you ate replying to. This is third time you’ve made claims like that. We were not talking about the SEM. We were talking about whether the mean function defined a functional relationship. This was the question your brother put to the chatbot

Is an average or mean considered a functional relationship

Do you see anything there about the SEM?

Reply to  Bellman
July 4, 2026 3:45 am

It’s a good function for determining the mean. I’m sorry if your engineering handbooks don’t mention that.”

The SEM doesn’t even determine the mean. It only indicates the sampling uncertainty of the mean. It is *NOT* the measurement uncertainty of the mean. The measurement uncertainty of the mean is the propagated measurement uncertainty of the data, i.e. Eq 10, and Eq 10 has no sqrt(n) in it.

Σ(u_i)^2/n^2 is the SAMPLING UNCERTAINTY, not the measurement uncertainty. The SEM only tells you how well you have estimated the average value, not the accuracy of the average value.

I keep pointing this out and you and the rest of the supporters of the SEM being the measurement uncertainty just ignores it: The SEM simply can *not* be used for anything in the real world. It can’t be used for a go-nogo decision. it can’t be used for calibration purposes, it can’t be used for reliability determination, it can’t be used for safety factors, it can’t be used for control limits, it can’t be used for distribution shape, etc.

The SEM is meant for describing the variability of the mean across multiple samples. That’s all. It does *NOT* characterize the dispersion of the values that could reasonably be attributed to the measurand.

For temperatures from multiple instruments under different environments the measurement uncertainty is:

u_c^2(y) = Σu^2(x_i)

This applies for the best estimate of the average of the x_i. You can refine the best estimate of the average by increasing sample size and/or increasing the number of samples BUT that does *NOT* change what the propagated measurement uncertainty is. One is calculated based on the standard deviation of the stated values of the measurements in the samples and the other is based on propagating the measurement uncertainty values of the measurements.

They are *NOT* the same thing no matter how much you and climate science wishes they are.

Your focus on the SEM just demonstrates your willful ignorance of metrology basic concepts. They have been given to you multiple times. References have been provided. And yet you *still* continue to try and imply that the SEM has something to do with measurement uncertainty. IT DOESN’T. It is the sampling uncertainty associated with a statistical descriptor. It has absolutely no impact on the measurement uncertainty interval.

Reply to  Tim Gorman
July 4, 2026 4:15 am

“The SEM doesn’t even determine the mean. X

Talk about lousy reading comprehension. I said nothing about the SEM. I was replying to this insane comment

I know you think your function of f(X1, X2, …, Xn) = (x1 + x2 + …, + xn)/n is a good function for determining temperature

And pointed out it was the function for the mean.

“The measurement uncertainty of the mean is the propagated measurement uncertainty of the data, i.e. Eq 10, and Eq 10 has no sqrt(n) in it. ”

This conversation is circling the drain even faster than usual. Do you want to use equation 10 to calculate the measurement uncertainty of a mean, or do you not?

I explain the result of using equation 10. You then claim it doesn’t count as the mean function isn’t really a function, but a statistic, so can’t have a measurement uncertainty.

I point out that usually the point of taking a mean is as a sample, in order to estimate the mean of a population, and a better estimate of the uncertainty is the SEM, and you complain that is not measurement uncertainty.

And all this confusion because you just don’t want to accept the inevitable conclusion that the larger your sample the less uncertainty there is, whatever sort of uncertainty you are talking about.

Reply to  Bellman
July 4, 2026 4:43 am

“For temperatures from multiple instruments under different environments the measurement uncertainty is:

u_c^2(y) = Σu^2(x_i)”

When all else fails, just make things up. This is exactly the same claim you made over 5 years ago. You have not got a single coherent arguments for why it should be that, or how it could possibly make sense. But it’s impossible to reason with you as that would mean admitting you made a mistake.

Reply to  Bellman
July 4, 2026 8:32 am

When all else fails, just make things up.”

You truly are willfully ignorant.

If a temperature data set is a single sample then the standard deviation of the sample means doesn’t exist and the standard deviation of the data is the measurement uncertainty. If a temperature data set is a collection of single samples of size 1 then the SEM for each is u(x_i)/1 = u(x_i) and u_c(y) =  Σu^(x_i).

You simply do not have ANY MATH that shows any differently. All you have is the unsupported claim that the SEM is the measurement uncertainty. I have shown you the MATH that this isn’t the case.

For a single sample, SD of the sample means is undefined.

SEM = SD/sqrt(n) means SEM is undefined if SD is undefined.

For multiple samples of size 1

SEM = SD/1 = SD

Simple math. Simple explanation.

Why you continue to say that the SEM is *not* an estimator for the SD of the sample means is beyond me. EVERYONE defines the standard error as the standard deviation of the sample means. I’ve given you the references, authoritative references! And they all give the formula SEM = SD/sqrt(n) as an ESTIMATE of that. But the concept of the standard error remains the variability of the sample means as defined by their standard deviation. And the SEM remains a statistical descriptor of the sample means distribution, which is *NOT* a measurement uncertainty.

Reply to  Bellman
July 4, 2026 8:16 am

This conversation is circling the drain even faster than usual. Do you want to use equation 10 to calculate the measurement uncertainty of a mean, or do you not?”

EQ 10 is *NOT* appropriate for calculating the variability of the statistical descriptor known as the SEM – your “uncertainty of the mean”. Eq 10 is for MEASUREMENT uncertainty, not SAMPLING uncertainty.

Your formula Σu(x_i)/sqrt(n) is the estimator for the SEM, i.e. sampling uncertainty as represented by the variation in the means of different samples. It doesn’t exist for when you have only one sample since there is no standard deviation of the sample means.

 You then claim it doesn’t count as the mean function isn’t really a function, but a statistic, so can’t have a measurement uncertainty.”

It is *NOT* an estimator of measurement uncertainty. Your formula is an estimator for a statistical descriptor determined from multiple samples which form a distribution of sample means. The distribution of sample means is not a general formula for measurement uncertainty so Eq 10 simply doesn’t apply.

q = Σx_i/n is a STATISTICAL DESCRIPTOR, it is *NOT* a measurement. Eq 10 is for MEASUREMENTS. Measurements are physical descriptors of actual objects. Their average is a statistical descriptor for the distribution of those physical descriptors, their average is *NOT* a replacement for the measurement data itself.

I point out that usually the point of taking a mean is as a sample, in order to estimate the mean of a population, and a better estimate of the uncertainty is the SEM, and you complain that is not measurement uncertainty.”

This is just conceptually wrong. You are trying to substitute a statistical descriptor of the distribution of the measurements for the measurements themselves. Again, the SEM is the standard deviation of the sample means, “meanS” as in plural. The SEM is not even defined for a single sample. It is a metric for SAMPLING uncertainty, not for measurement uncertainty.

I am not complaining. I am trying to explain to you the difference between statistical descriptors of the shape of a distribution and the physical descriptors represented by the measurements themselves.

When it comes to measurements, the mean is only a “best estimate” of the value of the measurand. There is no need for estimating that value beyond what the measurement uncertainty provides actual knowledge of. You can’t specify the mean beyond what the measurement uncertainty provides for.

The GUM says:
——————–
4.2.1 In most cases, the best available estimate of the expectation or expected value μ_q of a quantity q that varies randomly [a random variable (C.2.2)], and for which n independent observations q_k have been obtained under the same conditions of measurement
———————

This includes the unstated assumption that this kind of measurement process will always produce a Gaussian distribution. It simply does not apply in the general case of averaging single measurements of multiple things using different instruments under different environments. Yet climate science *always* just carries over the assumption of a special case into the general case in order to “simplify”. It’s actually not even always the case for “n” observations under the same conditions of measurement since systematic effects can make the mean inaccurate even if the distribution of measurements is Gaussian. It’s actually just one more instance of the garbage assumption that measurement uncertainty is always random, Gaussian, and cancels. It’s why measurement uncertainty budgets exist! They allow a better determination of whether or not the mean is actually the “best estimate” or not.

And all this confusion because you just don’t want to accept the inevitable conclusion that the larger your sample the less uncertainty there is”

There you go again with using the term “uncertainty” without specifying *what* uncertainty you are speaking of – sampling uncertainty or measurement uncertainty.

It is a TRUISM that measurement uncertainty does *NOT* decrease with more measurements, only sampling uncertainty decreases. And they are not the same. The standard deviation of the results of a measurement process does *NOT* reduce with more measurements. At best the SD will remain constant with more measurements. The more general case is that the SD will increase with more measurements. It’s why Bevington discusses that it can be counterproductive to increase sample size beyond a certaint point since even the SEM can go up because of increased SD.

You remain WILLFULLY IGNORANT of metrology concepts, the absolute worst kind of ignorance.

Reply to  Tim Gorman
July 4, 2026 8:39 am

“EQ 10 is *NOT* appropriate for calculating the variability of the statistical descriptor known as the SEM ”

Try reading what I say. I am not talking about the SEM. At this point it’s not worth answering your rants. You are just being willfully dishonest.

Reply to  Bellman
July 4, 2026 9:39 am

If you keep getting confused between combined measurement uncertainty as calculated by equation 10, and the SEm, it’s because the math involving random variables is the same.

Add any number of random variables and their variance add. If all the variables have the same variance then this becomes N times the variance.

Var(X1 + X2 + … + XN) = N(Var(X))

Where var(X) is the common variance.

And taking the square root we have

SD(X1 + X2 + … + XN) = √N(SD(X))

Then for an average you have the same, but have to divide by N².

Var((X1 + X2 + … + XN)/N) = (Var(X))/N

And

SD((X1 + X2 + … + XN)/N)= (SD(X))/√N

That’s the common equation. The difference is in what random variables you are talking about.

For a SEM the individual variables are based on taking one value from the population. In that case the SD is the standard deviation of the population.

For the measurement uncertainty in equation 10 we are only interested in the random variable representing the uncertainty and so the standard deviation is the standard uncertainty for each measurement. Hence,

u(x) / √N.

And of course all this is assuming I’d random variables so that the SD is the same for each. The more general result for different uncertainties is

Σu(xi)/n

Reply to  Bellman
July 4, 2026 11:54 am

If you keep getting confused between combined measurement uncertainty as calculated by equation 10, and the SEm, it’s because the math involving random variables is the same.”

I am not confused at all. Measurement uncertainty is variance of the data. The SEM is the variance of the sampling distribution. TWO ENTIRELY DIFFERENT DATA SETS.

One can’t be substituted for the other! It’s just that simple.

“Where var(X) is the common variance.”

Measurement uncertainties are *NOT* common when measuring different things using different instruments.

In addition, for the SEM the variance is *NOT* X1, X2, …., XN measurements, it is the variance of the sample means which do *NOT* have to be the same as the variance of the population. The sample means generate a sampling distribution that is DIFFERENT from the population distribution. The CLN and LLN, if their restrictions are met, will have the sampling distribution tend toward Gaussian while the population distribution is skewed. The variance of the sampling distribution cannot be substituted for the variance of the population. That’s why small numbers of samples generate such a large uncertainty in the SEM value.

For a SEM the individual variables are based on taking one value from the population. In that case the SD is the standard deviation of the population.”

This makes no sense. One value from the population doesn’t generate a standard deviation. One *sample* from the population doesn’t generate a sampling distribution with a standard deviation. In each case the SEM is not even defined since there is no standard deviation.

You seem to be fixated on the ESTIMATOR for the SEM always being appropriate. It isn’t. It doesn’t work at all for one sample. For two samples the uncertainty in the SEM is 20%, a huge value. For ten samples the SEM is 3% underestimated.

The SEM equation SEM = SD/sqrt(n) is a SHORTCUT with error. The actual standard error *IS* the standard deviation of the sample means distribution. If you focus on that definition perhaps it will be more clear. With one sample there is no sample means distribution and , therefore, no standard deviation, the standard error doesn’t apply, it is undefined.

You need at least 30 samples and preferably more to get a reasonably accurate standard error. When climate science has at least 30 independent measuring systems set up creating 30 samples they will have a sampling distribution whose standard deviation is pretty accurate. It will *still not be* the measurement uncertainty, it will only make the mean more precisely located.

(note: in your equations the SD(X) is only used in the SEM ESTIMATOR and is only the SD of a single sample. You yourself said the temperature data sets are a single sample. A single sample does not generate a sampling distribution from which an SEM can be found. The SEM = SD/sqrt(n) just simply doesn’t apply, it is undefined. You can’t fix that with a shortcut equation being used as an estimator. )

Reply to  Tim Gorman
July 4, 2026 12:31 pm

I am not confused at all. Measurement uncertainty is variance of the data.

According to bellman, it is something entirely different:

“uncertainty as defined by the deviation of possible error”

He apparently believes this is a rational idea.

Reply to  karlomonte
July 4, 2026 1:54 pm

Let alone understand that the deviation need not be Gaussian!

Reply to  Tim Gorman
July 4, 2026 4:48 pm

I am not confused at all. Measurement uncertainty is variance of the data.

You keep confusing what data you are talking about. Measurement uncertainty is the standard deviation of measurements of the same thing. That’s on the assumption that the errors vary randomly about a true value, and by extension that these are the range of values it’s reasonable to attribute to the measurand.

When you average different things the standard deviation of those things is not the measurement uncertainty. Things vary in size because they vary in size, not because of measurement uncertainty.

One can’t be substituted for the other!

So why do you keep doing that. When I say you can use equation 10 to get the combined measurement uncertainty of a mean of N inputs, I am treating the mean as an exact mean of those inputs, not as a random sample from a population.

Measurement uncertainties are *NOT* common when measuring different things using different instruments.

Which why I gave the general result at the end.

In addition, for the SEM the variance is *NOT* X1, X2, …., XN measurements,

It’s so difficult to explain things to someone who is incapable of keeping track.

the variance of the sample means which do *NOT* have to be the same as the variance of the population.

The SEM is not the variance – it’s the standard deviation. And I’ve no idea what other point you think you are making. The SEM is not the variance of the population, nor is it the standard deviation of the population.

If you think the sampling distribution is supposed to be the same as the population distribution, I’m not surprised you are so confused.

This makes no sense.

Obviously not to you, as you keep demonstrating.

One value from the population doesn’t generate a standard deviation.

Read what I said. Each individual variable is based on one item chosen at random from the population. Nobody said that one value has to generate a standard deviation. The standard deviation for that variable is the population standard deviation.

One *sample* from the population doesn’t generate a sampling distribution with a standard deviation.

If you don’t understand what a random variable is, maybe you should just avoid getting into these conversations.

You seem to be fixated on the ESTIMATOR for the SEM always being appropriate

Stop putting words into my mouth. I’ve never said it’s always appropriate. It’s an estimator based on a lot of assumptions – in particular that your all your variables are iid.

It doesn’t work at all for one sample.

Maybe I should just accept that you are beyond redemption. I keep explaining why this is not true. You can find any number of simple explanations of how to calculate a SEM from a single sample. It’s one of the most fundamental exercises of simple statistics. Yet, you will never except it because you refuse to accept that the sampling distribution is an abstract concept describing random variables and not something that actually exits in the form of an infinite number of actual random samples.

You need at least 30 samples and preferably more to get a reasonably accurate standard error.

Which is a good illustration of why you would not actually want to take multiple samples in order to estimate the SEM. Ideally you want a sample size of at least 30, so you are arguing for taking 900 measurements, just to estimate how uncertain you are about a sample of size 30.

When climate science has at least 30 independent measuring systems set up creating 30 samples they will have a sampling distribution whose standard deviation is pretty accurate.

They have rather more than 30 – but you still can;t ignore they are not a random sample.

It will *still not be* the measurement uncertainty

It is not the measurement uncertainty – unless you want to treat the population mean as a measurand, which you keep denying, or you are measuring the same thing multiple times, as in the GUM’s Experimental standard deviation of the sample mean.

A single sample does not generate a sampling distribution from which an SEM can be found.

Well, there are techniques that allow you do thing like that. E.g. bootstrapping.

Reply to  Bellman
July 5, 2026 7:09 pm

Measurement uncertainty is the standard deviation of measurements of the same thing. That’s on the assumption that the errors vary randomly about a true value, and by extension that these are the range of values it’s reasonable to attribute to the measurand.”

No kidding! What in Pete’s name do you think we’ve been telling you?

It is *NOT* the standard deviation of the sample means as you claim!

When you average different things the standard deviation of those things is not the measurement uncertainty. Things vary in size because they vary in size, not because of measurement uncertainty.”

You are in the bottle again, aren’t you?

  1. SIZE IS THE STATED ESTIMATED VALUE.
  2. MEASUREMENT UNCERTAINTY IS THE DISPERSION OF REASONABLE VALUES THAT COULD BE ASSIGNED TO THE MEASURAND!

1 AND 2 are not the same thing!

The estimated value of the mass for each of two rocks can be added together to get a total, i.e. the sum describes a larger system. The variances associated with the estimated values give the dispersion of the values that can reasonably assigned as an estimated value. Those variances ADD when describing the properties of the larger system.

Are we REALLY going to have to start over from scratch trying to explain to you how measurement uncertainty works?

If you don’t understand what a random variable is, maybe you should just avoid getting into these conversations.”

One sample pulled from a population has nothing to do with what a random variable is. Even if the measurement uncertainty of each sample data point was 0 (zero), i.e. perfectly accurate out to infinity, that one sample would *NOT* generate a distribution of sample means. A single mean from a single sample is *NOT* a distribution of valueS (plural). Meaning there *IS* not an SEM. There is no SD/sqrt(n). It’s undefined for a single sample!

*YOU ARE THE ONE THAT SAID THE TEMPERATURE DFATA BASE IS A SINGLE SAMPLE.”

Are you now going to run away from that? Are you going to say that the temperature data base is a collection of samples of size 1?

You can have it one way or the other but not both. Pick one and stick with it.

Which is a good illustration of why you would not actually want to take multiple samples in order to estimate the SEM. Ideally you want a sample size of at least 30, so you are arguing for taking 900 measurements, just to estimate how uncertain you are about a sample of size 30.”

Are you just beginning to see the problem with climate science and its data?

I don’t think you actually understand how to calculate the uncertainty associated with the NUMBER OF SAMPLES which is different than the uncertainty associated with the sample size!

Two samples have an in-built uncertainty of 20% when it comes to locating the mean. And that uncertainty does *NOT* depend on sample size. It is separate from the uncertainty generated from sample size.

They have rather more than 30 – but you still can;t ignore they are not a random sample.”

So you think UAH has 30 separate, independent data bases of temperature measurements? Just how many satellites do you think they have available?

The number of samples and the sample size are two different things. And each contribute their own measurement uncertainty.

With one sample you can’t even know what the sampling uncertainty of the mean is since it is undefined! And it doesn’t matter how many data points you have in the sample!

=================
I am tired of trying to teach you even the basics of metrology in this thread.

You won’t listen. You won’t read ANY metrology textbooks or the GUM for meaning and context. And it’s not obvious that you have any actual training in statistics since you can’t even get the basics of sampling right.

At this point I am unsubscribing from this article. You are own your own. Post all the garbage memes you want. I won’t see any of it!

=============

Reply to  Tim Gorman
July 5, 2026 7:56 pm

No kidding! What in Pete’s name do you think we’ve been telling you?

You were disputing my suggestion that the GUM was trying to move away from a frequentist probability paradigm, in favour of a Bayesian one.

Not sure if it’s work going through the rest of your rant at this moment.

Reply to  Bellman
July 5, 2026 9:27 pm

“Not sure if it’s work going through the rest of your rant at this moment.”

Not much is.

Reply to  Bellman
July 4, 2026 2:49 pm

“Σu(xi)/n”

Correctivo that should be

√(Σu(xi))/n

Reply to  Bellman
July 4, 2026 2:16 pm

Try reading what I say. I am not talking about the SEM”

Of course you are talking about the SEM.

SEM = SD/sqrt(n). EXACTLY what your formula is.

SD/sqrt(n) doesn’t equal both the SD of the measurement values (i.e. the measurement uncertainty) AND the SD of the sampling distribution.

And it is the SD of the measurement values that determines the measurement uncertainty. It’s the NUMERATOR in SD/sqrt(n) that determines measurement uncertainty.

Reply to  Tim Gorman
July 4, 2026 2:48 pm

“Of course you are talking about the SEM.

SEM = SD/sqrt(n). EXACTLY what your formula is”

Sigh, you are just confusing yourself again. The SD in the SEM equation is the standard deviation of the sample. The standard deviation in equation 10 is the measurement uncertainty of each individual measurement.

And if you are going to call an equation “your formula” could you at least quote it so we don’t keep talking past each other.

Reply to  Bellman
July 5, 2026 7:33 am

If you have a measurement uncertainty interval for each stated value in the distribution then you propagate those measurement uncertainties to find the combined measurement uncertainty interval using Eq 10.

If you only have a stated value then that represents experimental observations used to determine the measurement uncertainty of the object involved. The measurement uncertainty is the standard deviation of the stated values, i.e. a Type A evaluation.

What you seem to be saying is that you should SCALE the measurement uncertainties of the stated values in order to find the combined measurement uncertainty. That makes *NO* sense at all. Why would you scale the measurement uncertainties? Especially by (1/n)?

If you have observations [x1 +/- u1, x2 +/- u2, …, xn +/- un ] you are saying the average of the measurement is q_avg = Σx_i / n. This is the BEST ESTIMATE for the value of the property be measured. It has nothing to do with the measurement uncertainty of the data.

Then the question becomes “how do you propagate the measurement uncertainties associated with x_i”?

The proper method in the GUM is to use Eq 10 to add the measurement uncertainties. The measurement stated values do *NOT* have a factor of (1/n) associated with them. So the ∂f/∂x_i is *NOT* (1/n). It is just “1”.

You seem to be wanting to *ADD* a division by n to the measurements.

q = Σ(x_i)/n to find the average value, the *BEST ESTIMATE*,

and then say u(q) = Σu(x_i)/n as well.

It just doesn’t follow.

This is the AVERAGE of the stated values. It is the best estimate of the measured value for the property being measured. It has nothing to do with measurement uncertainty. The measurement uncertainty of the data set is Σu(x_i). (I am omitting the step of actually using variance) If you want to use RSS then u(q) = sqrt[Σ u^2(x_i) ]

Nor is the measurement uncertainty of the average related to the standard deviation of the stated values. It is the sum of the individual measurement uncertainties. Thus dividing the SD of the stated values by sqrt(n) is not indicated in order to get an SEM. That only applies in the case of a Type A evaluation where the individual measurements consist only of a stated value and do not include individual measurement uncertainties.

Bottom line:

-1. If you have measurements of x_i +/- u_i then

  • a. Σx_i/n is the best estimate of the value of the measurand
  • b. Σu(x_i) is the measurement uncertainty of that best value

-2. If you have measurements of just x_i then

  • a. Σx_i/n is the best estimate of the value of the measurand
  • b. SD(x_i) is the measurement uncertainty .

No. 1 is many times is used with a Type B measurement uncertainty. No. 2 is typically considered to be a Type A measurement uncertainty.

You are confusing what the GUM calls “Experimental standard deviation of the mean” [ s^2(q_k) /sqrt(n)] with the “Experimental standard deviation” [s^2(q_k)]

GUM:
————————
3.3.5 The estimated variance u^2 characterizing an uncertainty component obtained from a Type A evaluation is calculated from series of repeated observations and is the familiar statistically estimated variance s^2 (see 4.2). The estimated standard deviation (C.2.12, C.2.21, C.3.3) u, the positive square root of u^2, is thus u = s and for convenience is sometimes called a Type A standard uncertainty. For an uncertainty component obtained from a Type B evaluation, the estimated variance u^2 is evaluated using available knowledge (see 4.3), and the estimated standard deviation u is sometimes called a Type B standard uncertainty.
———————–(bolding mine, tpg)

In neither case is the Experimental Standard deviation of the mean used for the measurement uncertainty of a measurand.

Again, the mean of the stated values is the BEST ESTIMATE for the value of the measurand. The measurement uncertainty of that BEST ESTIMATE for the value of the measurand is the square root of the variance s^2 of the input data.

The experimental standard deviation of the mean is *NOT* used for the uncertainty of the measurand.

  1. If the input quantities have measurement uncertainties specified they add using Eq 10.
  2. If the input quantities do not have measurement uncertainty specified then you use the variance of the input data.

note: see the bolded portion of Section 3.3.5.

Reply to  Tim Gorman
July 5, 2026 8:10 am

“What you seem to be saying is that you should SCALE the measurement uncertainties of the stated values in order to find the combined measurement uncertainty.”

Yes, when the function involves scaling it scales the relative uncertainties. That’s the point if equation 10. The partial derivative is the scaling factor. And it’s stated explicitly when you use the specific rules, as say given by Taylor. I’ve quoted the exact equation in Taylor enough times. If q = Bx, then u(q) = Bu(x).

“That makes *NO* sense at all.”

Obviously not to you. To me it makes perfect sense, and the maths agrees with my sense.

“Why would you scale the measurement uncertainties? Especially by (1/n)?”

Say you take a measurement of 10cm, and the expanded uncertainty interval is ±1cm. That means you think it’s reasonable that that the measurand could be somewhere between 9 and 11cm.

Now take that measurement and multiply it by a scaling factor. Maybe the measurement was a diameter and you want to know the circumference, so you multiply the by π. The best estimate is now 10π = 31.4cm. What happens to the uncertainty interval? The lowest reasonable value for the diameter is 9cm. That translates to a lower bound of 9π = 28.3cm, and the upper bound is 11π = 34.6cm. The uncertainty interval is now ±1π = ±3.1cm.

Reply to  Bellman
July 5, 2026 8:21 am

“The measurement stated values do *NOT* have a factor of (1/n) associated with them. So the ∂f/∂x_i is *NOT* (1/n). It is just “1”. ”

You just keep switching terms. It is the stated values that have a factor if 1/n, it’s the function you are using on the inputs. f(x1, x2… xn) = (x1 + x2 + … + xn) / n = x1/n + x2/n + … + xn/n.

For xi the function is xi + C, for some constant C.

∂f/∂x_i = 1/n

It’s the most basic calculus, multiplication by a constant.

Reply to  Bellman
July 5, 2026 8:23 am

“q = Σ(x_i)/n to find the average value, the *BEST ESTIMATE*,

and then say u(q) = Σu(x_i)/n as well.

It just doesn’t follow.”

Yes, that was a mistake on my part. It should be

u(q) = √[Σu(x_i)²]/n

Reply to  Bellman
July 5, 2026 8:34 am

“Again, the mean of the stated values is the BEST ESTIMATE for the value of the measurand. The measurement uncertainty of that BEST ESTIMATE for the value of the measurand is the square root of the variance s^2 of the input data. ”

You keep confusing two things.

1. Measuring the same thing multiple times to get the best estimate if a single thing.
2. Applying an average to a set of different things and using equation 10 to get the measurement uncertainty of that average.

The first is equivalent to sampling, and the uncertainty of the mean is the standard deviation divided by √N. The second is only applicable when you only want to know the measurement uncertainty of that average, rather than treating your values as a sample.

I don’t know how often you would actually want an exact average like that, but it’s a useful exercise to illustrate that in abybaveragevtge measurement uncertainty decreases with the number of measurements.

Reply to  Bellman
July 5, 2026 8:49 am

“In neither case is the Experimental Standard deviation of the mean used for the measurement uncertainty of a measurand.”

You need to look at the examples in the GUM. E.g.

H.2 Simultaneous resistance and reactance measurement

Reply to  Bellman
July 5, 2026 7:50 am

The experimental standard deviation of the mean is *ONLY* used for measurement uncertainty when the repeatable observations form a Gaussian distribution AND it can be assumed that *NO* systematic uncertainty is involved. This typically *ONLY* occurs in situations where you have a controlled environment, accurately calibrated instruments, and a single object as a measurand. I have *NEVER* seen these conditions met in unmanned field measurement stations.

The fact that you think the experimental standard deviation of the mean associated with temperature measurements made in the field is the measurement uncertainty of the BEST ESTIMATE of the value of the property being measured tells me that 1. you have never actually done anything in the field and, 2. you’ve NEVER had to develop a measurement uncertainty budget for *ANYTHING*.

There is a reason why so many teaching examples for measurement procedures include the assumption of no systematic uncertainty. It’s not that there isn’t any systematic uncertainty but that it is a “fog” factor for teaching the basics. That assumption very, VERY seldom applies in the field. It’s only blackboard statisticians and mathematicians that think systematic uncertainty can be ignored using the meme: “all measurement uncertainty is random, Gaussian, and cancels”.

Reply to  bigoilbob
July 1, 2026 4:39 pm

Not if the systemic errors were from bad sites, urban encroachment .

Then the trend would tend to INCREASE.

Reply to  bigoilbob
July 1, 2026 12:50 pm

Doesn’t solve the basic problem that the statement is nonsensical,

Sparta Nova 4
Reply to  bigoilbob
July 1, 2026 12:54 pm

Upvote for admitting an error.

Reply to  bigoilbob
July 1, 2026 12:46 pm

The “measurement uncertainty” for the individual measurements would tend to diminish with the more of them used to calculate the resulting trends. 

Well this is certainly nonsense, uncertainty increases, always.

Why is the phrase in quotes?

Reply to  Bellman
July 1, 2026 6:44 am

Of course all of this is ignoring uncertainties, and for daily CET the uncertainties will be large owing to the small sample size. None of these records will be statistically significant, but that’s not really the point. Records are just a bit of fun, they may indicate a general worming but you really need to look at the long term trends in monthly or annual temperatures.

Reply to  Bellman
July 1, 2026 7:55 am

daily CET the uncertainties will be large owing to the small sample size.

Sample size has nothing to do with single measurements. The uncertainty of a single measurement is determined from an uncertainty budget whose components might be determined by statistical analysis (Type A) or other means (Type B). The additional uncertainty due to siting is a Type B uncertainty and is appropriate to include.

Reply to  Jim Gorman
July 1, 2026 3:10 pm

Sample size has nothing to do with single measurements.

It’s not a single measurement. It’s three measurements.

Reply to  Bellman
July 2, 2026 4:33 am

Exactly what ‘three’ measurements are you referencing?

You do realize that a single measurement is a single measurement under repeatable conditions as defined in GUM B.2.15, right?

Reply to  Jim Gorman
July 2, 2026 6:19 am

The three stations currently used to calculate the daily CET of course.

Reply to  Bellman
July 1, 2026 4:58 pm

Just a bit of fun, but as people keep asking about uncertainty, I though I’d run a Monte Carlo simulation to see what the probability of each day being the hottest was, assuming a random standard uncertainty 0.7°C for daily maximums. (I think that may be a quoted uncertainty somewhere, but I’ll have to try to find the source.)

For the warmest June day (Max temperature), the probabilities (as percentages) are estimated as

1 26-Jun-2026 76.2
2 25-Jun-2026 15.8
3 23-Jun-2026 6.5
4 24-Jun-2026 0.7
5 29-Jun-2019 0.3
6 28-Jun-1976 0.2
7 19-Jun-2005 0.1
8 29-Jun-1976 0.1
Very likely that the warmest June day for CET was from 2026, though which is more difficult to say.

For all year warmest day,

1 19-Jul-2022 99.3
2 18-Jul-2022 0.6
3 25-Jul-2019 0.1
4 03-Aug-1990 0.003
5 01-Jul-2015 0.002
6 31-Jul-2020 0.001
Not surprisingly, it’s almost certainly the 19th of July 2022, and if not that, it was the probably the day before. And from my own experience, if the 18th was actually warmer than the 19th, I’d demand a recount.

Reply to  Bellman
July 1, 2026 8:47 am

CET Mean temperature was the 3rd warmest for June, which is surprising given how cold the first half of the month was.

1 1846 18.2
2 1676 18.0
3 2026 17.4
4 1826 17.3
5 1822 17.1
=6 2023 17.0
=6 2025 17.0
=8 1762 16.9
=8 1798 16.9
=8 1976 16.9

Obviously there is still much uncertainty with these figures, especially the 1676 one. It’s noticeable how many pre-19th century records there are.

Max temperatures, which only go back to 1880, which just 3rd warmest, still cooler than 1976.

1 2023 22.6
2 1976 22.5
3 2026 22.1
4 2025 22.0
5 1940 21.9
6 1970 21.7
7 2018 21.5
8 1899 21.3
9 1960 21.2
10 1957 21.1

Min Temperatures, also only going back to 1880 was the warmest June

1 2026 12.6
2 2025 12.0
3 2017 11.8
4 1950 11.5
=5 1896 11.4
=5 2023 11.4
=7 1976 11.3
=7 1982 11.3
9 2016 11.2
=10 1947 11.1
=10 1970 11.1
=10 2005 11.1
=10 2007 11.1

Reply to  Bellman
July 1, 2026 3:07 pm

Sorry about the formatting. I keep using the code block formatting, and it looks fine, but when I come back later it’s been turned into plane text.

Reply to  Bellman
July 1, 2026 1:28 pm

CET is heavily affected by urban warming and bad sites.

UK-3-x-3-new-1749654188.1402
paul courtney
Reply to  Bellman
July 1, 2026 2:33 pm

Anyone else here recall CliScis claiming every year since 2016 has been getting hot-hot-hotter for the globe? Not in these June numbers, though.
How can anybody create that list and not see how scattered the “global warming” (scare quotes appropriate, Mr. Hunt) is going back decades?

Reply to  paul courtney
July 1, 2026 2:45 pm

UK TMAX in June – a local measure of daytime maximum temperatures.

Global June avg – a worldwide average that includes not only summer across the rest of the much larger NH but also winter across the SH.

Very different.

Reply to  paul courtney
July 1, 2026 3:09 pm

Anyone else here recall CliScis claiming every year since 2016 has been getting hot-hot-hotter for the globe?

No. Must be a figment of your imagination.

How can anybody create that list and not see how scattered the “global warming”

The Central England Temperature set is not “global”. There’s a clue in the name.

Reply to  Bellman
July 1, 2026 3:13 pm

Lol, how do these people think they’ve outsmarted the entire field of climate science?

paul courtney
Reply to  Eldrosion
July 2, 2026 5:09 am

Mr. ‘nother fakename: We think that we will read your comments and expose the errors underlying your moral cause. Bellman thinks, because he was blind to it, it was imagined. His blindness exposed, I move on.

Mr.
July 1, 2026 5:53 am

So these Class 5 sites can have error factors of 5C up or down?

For all the use they are, we may as well treat them as suppositories and stick them up our arse.

July 1, 2026 6:09 am

I’m on the South Coast of UK and last Thursday I recorded 37.4 C for a few minutes and 37C for several hours , the digital thermometer is in the shade next to the house , which backs onto woodland . It must be remembered that we haven’t seen temperatures this high since 1976 , yes we’ve regularly over the years had temperatures in the 30+C range but not quite this high . The digital thermometer normally agrees with the forecasted temperatures and whilst we were out driving around the temperature readout in the car was showing 37.5C . But as the old English saying goes “ one swallow doesn’t make it summer “ I think one isolated high temperature in 50 years doesn’t prove global warming ( a swallow is an African migratory bird seen in Uk during summer ) . In days of this high temperature weather returned to normal Uk summer temps .

IMG_3433
Reply to  Northern Bear
July 1, 2026 6:12 am

What is the uncertainty of your device?

strativarius
Reply to  Jim Gorman
July 1, 2026 6:40 am

Nobody checks calibration these days.

MrGrimNasty
Reply to  Northern Bear
July 1, 2026 7:46 am

Yes nothing to see, record heat May, then June, another heatwave coming, by far warmest year in 400 odd years so far. All down to dodgy thermometers/sites and lack of declaring uncertainties. Risible.

Sparta Nova 4
Reply to  MrGrimNasty
July 1, 2026 12:57 pm

400 odd years.

Wow.

Humor – a difficult concept.
— Lt. Saavik

Reply to  Sparta Nova 4
July 1, 2026 1:37 pm

Seems to have forgotten that the LIA was the COLDEST period in some 10,000 years.

cartoss
July 1, 2026 6:10 am

Anyone ever seen Stokes, Banton and Nail in the same room at the same time? Or any pair of them? Thought not. They are the same animal and I claim my $50 dollars prize!

strativarius
Reply to  cartoss
July 1, 2026 6:41 am

The Unholy Triumvirate – unmasked.

Reply to  cartoss
July 1, 2026 7:20 am

Nick’s from down under, whereas I am from out West.

I can assure you that we are unrelated!

paul courtney
Reply to  Jim Hunt
July 2, 2026 5:11 am

Mr. Hunt: Did he say your name? Why’d you think he included you in the troll list??

Reply to  paul courtney
July 2, 2026 8:23 am

I commented “supportively” on Nick’s first comment above.

It seemed to me inevitable that I would be added to “the troll list” you refer to in due course.

It seems I was mistaken!

Reply to  cartoss
July 1, 2026 12:55 pm

The trendologists are all up-in-arms defending the honor of the UK MO.

July 1, 2026 8:54 am

I notice that our resident climate alarmists quite forgot to mention what caused this unusual heat.
Let me remind them.
There was an Omega block over most of Europe.
This crashed wind power in Europe.
In Britain we went down at times to 3GW of wind power.
As a result, Britain had to burn more fossil fuels to keep the lights on.
Can we expect more crashes of renewable energy at just the times that we want to rely on them?

Reply to  stevencarr
July 1, 2026 9:23 am

There was an Omega block over most of Europe.”

Not the whole story.

You’re forgetting the role of dry soils reducing evaporative cooling (a direct consequence of global warming) which primes the land for hotter temperatures.

Mediterranean climates are becoming increasingly North African in character…

Global warming is so scary.

Reply to  Eldrosion
July 1, 2026 9:29 am

Dry soils stopped the wind from blowing?
Dry soil meant we had to burn more gas , because solar power was reduced on one of the sunniest days in England very close to the solstice where there is 17 hours of sunlight?

Reply to  stevencarr
July 1, 2026 9:36 am

Please stop denying the seriousness of human caused climate change.

Reply to  Eldrosion
July 1, 2026 9:44 am

It’s serious is it?
There are going to be more times when we need renewables to step up to meet extra demand?
If it is so serious, then why do you deny that renewables failed at the very time we needed them most, and Britain had to step on the gas?

Are people dying of heatstroke because wind of 3GW is not enough to power air conditioning serious enough for you?

Reply to  stevencarr
July 1, 2026 11:24 am

If it is so serious, then why do you deny that renewables failed at the very time we needed them most, and Britain had to step on the gas?”

I don’t know. The same reason you do?

Reply to  Eldrosion
July 1, 2026 3:41 pm

Urbanisation, airport expansion and site degradation are the cause of a large proportion of warming in the fabricated surface temperature pseudo-data.

Humans have not changed the “global climate” in any appreciable or measurable way.

The really serious damage to society is being done by things like Net-Zero agendas.

Reply to  bnice2000
July 1, 2026 5:03 pm

Urbanisation, airport expansion and site degradation are the cause of a large proportion of warming in the fabricated surface temperature pseudo-data.”

and yet the UAH LT grid cell covering the UK also shows warming:

comment image

The really serious damage to society is being done by things like Net-Zero agendas.”

I’ll tell you where the real energy crisis is: lower income households across Europe are increasingly being forced to choose between keeping cool during heat waves and keeping their electricity bills affordable.

Reply to  Eldrosion
July 1, 2026 5:43 pm

1979 was the coldest period since 1900, down from the peak in 1940.
That is were alarmist stooges love to start.

Net-zero is causing high and erratic electricity prices. If Germany still had its nuclear power stations running, and had built new coal and gas fired stations using their own coal and gas, instead of wasting money on unreliable erratic pseudo supply of wind and solar.. none of this would be happening. European grid is in a total mess because of anti-CO2 agendas.

In UAH, none of the warming is caused by humans, just solar charged El Nino events.
This becomes very obvious when you look at the periods between El Nino events.
No warming in earlier periods….. cooling from 2017 to beginning of 2023.4 El Nino

UAH-global-with-near-zero-trend-sections
Reply to  bnice2000
July 1, 2026 7:47 pm

In UAH, none of the warming is caused by humans, just solar charged El Nino events.
This becomes very obvious when you look at the periods between El Nino events.
No warming in earlier periods….. cooling from 2017 to beginning of 2023.4 El Nino”

#1) That isn’t true. You’re simply cherry-picking time intervals to support your argument.

#2) This is a deflection from your original claim that the UK surface station record is corrupted and that this accounts for most of the observed warming. You shifted the discussion because LT warming over the UK significantly undermines that hypothesis.

Reply to  Eldrosion
July 1, 2026 8:00 pm

If you don’t now when El Nino events were.. No-one can help you. !

Yes, UK temperature sites are totally corrupted. This adds on top of large urban effects, and and natural atmospheric warming by solar charged El Nino events and increase sunshine over the UK

A very large proportion of Met-Office sites are class 4 or 5 and thus totally unfit for “climate” uses.

uk-temp-stations-2
Reply to  bnice2000
July 2, 2026 4:47 pm

Yes, UK temperature sites are totally corrupted. This adds on top of large urban effects”

No. You also need to explain why the lower troposphere is also warming significantly. Otherwise, your claim can be dismissed.

Reply to  Eldrosion
July 2, 2026 5:57 am

That isn’t true. You’re simply cherry-picking time intervals to support your argument.

This is showing that there are “step increases” that coincide with El Nino’s. In time series, these are known as shocks. If shocks are not adequately dealt with to insure proper decay they can easily remain for long periods of time, even permanently. The intervals between shocks that have little change is evidence of shocks not decaying.

Ask your preferred AI to define a time series shock and to discuss how long they can persist. I think you’ll find that they can decay quickly on their own to permanent depending on the treatment of the data.

0perator
Reply to  Eldrosion
July 1, 2026 9:57 pm

I’ll tell you where the real energy crisis is: lower income households across Europe are increasingly being forced to choose between keeping cool during heat waves and keeping their electricity bills affordable.

Yeah, because they’re retarded socialists like you.

Reply to  0perator
July 1, 2026 10:07 pm

Thank you.

MrGrimNasty
July 1, 2026 9:55 am

To those who think the warming is a result of MO fraud and general dodgy stuff, please answer this.

How have they manage to maintain a steady warming of over 1C in the annual mean CET in merely the last 15 years?

How have they managed to make 2026 almost another 0.7C warmer than the previously warmest year, so far this year?

Or are we just actually experiencing one of the fastest warming periods in the best part of 400 years and currently warmer than any other time in that record. And 2026 is an extraordinarily warm year on top?

There’s only one credible answer.

Reply to  MrGrimNasty
July 1, 2026 10:03 am

How can temperatures be increasing when Britain has a Net Zero program resulting in huge amounts of renewable power and a reduction in our CO2 emissions by over 50%?

Sparta Nova 4
Reply to  MrGrimNasty
July 1, 2026 1:01 pm

400 years.

Wow

Humor – a difficult concept
— Lt. Saavik

Reply to  MrGrimNasty
July 1, 2026 3:28 pm

There’s only one credible answer.”

Yep.. Met-Office took over control of CET !

0perator
July 1, 2026 10:22 am

They lie. We know they lie. They know we know they lie. And these feeble minded mouthbreathing propagandists still run around screaming “the sky is falling!” Insufferable cockwombles pushing their communism under the guise about caring about the planet and people.

July 1, 2026 11:59 am

It really is difficult imagining a worse site for measuring temperatures.”

How about this one?

comment image?ve=1&tl=1

Reply to  Phil.
July 1, 2026 1:45 pm

There area HUGE number of really bad sites to chose from in the UK !!!

This one is a wee ripper, overgrown, un-maintained, and metal boxes on metal plates either side of the legs of the screen, creating an enclose oven in hot weather.

Kew-Gardens
Reply to  bnice2000
July 1, 2026 7:38 pm

The one I gave the photo isn’t in the UK!

Reply to  Phil.
July 1, 2026 8:02 pm

Thanks for showing the temperature site corruption isn’t just in the UK. ! 🙂

Reply to  bnice2000
July 3, 2026 6:19 am

Yesterday that site recorded its first 100ºF temperature since July 2012. It holds the WMO ‘Centennial Station’ designation, awarded for maintaining continuous, high-quality weather records for over a century. 

Reply to  Phil.
July 2, 2026 5:44 am

How the h*ll do you ever get an accurate wind measurement from this?

Reply to  Jim Gorman
July 2, 2026 6:27 am

This is what it looked like before the trees grew.

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Bob
July 1, 2026 5:22 pm

Man, I wish I had firing authority for the Met.

George Kaplan
July 1, 2026 10:16 pm

Shouldn’t some Junk 5 sites also offer record (low) temperatures suggesting a coming Ice Age? Or does that data simply get ignored?

July 3, 2026 6:55 am

Here’s a better view of the Lingwood site.
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And its owner who has been operating it for 50 years.

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