# Durable Original Measurement Uncertainty

### Guest Essay by Kip Hansen

Introduction:

Temperature and Water Level (MSL) are two hot topic measurements being widely bandied about and vast sums of money are being invested in research to determine whether, on a global scale, these physical quantities — Global Average Temperature and Global Mean Sea Level — are changing, and if changing, at what magnitude and at what rate. The Global Averages of these ever-changing, continuous variables are being said to be calculated to extremely precise levels — hundredths of a degree for temperature and millimeters for Global Sea Level — and minute changes on those scales are claimed to be significant and important.

In my recent essays on Tide Gauges, the question of the durability of original measurement uncertainty raised its toothy head in the comments section.

Here is the question I will try to resolve in this essay:

If original measurements are made to an accuracy of +/- X (some value in some units), does the uncertainty of the original measurement devolve on any and all averages – to the mean –  of these measurements?

Does taking more measurements to that same degree of accuracy allow one to create more accurate averages or “means”?

My stated position in the essay read as follows:

If each measurement is only accurate to ± 2 cm,  then the monthly mean cannot be MORE accurate than that — it must carry the same range of error/uncertainty as the original measurements from which it is made.   Averaging does not increase accuracy.

It would be an understatement to say that there was a lot of disagreement from some statisticians and those with classical statistics training.

I will not touch on the subject of precision or the precision of means.  There is a good discussion of the subject on the Wiki page: Accuracy and precision .

The subject of concern here is plain vanilla accuracy:  “accuracy of a measurement is the degree of closeness of measurement of a quantity to that quantity’s true value.” [ True value means is the actual real world value — not some cognitive construct of it.)

The general statistician’s viewpoint is summarized in this comment:

“The suggestion that the accuracy of the mean sea level at a location is not improved by taking many readings over an extended period is risible, and betrays a fundamental lack of understanding of physical science.”

I will admit that at one time, fresh from university, I agreed with the StatsFolk.  That is, until I asked a famous statistician this question and was promptly and thoroughly drummed into submission with a series of homework assignments designed to prove to myself that the idea is incorrect in many cases.

### First Example:

Let’s start with a simple example about temperatures.   Temperatures, in the USA, are reported and recorded in whole degrees Fahrenheit.  (Don’t ask why we don’t use the scientific standard.  I don’t know).  These whole Fahrenheit degree records are then machine converted into Celsius (centigrade) degrees to one decimal place, such as 15.6 °C.

This means that each and every temperature between, for example, 72.5 and 71.5 °F is recorded as 72 °F.  (In practice, one or the other of the precisely .5 readings is excluded and the other rounded up or down).  Thus an official report for the temperature at the Battery, NY at 12 noon of “72 °F” means, in the real world, that the temperature, by measurement, was found to lie in the range of 71.5 °F and 72.5 °F — in other words, the recorded figure represents a range 1 degree F wide.

In scientific literature, we might see this in the notation:  72 +/- 0.5 °F.  This then is often misunderstood to be some sort of “confidence interval”, “error bar”, or standard deviation.

It is none of those things in this specific example of temperature measurements.  It is simply a form of shorthand for the actual measurement procedure which is to represent each 1 degree range of temperature as a single integer — when the real world meaning is “some temperature in the range of 0.5 degrees above or below the integer reported”.

Any difference of the actual temperature, above or below the reported integer is not an error.   These deviations are not “random errors” and are not “normally distributed”.

Repeating for emphasis:  The integer reported for the temperature at some place/time is shorthand for a degree-wide range of actual temperatures, which though measured to be different, are reported with the same integer.  Visually:

Even though the practice is to record only whole integer temperatures, in the real world, temperatures do not change in one-degree steps — 72, 73, 74, 72, 71, etc.  Temperature is a continuous variable.  Not  only is temperature a continuous variable, it is a constantly changing variable.  When temperature is measured at 11:00 and at 11:01, one is measuring two different quantities; the measurements are independent of one another.  Further, any and all values in the range shown above are equally likely — Nature does not “prefer” temperatures closer to the whole degree integer value.

[ Note:  In the U.S., whole degree Fahrenheit values are converted to Celsius values rounded to one decimal place –72°F is converted and also recorded as  22.2°C.  Nature does not prefer temperatures closer to tenths of a degree Celsius either. ]

While the current practice is to report an integer to represent the range from integer-plus-half-a-degree to integer-minus-half-a-degree, this practice could have been some other notation just as well.  It might have been just report the integer to represent all temperatures from the integer to the next integer, as in 71 to mean “any temperature from 71 to 72” — the current system of using the midpoint integer is better because the integer reported is centered in the range it represents — this practice, however,  is easily misunderstood when notated 72 +/- 0.5.

Because temperature is a continuous variable,  deviations from the whole integer are not even “deviations” — they are just the portion of the temperature measured in degrees Fahrenheit normally represented by the decimal fraction that would follow the whole degree notation — the  “.4999” part  of 72.4999°F.  These decimal portions are not errors, they are the unreported, unrecorded part of the measurement and because temperature is a continuous variable, must be considered evenly spread across the entire scale — in other words, they are not, not, not “normally distributed random errors”.  They only reason they are uncertain is that even when measured, they have not been recorded.

### So what happens when we now find the mean of these records, which, remember, are short-hand notations of temperature ranges?

Let’s do a basic, grade-school level experiment to find out…

We will find the mean of a whole three temperatures; we will use these recorded temperatures from my living room:

```11:00     71 degrees F

12:00     72 degrees F

13:00     73 degrees F```

As discussed above, each of these recorded temperatures really represent any of the infinitely variable intervening temperatures, however I will make this little boxy chart:

Here we see each hour’s temperature represented as the highest value in the range, the midpoint value of the range (the reported integer), and as the lowest value of the range.  [ Note: Between each box in a column, we must remember that there are an infinite number of fractional values, we just are not showing them at this time. ]   These are then averaged — the mean calculated — left to right:  the three hour’s highest values give a mean of 72.5, the midpoint values give a mean of 72, and the lowest values give a mean of 71.5.

The resultant mean could be written in this form:  72 +/- 0.5 which would be a short-hand notation representing the range from 71.5 to 72.5.

The accuracy of the mean, represented in notation as +/- 0.5, is identical to the original measurement accuracy — they both represent a range of possible values.

Note:  This uncertainty stems not from the actual instrumental accuracy of the original measurement, which is a different issue and must be considered additive to the accuracy discussed here which arises solely from the fact that measured temperatures are recorded as one-degree ranges with the fractional information discarded and lost forever, leaving us with the uncertainty — a lack of knowledge — of what the actual measurement itself was.

Of course, the 11:00 actual temperature might have been 71.5, the 12:00 actual temperature 72, and the 13:00 temperature 72.5.  Or it may have been 70.5, 72, 73.5.

Finding the means kiddy-corner gives us 72 for each corner to corner, and across the midpoints still gives 72.

Any combination of high, mid-, and low, one from each hour, gives a mean that falls between 72.5 and 71.5 — within the range of uncertainty for the mean.

Even for these simplified grids, there are many possible combinations of one value from each column.  The means of any of these combinations falls between the values of 72.5 and 71.5.

There are literally an infinite number of potential values between 72.5 and 71.5 (someone correct me if I am wrong, infinity is a tricky subject) as temperature is a continuous variable.  All possible values for each hourly temperature are just as likely to occur — thus all possible values, and all possible combinations of one value for each hour, must be considered. Taking any one possible value from each hourly reading column and finding the mean of the three gives the same result — all means have a value between 72.5 and 71.5, which represents a range of the same magnitude as the original measurement’s, a range one degree Fahrenheit wide.

The accuracy of the mean is exactly the same as the accuracy for the original measurement — they are both a 1-degree wide range.     It has not been reduced one bit through the averaging process.  It cannot be.

Note: For those who prefer a more technical treatment of this topic should read Clyde Spencer’s “The Meaning and Utility of Averages as it Applies to Climate” and my series “The Laws of Averages”.

### And Tide Gauge Data?

It is clear that the original measurement accuracy’s uncertainty in the  temperature record arises from the procedure of reporting only whole degrees F or degrees C to one decimal place, thus giving us not measurements with a single value, but ranges in their places.

But what about tide gauge data?  Isn’t it a single reported value to millimetric precision, thus different from the above example?

The short answer is NO, but I don’t suppose anyone will let me get away with that.

What are the data collected by Tide Gauges in the United States (and similarly in most other developed nations)?

The Estimated Accuracy is shown as +/- 0.02 m (2 cm) for individual measurements and claimed to be +/- 0.005 m (5 mm) for monthly means. When we look at a data record for the Battery, NY tide gauge we see something like this:

 Date Time Water Level Sigma 9/8/2017 0:00 4.639 0.092 9/8/2017 0:06 4.744 0.085 9/8/2017 0:12 4.833 0.082 9/8/2017 0:18 4.905 0.082 9/8/2017 0:24 4.977 0.18 9/8/2017 0:30 5.039 0.121

Notice that, as the spec sheet says, we have a record every six minutes (1/10th hr), water level is reported in meters to the millimeter level (4.639 m) and the “sigma” is given.  The six-minute figure is calculated as follows:

“181 one-second water level samples centered on each tenth of an hour are averaged, a three standard deviation outlier rejection test applied, the mean and standard deviation are recalculated and reported along with the number of outliers. (3 minute water level average)”

Just to be sure we would understand this procedure, I emailed CO-OPS support [ @ co-ops.userservices@noaa.gov ]:

To clarify what they mean by accuracy, I asked:

When we say spec’d to the accuracy of +/- 2 cm we specifically mean that each measurement is believed to match the actual instantaneous water level outside the stilling well to be within that +/- 2 cm range.

That is correct, the accuracy of each 6-minute data value is +/- 0.02m (2cm) of the water level value at that time.

[ Note:  In a separate email, it was clarified that “Sigma is the standard deviation, essential the statistical variance, between these (181 1-second) samples.” ]

The question and answer verify that both the individual 1-second measurements and the 6-minute data value represents a range of water level 4 cm wide, 2 cm plus or minus of the value recorded.

This seemingly vague accuracy — each measurement actually a range 4 cm or 1 ½ inches wide — is the result of the mechanical procedure of the measurement apparatus, despite its resolution of 1 millimeter.  How so?

NOAA’s illustration of the modern Acoustic water level tide gauge at the Battery, NY shows why this is so.  The blow-up circle to the top-left shows clearly what happens at the one second interval of measurement:  The instantaneous water level inside the stilling well is different than the instantaneous water level outside the stilling well.

This one-second reading, which is stored in the “primary data collection platform” and later used as part of the 181 readings averaged to get the 6-minute recorded value, will be different from the actual water level outside the stilling well, as illustrated.  Sometimes it will be lower than the actual water level, sometimes it will be higher.  The apparatus as a whole is designed to limit this difference, in most cases, at the one second time scale, to a range of 2 cm above or below the level inside the stilling well  — although some readings will be far outside this range, and will be discarded as “outliers” (the rule is to discard all 3-sigma outliers — of the set of 181 readings — from the set before calculating the mean which is reported as the six-minute record).

We cannot regard each individual measurement as measuring the water level outside the stilling well — they measure the water level inside the stilling well. These inside-the-well measurements are both very accurate and precise — to 1 millimeter. However, each 1-second record is a mechanical approximation of the water level outside the well — the actual water level of the harbor, which is a constantly changing continuous variable  — specified to the accuracy range of +/- 2 centimeters. The recorded measurements represent ranges of values.  These measurements do not have “errors” (random or otherwise) when they are different than the actual harbor water level.  The water level in the harbor or river or bay itself was never actually measured.

The data recorded as “water level” is a derived value – it is not a direct measurement at all.  The tide gauge, as a measurement instrument, has been designed so that it will report measurements inside the well that will be reliably within 2 cm, plus or minus,  of the actual instantaneous water level outside the well – which is the thing we wish to measure.  After taking 181 measurements inside the well, throwing out any data that seems too far off, the remainder of the 181 are averaged and reported as the  six-minute recorded value, with the correct accuracy notation of +/- 2 cmthe same accuracy notation as for the individual 1-second measurements.

The recorded value denotes a value range – which must always be properly noted with each value — in the case of water levels from NOAA tide gauges, +/- 2 cm.

NOAA quite correctly makes no claim that the six-second records, which are the means of 181 1-second records, have any greater accuracy than the original individual measurements.

Why then do they make a claim that monthly means are then accurate to +/- 0.005 meters (5 mm)?    In those calculations, the original measurement accuracy is simply ignored altogether, and only the reported/recorded six-minute mean values are considered (confirmed by the author) — the same error that is made as with almost all other large data set calculations, applying the inapplicable Law of Large Numbers.

Accuracy, however, as demonstrated here, is determined by the accuracy of the original measurements when measuring a non-static, ever-changing,   continuously variable quantity and which is then recorded as a range of possible values — the range of accuracy specified for the measurement system —  and cannot be improved when (or by) calculating means.

### Take Home Messages:

1. When numerical values are ranges, rather than true discrete values, the width of the range of the original value (measurement in our cases) determines the width of the range of any subsequent mean or average of these numerical values.
2. Temperatures calculated from ASOS stations however are recorded and reported temperatures as ranges 1°F wide (0.55°C), and such temperatures are correctly recorded as “Integer +/- 0.5°F”. The means of these recorded temperatures cannot be more accurate than the original measurements –because the original measurement records themselves are ranges,  the means must be denoted with the same +/- 0.5°F.
3. The same is true of Tide Gauge data as currently collected and recorded. The primary record of 6-minute-values, though recorded to millimetric precision, are also ranges with an original accuracy of +/- 2 centimeters.  This is the result of the measurement instrument design and specification, which is that of a sort-of mechanical averaging system.  The means of tide gauge recorded values cannot be made more accurate the +/- 2 cm — which is far more accurate than needed for measuring tides and determining safe water levels for ships and boats.
4. When original measurements are ranges, their means are also ranges of the same magnitude. This fact must not be ignored or discounted; doing so creates a false sense of the accuracy of our numerical knowledge.  Often the mathematical precision of a calculated mean overshadows its real world, far fuzzier accuracy, leading to incorrect significance being given to changes of very small magnitude in those over-confident means.

# # # # #

Author’s Comment Policy:

Thanks for reading — I know that this will be a difficult concept for some.   For those, I advise working through the example themselves.  Use as many measurements as you have patience for. Work out all the possible means of all the possible values of the measurements, within the ranges of those original measurements, then report the range of the means found.

I’d be glad to answer your questions on the subject, as long as they are civil and constructive.

# # # # #

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TG
October 14, 2017 10:28 pm

Thank you for all the hard work.

Greg
October 15, 2017 12:27 am

these physical quantities — Global Average Temperature and Global Mean Sea Level

The first place to start is to point out that Global Average Temperature is NOT a “physical quantity”. You can not take the average of temperature, especially across vastly different media like land sea and ice. It’s scientific bullshit.

Are land + sea temperature averages meaningful?
https://judithcurry.com/2016/02/10/are-land-sea-temperature-averages-meaningful/

Before you start arguing about uncertainty ( which is a very good argument to get into ) you need to make sure are measuring something that is physically meaningful.

October 15, 2017 12:45 am

Greg, if you don’t think there is a physical “global temperature” what is your opinion of the global average of temperature anomalies? Ditto for sea surface levels.

Greg
October 15, 2017 12:45 am

This whole subject of uncertainty and measurement error is very complex out side a carefully constructed lab experiment. It is certainly key to the whole climate discussion and is something that Judith Curry has been pointing out fro at least a decade now.

However, this simplistic article by Kip does not really advance the discussion and sadly is unlikely to get advanced very much an anarchic chain of blog posts.

Kip clearly does not have the expertise to present a thorough discussion. It would be good if someone like his stats expert could have would have written it. This definately does need a thorough treatment and the currently claimed uncertainties are farcical, I will second him on that point.

Don K
October 15, 2017 1:22 am

Greg. You won’t get any argument from me that “Global Average Temperature” isn’t a poor metric. It’s very sensitive to the constantly changing distribution of warm water in the Pacific Ocean basin. Why would anyone not working on ENSO want a temperature metric that behaves like that? But it really is a physical quantity — if an inappropriate one for the purposes it’s being used for. Don’t you think it was almost certainly lower at the height of the last glaciation, or higher during the Cretaceous?

October 15, 2017 1:37 am

“if you don’t think there is a physical “global temperature”” – It’s not an opinion. It stems from the definition of temperature. They do indeed extend the notion of temperature in some very special cases for systems out of thermodynamic equilibrium, but typically it’s for dynamical equilibrium and they do lead to nonsense when taking out of context (such as absolute negative temperature). But for systems that are not even in dynamical equilibrium, such as Earth, it’s pure nonsense to average an intensive value that can be defined only locally, due of cvasiequilibrium. It’s not only pure nonsense, but it’s very provable that if you still insist of using such nonsense, you’ll get the wrong physical results out of calculation, even for extremely simple systems.

Greg
October 15, 2017 3:14 am

Don , maybe you should read the link in my first comment. There is a whole article explaining why global mean temperature is not physically meaningful.

oeman50
October 15, 2017 9:13 am

Dr. Curry’s point are well taken, many people do not understand the differences between energy and temperature. I also point out that “average daily temperature,” which has been interpreted as the average of the daily maximum and minimum is also misunderstood. We are now able to take temperature at the interval of our choice and come up with a weighted average. The average computed from just one daily maximum and one daily minimum assumes the temperatures spend equal amount of time clustered around the average. This is clearly not the case. So when comparing historical temperatures to newer values, it is important to realize the differences.

Greg
October 15, 2017 9:38 am

just to be clear oeman50, that was my article that Judith Curry published on here site. Note the credit just below the title. 😉

Hoyt Clagwell
October 15, 2017 10:04 am

The main problem with averaging anything globally is that no living thing on Earth actually experiences the global average. Additionally, the average temperature tells us nothing about the daily range of temperatures. If I experience a day which is 60 degrees in the morning, and 100 degrees in the afternoon, is it not hotter than a day which starts out at 75 and reaches a high of 95? Yet once averaged, the 95 degree day is reported as 5 degrees hotter than the 100 degree day. Of course it gets more complex, but it would be like calculating a globally averaged per capita crime rate. You could do it, but it would be a useless number because the only thing that is important is the criime rate where you are or plan to be. Same with temperature. If we experience a decade where the global average temperature goes up a small amount, was it higher daytime highs that caused it? Was it higher daytime lows that caused it? Was the range the same, but the heat lingered on a little longer after sunset? You can’t tell what is happening unless you look at local specifics, hour by hour. It would be like trying to tell me what song I’m thinking of if I just told you what the average musical note was. Meaning is in the details.
In the same vein, I’ve always wondered why we track the CO2 content of the atmosphere without tracking all of the other greenhouse gases as closely. If CO2 concentration goes up, do we know for a fact that that increases the total amount of greenhouse gases? Could another gas like water vapor decrease at times to balance out or even diminish the total?
It just seems to me that we are standing so far back trying to get the “big picture” that we are missing the details that would have told us the picture was a forgery.
I’m no scientist, so blast me if I’m wrong, but the logic of it all seems to be lost.

Dave Fair
October 15, 2017 10:23 am

Which is why only satellite, radiosonde and atmospheric reanalysis information [I hesitate to use “data.”] are appropriate for use in determining any averages, trends, etc.

In a few [number of?] years ARGO may be useful. Early ARGO information shows no worrisome patterns.

Science or Fiction
October 16, 2017 2:34 pm

@ Greg “This whole subject of uncertainty and measurement error is very complex”

Yes it is: “In 1977, recognizing the lack of international consensus on the expression of uncertainty in measurement, the world’s highest authority in metrology, the Comité International des Poids et Mesures (CIPM), requested the Bureau International des Poids et Mesures (BIPM) to address the problem in conjunction with the national standards laboratories and to make a recommendation.”

It took 18 years before the first version of a standard that deals with these issues in a successful way, was finally published. That standard is called: ´Guide to the expression of uncertainty in measurement´. There now exists only this one international standard for expression of uncertainty in measurement.

“The following seven organizations supported the development of the Guide to expression of uncertainty, which is published in their name:
BIPM: Bureau International des Poids et Measures
IEC: International Electrotechnical Commission
IFCC: International Federation of Clinical Chemistry
ISO: International Organization for Standardization
IUPAC: International Union of Pure and Applied Chemistry
IUPAP: International Union of Pure and Applied Physics
OlML: International Organization of Legal Metrology ..”

The standard is freely available. I think of it as a really good idea to use that standard for what should be obvious reasons. Even some climate scientists are now starting to realize that international standards should be used. See:
Uncertainty information in climate data records from Earth observation:
“The terms “error” and “uncertainty” are often unhelpfully conflated. Usage should follow international standards from metrology (the science of measurement), which bring clarity to thinking about and communicating uncertainty information.”

Robert B
October 17, 2017 7:47 pm

“Before you start arguing about uncertainty ( which is a very good argument to get into ) you need to make sure are measuring something that is physically meaningful.”
They are connected. The mean of an infinite number of measurements should give you the true value if individual measurements were only off due to random error. You need precise measurements to be sure that the distribution is perfect if you want others to believe that 10 000 measurements has reduced the error by √100. Even the act of rounding up or down means that you shouldn’t pretend that the errors were close to a symmetrical distribution and definitely not close enough to attribute meaning to a difference of 1/100th of the resolution. How anyone could argue against it is beyond me.
To then do it for something that it not an intrinsic property is getting silly. I know what people are thinking but the air around a station in the morning is not the same as that around it when the max is read.

ATheoK
October 15, 2017 9:10 am

Agreed, TG!

An excellent essay Kip!

Dr. S. Jeevananda Reddy
October 14, 2017 10:29 pm

I worked with IMD in Pune/India [prepared formats to transfer data on to punched cards as there was no computer to transfer the data directly]. There are two factors that affect the accuracy of data, namely:

Prior to 1957 the unit of measurement was rainfall in inches and temperature in oF and from 1957 they are in mm and oC. Now, all these were converted in to mm and oC for global comparison.

The second is correcting to first place of decimal while averaging: 34.15 is 34.1; 34.16 is 34.2; 34.14 is 34.1 and 34.25 is 34.3; 34.26 is 34.3; 34.24 is 34.2

Observational error: Error in inches is higher than mm and Error in oC is higher than oF

These are common to all nations defined by WMO

Dr. S. Jeevananda Reddy

Don Bennett
Reply to  Dr. S. Jeevananda Reddy
October 16, 2017 5:36 am

Dr. Reddy, Very interesting. By the way, you can use alt-248 to do the degree symbol, °.

Take care,

Robert of Texas
October 14, 2017 10:45 pm

Thank you for this information. I have always suspected the reported accuracy of many averaged numbers were simply impossible. This helps to clarify my suspicions. I also do not understand how using 100 year old measurements mixed with modern ones can result in the high accuracy stated in many posts. They seem to just assume that a lot of values increases the final accuracy regardless of the origin and magnitude of the underlying uncertainties.

October 15, 2017 1:43 am

Only bullshit results. Even for modern measurements, it’s the hasty generalization fallacy to claim that it applies to the whole Earth. Statisticians call it a convenience sampling. And that is only for the pseudo-measurement that does not evolve radically over time. Combining all together is like comparing apples with pears to infer things about a coniferous forest.

Robert McCarter
October 15, 2017 3:05 pm

Standard calculations in Chemistry carefully watch the significant digits. 5 grams per 7 mililiters is reported as 0.7 g/mL. Measuring several times with such low precision results in an answer with equally low precision. The extra digits spit out by calculators are fanciful in the real world.

John Harmsworth
October 15, 2017 10:38 am

People assume that modern digital instruments are inherently more accurate than old-style types. In the case of temperature at least this is not necessarily so. When temperature readings are collated and processed by software yet another confounding factor is introduced.
With no recognition of humidity, differing and changing elevation, partial sampling and other data quality issues, the idea that we could be contemplating turning the world’s function inside out over a possible few hundredths of a degree in 60 years of the assumed process is plainly idiotic.
AGW is an eco Socialist ghost story designed to destroy Capitalism and give power to those who can’t count and don’t want to work. I’m hardly a big fan of Capitalism myself but I don’t see anything better around. Socialism has failed everywhere it’s been tried.

October 15, 2017 3:56 pm

If quantization does not deceive you Nyqust will.

October 14, 2017 10:56 pm

Kip says: “If each measurement is only accurate to ± 2 cm, then the monthly mean cannot be MORE accurate than that — it must carry the same range of error/uncertainty as the original measurements from which it is made. Averaging does not increase accuracy.”

WRONG!

the +/- 2cm is the standard deviation of the measurement. This value is “sigma of x ” in the equation for the standard error of the estimator of the mean:

https://www.bing.com/images/search?view=detailV2&ccid=CYUOXtuv&id=B531D5E2BA00E15F611F3DAEC1B85110014F74C6&thid=OIP.CYUOXtuvcFogpL3jEnQw_gEsBg&q=standard+error&simid=608028072239301597&selectedIndex=1

The error bars for the mean estimator depends on the sqrt of “N”

AndyG55
October 14, 2017 11:01 pm

roflmao..

You haven’t understood a single bit of what was presented, have you johnson

You have ZERO comprehension when that rule can and can’t be used, do you. !!

(Andy, you need to do better than this when you think Johnson or anyone else is wrong. Everyone here is expected to moderate themselves according to the BOARD rules of conduct. No matter if Johnson is right or wrong,being rude and confrontative without a counterargument,is not going to help you) MOD

October 14, 2017 11:56 pm

I know perfectly well when to use standard error for the estimator of the mean.

See comment by Nick Stokes below.

Greg
October 15, 2017 12:32 am

Andy, how about you drop the aggressive, insulting habit of addressing all you replies to “johnson”. If you don’t agree with him, make you point. Being disrespectful does not give more weight to your point of view.

Also getting stroppy from the safely of your keyboard is a bit pathetic.

MarkW
October 15, 2017 7:25 am

lighten up greg

October 15, 2017 9:08 am

ROFL^2

You are a bit rude, Andy, but you are right.

Can we all TRY to be both polite and scientifically /mathematically correct please. It makes for a better blog all round.

ATheoK
October 15, 2017 9:19 am

“Greg October 15, 2017 at 12:32 am
Andy, how about you drop the aggressive, insulting habit of addressing all you replies to “johnson”. If you don’t agree with him, make you point. Being disrespectful does not give more weight to your point of view.

Also getting stroppy from the safely of your keyboard is a bit pathetic.”

“MarkW October 15, 2017 at 7:25 am
lighten up greg”

“The Reverend Badger October 15, 2017 at 9:08 am
ROFL^2

You are a bit rude, Andy, but you are right.

Can we all TRY to be both polite and scientifically /mathematically correct please. It makes for a better blog all round.”

Is Andy any ruder than Johnson was?

Especially when Johnson ignores facts, documentation and evidence presented in order to proclaim his personal bad statistics superior.

AndyG55
October 15, 2017 4:07 pm

Sorry, but it very obvious that mark DID NOT understand the original post.

When their baseless religion relies totally on a shoddy understand of mathematical principles, is it any wonder the AGW apostles will continue to dig deeper?

“I know perfectly well when to use standard error for the estimator of the mean.”

Again. it is obvious that you don’t !!

AndyG55
October 15, 2017 7:03 pm

For those who are actually able to comprehend.

Set up a spreadsheet and make a column as long as you like of uniformly distributed numbers between 0 and 1, use =rand(1)

Now calculate the mean and standard deviation.

The mean should obviously get close to 0.5..

but watch what happens to the deviation as you make “n” larger.

For uniformly distributed numbers, the standard deviation is actually INDEPENDENT of “n”

AndyG55
October 15, 2017 7:05 pm

darn typo..

formula is ” =rand()” without the 1, getting my computer languages mixed up again. !!

AndyG55
October 15, 2017 7:10 pm

Furthermore, since ALL temperature measurements are uniformly distributed within the individual ranged used for each measurement, they can all be converted to a uniform distribution between 0 and 1 and the standard deviation remains INDEPENDENT OF “n”</strong)

AndyG55
October 15, 2017 7:11 pm

Obviously, that means that the standard error is also INDEPENDENT of n

October 15, 2017 7:15 pm

Andy, standard deviation and sampling error are not the same things, so please tell me what you think your example is showing?

AndyG55
October 15, 2017 7:18 pm

Sorry you are having problems understanding, Mark.. Your problem, not mine.

Another simple explanation for those with stuck and confused minds.

Suppose you had a 1m diameter target, and, ignoring missed shots”, they were random uniformly distributed on the target.

Now, the more shots you have, the closer the mean will be to bulls eye..

But the error from that mean with ALWAYS be approximately +/- 0.5m uniformly distributed.

Nick Stokes
October 15, 2017 7:38 pm

“The mean should obviously get close to 0.5.”
“Obviously, that means that the standard error is also INDEPENDENT of n”
Those statements are contradictory. Standard error is the error of the mean (which is what we are talking about). If it’s getting closer to 0.5 (true) then the error isn’t independent of n. In fact it is about sqrt(1/12/n).

I did that test with R : for(i in 1:10)g[i]=mean(runif(1000))
The numbers g were
0.5002 0.5028 0.4956 0.4975 0.4824 0.5000 0.4865 0.5103 0.5106 0.5063
Standard dev of those means is 0.00930. Theoretical is sqrt(1/12000)=0.00913

Old England
October 15, 2017 2:22 am

Seems to me that no matter how data is treated or manipulated there is nothing that can be done to it which will remove the underlying inaccuracies of the original measurements.

If the original measurements are +/- 2cm then anything resulting from averaging or mean is still bound by that +/- 2cm.

Mark, could you explain why you believe that averagaing or the mean is able to remove the original uncertainty ? because I can’t see how it can.

Old England
October 15, 2017 2:30 am

Btw I can see how a trend might be developed from data with a long enough time series – But until the Trend is greater than the uncertainty it cannot constitute a valid trend.

e.g. In temperature a trend showing an increase of 1 deg C from measurements with a +/- 0.5 deg C (i.e. 1 deg C spread) cannot be treated as a valid trend until it is well beyond the 1 deg C, and even then it remains questionable.

I’m no mathematician or statistician but to me that is plain commonsense despite the hard-wired predilection for humans to see trends in everything ………

October 15, 2017 9:18 am

Maybe someone here has experience with information theory, I did some work with this years ago in relation to colour TV transmissions and it is highly relevant to digital TV . All about resolution and what you need to start with to get a final result. I am quire rusty on it now but think it is very relevant here, inability to get out more than you start with.

ATheoK
October 15, 2017 9:38 am

Old England:

Consider this; you take your temperature several times a day for a period of time.
Emulating NOAA, use a variety of devices from mercury thermometers, alcohol thermometers, cheap digital thermistors and infra red readers.

Sum various averages from your collection of temperatures. e.g.;
Morning temperature,
Noon temperature,
Evening temperature,
Weekly temperature,
Monthly temperature,
Lunar cycle temperatures, etc.

Don’t forget to calculate anomalies from each average set. With such a large set of temperatures you’ll be able to achieve several decimal places of precision, though of very dubious accuracy.

Now when your temperature anomaly declines are you suffering hypothermia?
When your temperature anomaly is stable are you healthy?
When your temperature anomaly increases, are you running a fever or developing hyperthermia?

Then after all that work, does calculating daily temperatures and anomalies to several decimal places really convey more information than your original measurement’s level of precision?

Then consider; what levels of precision one pretends are possible within a defined database are unlikely to be repeatable for future collections of data.
i.e. a brief window of data in a cycle is unlikely to convey the possibilities over the entire cycle.

Nor do the alleged multiple decimals of precision ever truly improve the accuracy of the original half/whole degree temperature reading.

Then, consider the accuracy of the various devices used; NOAA ignores error rates inherent from equipment, readings, handlings, adjustments and calculations.

tty
October 15, 2017 4:06 am

“The error bars for the mean estimator depends on the sqrt of “N””

Only true if the measured quantity consists of independent and identically distributed random variables. Amazing how few people seem to be aware of this.

Good luck in proving that there is no autocorrelation between sea-level measurements Mark!

October 15, 2017 8:31 am

Kip: ” In actual fact, it is the uncertainty of the measurement ”
..
Maybe this can clear up your misunderstanding: https://explorable.com/measurement-of-uncertainty-standard-deviation
….
Just remember std deviation is defined independent of the underlying distribution…(i.e. normal, uniform, geometic, etc.)

Nick Stokes
October 15, 2017 10:33 am

“The +/-2cm is NOT a standard deviation, not an error bar, not a confidence interval”
Then what is “uncertainty”?

Rick C PE
October 15, 2017 7:40 pm

Kip/Nick: Actually a stated instrument MU is a confidence interval. It is defined in the ISO Guides and elsewhere (including NIST) as:

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

The default is a 95% confidence interval. Thus a measured value of 100 cm can be said to have a true value of between 98 and 102 cm with a 95% confidence if the instrument MU is +/- 2 cm. While it is indeed derived from the standard deviations of various factors that affect the measurement, it is actually a multiple of the combined SDs. Two times the SD for a 95% MU confidence. However, it is not related to the SD of multiple measurements of the measurand. This is a measure of the variability of the thing being measured and such variability is only partly the result of instrument MU. Proper choice of instruments should make instrument MU a negligible issue. Problems arise when the measurement precision required to make a valid determination is not possible with the equipment available. In short, if you want to measure sea level to +/- 1 mm you need a measuring device with an MU of less than 1 mm.

Put another way, you can’t determine the weight of a truck to the nearest pound by weighing it on a scale with a 10 pound resolution no matter how many times you weigh it.

Above I referred to multiple sources of MU that need to be combined. This is known as an uncertainty budget. As an example a simple screw thread micrometer includes the following items: repeatability, scale error, zero point error, parallelism of anvils, temperature of micrometer, temperature difference between micrometer and measured item. However, the vast majority of instrument calibrations are done by simple multiple comparisons of measured values of certified reference standards. In these calibrations there are always at least three sources of MU. The uncertainty of the reference standard, one half the instrument resolution and the standard deviation of the repeated comparison deviation from the reference value. In addition, to be considered adequate the Test Uncertainty Ratio (MU of device being calibrated divided by MU of reference) must be at least 4:1.

This is all basic metrology that should be well understood by any scientist or engineer. But I know from experience that it is not as is clearly evident in these discussions.

Greg
October 15, 2017 11:51 pm

Thanks again for you clear and well informed opinion on these matters.

The problem with using S.D as the basis for establishing “confidence intervals” is that it is based soley on statistics and addresses only the sampling error.

If global mean SST is given as +/-0.1 deg C then a “correction” is made due to a perceived bias of 0.05 deg and the error bars are the same ( because the stats are still the same ) then we realise that they are not including all sources of error and the earlier claimed accuracy was not correct.

The various iterations of hadSST have not changed notably in their claimed confidence levels yet at one point they introduced -0.5 deg step change “correction”. This was later backed out and reintroduced as a progressive change, having come up with another logic to do just about the same overall change of 0.5 deg C.

Variance derived confidence levels do NOT reflect the full range of uncertainty, only one aspect: sampling error.

Gary Pearse
October 15, 2017 7:21 am

Mark S, you missed the whole point of why this isn’t so in the case of temperatures and tide gauges. If you measure the length of a board a dozen times carefully, then you are right. But if the board keeps changing its own length, then multiple measurings are not going to prove more accurate or even representative of anything. I hope this helps.

MarkW
October 15, 2017 7:24 am

If the measurement is made of the same thing, the different results can be averaged to improve the accuracy.
Since the temperature measurements are being made at different times, they cannot be used to improve the accuracy.
That’s basic statistics.

October 15, 2017 8:20 am

Measuring an individual “thing” and sampling a population for an average are two distinct, and different things. You seem to be confusing the two.

October 15, 2017 8:06 am

Mark S Johnson,
You are quite wrong. If I handed you an instrument I calibrated to some specific accuracy, say plus or minus one percent of full scale for discussion purposes, you had better not claim any measurement made with it, or any averages of those values, is more accurate than what I specified. In fact, if the measurement involved safety of life, you must return the instrument for a calibration check to verify it is still in spec.

Where anyone would come up with the idea that an instrument calibration sticker that say something like “+/- 2 cm” indicates a standard deviation, I cannot imagine. In the cal lab, there is no standard deviation scheme for specifying accuracy. When we wrote something like “+/- 2 cm”, we meant that exactly. That was the sum of the specified accuracy of the National Bureau of Standards standard plus the additional error introduced by the transfer reference used to calibrate the calibration instrument plus the additional error introduced by the calibration instrument used on your test instrument.

Again, that calibration sticker does not say “+/- 2 cm” is some calculated standard deviation from true physical values. It means what at each calibration mark on the scale, the value will be within “+/- 2 cm” of true physical value. That does not, however specify the Precision of the values you read. That is determined by the way the instrument presents its values. An instrument calibrated to “+/- 2 cm” could actually have markings at 1 cm intervals. In that case, the best that can be claimed for the indication is +/- 0.5 cm. The claimed value would then be +/- 0.5 cm plus the +/- 2 cm calibration accuracy. Claiming an accuracy of better than +/- 2.5 cm would in fact be wrong, and in some industries illegal. (Nuclear industry for example.)

So drop the claims about standard deviation in instrument errors. It does not even apply to using multiple instrument reading the same process value at the same time. In absolutely no case can instrument reading values be assumed to be randomly scattered around true physical values within specified instrument calibration accuracy. Presenting theories about using multiple instruments from multiple manufacturers, each calibrated with different calibration standards by different technicians or some such similar example is just plain silly when talking about real world instrumentation use. You are jumping into the “How many angels can dance on the head of a pin” kind of argument.

October 15, 2017 8:16 am

Gary, they do not make an instrument that can measure “global temperature.”

Measuring “global temperature” is a problem in sampling a population for the population mean. Once you understand this, you may be able to grasp the concept of “standard error” which is comprised of the standard deviation of the instrument used for measurement, divided by the sqrt of the number of obs.

Now when/if they build an instrument that can measure the global temperature with one reading, then your argument might hold water.

October 15, 2017 8:36 am

Mark,

Where above do I mention “global temperature”? My statements were about the use of instrument readings (or observations to the scientific folks.) I would suggest that however that “global temperature” be derived, it cannot claim an accuracy better than the calibration accuracy of the instrumentation used. Wishful thinking and statistical averaging cannot change that.

Remember the early example of averages of large numbers was based upon farm folks at an agricultural fair guessing the weight of a bull. The more guesses that were accumulated, the closer the average came to the true weight. Somehow that has justified the use of averaging in many inappropriate situations. Mathematical proofs using random numbers do not justify or indicate the associated algorithms are universally applicable to real world situations.

October 15, 2017 8:57 am

Gary, the estimator of the population mean can be made more accurate with more observations. The standard error is inversely proportional to the sqrt of the number of obs.
…..
Here’s an example.
….
Suppose you wanted to measure the average daily high temperature for where you live on Oct 20th. You measure the temp on Oct 20th next Friday.

Is this measure any good?

Now, suppose you do the same measurement 10/20/2017, 10/20/2018, 10/20/2019 and 10/20/2020, then take the average of the four readings.
..
Which is more accurate?…..the single lone observation you make on Friday, or the average of the four readings you make over the next four years?
….
If you are interested in the real climatic average for your location on Oct 20th, you really need 30 years of data to be precise.

October 15, 2017 9:02 am

Gary, RE: weight of bull.

Here you go again with an incorrect analogy. The weight of an individual bull is not a population mean. Don’t confuse the two. The correct “bull” analogy would be to actually measure the weight of 100 bulls, to determine what the average weight of a bull is. The more bulls you measure, the closer you will get to what the “real” average bull weight is.

Retired Kit P
October 15, 2017 9:22 am

BZ!

October 15, 2017 9:27 am

There will be some of us (like Gary and myself) on here who have regularly sent instruments away to be calibrated and had to carefully consider the results, check the certificates etc. We appear to know rather more about this than some contributors today. I find it interesting that a simple experience like this can help a lot in an important discussion.

October 15, 2017 9:58 am

“the estimator of the population mean can be made more accurate with more observations. The standard error is inversely proportional to the sqrt of the number of obs.”

Two points here: 1. “estimator” mean guess. 2. your estimator may be made more precise according to a specified estimation algorithm. That does not relate to its accuracy. Your comment about standard deviation only applies to how you derive your guess.

“If you are interested in the real climatic average for your location on Oct 20th, you really need 30 years of data to be precise.”

Good now we are on the same page. You are achieving a desired PRECISION. Accuracy, however remains no better than the original instrumentation accuracy and often worse depending upon how the data is mangled to fit your algorithm. (F to C etc.)

“Here you go again with an incorrect analogy. The weight of an individual bull is not a population mean. Don’t confuse the two. The correct “bull” analogy would be to actually measure the weight of 100 bulls, to determine what the average weight of a bull is. The more bulls you measure, the closer you will get to what the “real” average bull weight is.”

Nope, the exercise was to determine the accuracy of guesses about the weight of a single bull tethered to a post at the fair. A prize was awarded to the person who guessed the closest. It was not about guessing the weight bulls as a population. The observation about that large numbers of guesses was that the average became closer to true weight of the bull as the number of guess increased, one guess per person. It was never claimed that random guess about random bulls would average to any meaningful or useful number.

October 15, 2017 10:05 am

Guessing the weight of an individual bull is not the same as sampling a population. Hey…..ever hear about destructive testing? It’s what happens when running the test obliterates the item “measured.” For example, how would you insure the quality of 1000 sticks of dynamite? Would you test each one, or would you take a representative random sample and test the smaller number?

Slacko
October 16, 2017 8:39 pm

Mark S Johnson October 15, 2017 at 9:02 am
“The weight of an individual bull is not a population mean. Don’t confuse the two.”

He didn’t confuse anything. He said “The more guesses that were accumulated, the closer the average came to the true weight. Somehow that has justified the use of averaging in many inappropriate situations.” But you like to fly off on your own illogical tangent, which just gets in the way of those of us trying to understand the arguments.

Phoenix44
October 15, 2017 8:33 am

Then explain how that applies if the measurements are not normally distributed? And if you have no idea if they are normally distributed?Let’s say the sides of the block of metal I have on my desk.

October 15, 2017 9:55 am

Just to clarify Andy’s concerns. Mark Johnson is confusing uncertainty of the estimate with accuracy of the measure; they’re two different things, something Kip attempts to point out in his essay and also something that anyone familiar with measurement theory and statistics would understand from his essay. It’s possible a person without much practical experience in numerical modeling might miss the distinction, but I can assure you it’s there.

While the “law of large numbers” will reduce the error of estimate as Mark describes, it does nothing to increase accuracy of the measure.

October 15, 2017 10:01 am

Maybe another example is in order?

If a single measure is accurate +/- 2cm, it has an uncertainty associated with it also, which may perhaps be +/- 5mm. As repeated measures are taken and averaged, the uncertainty (5mm) can be reduced arithmetically as Mark Johnson describes, but the result is a measure accurate +/- 2cm with a lower uncertainty (for example +/- .1 mm).

I hope that resolves the conflicting views expressed here. I agree there’s no reason for ad hominem by either party. It’s a very confusing subject for most people, even some who’ve been involved with it professionally.

October 15, 2017 10:08 am

When what you are measuring is a population mean, it most certainly does increase the accuracy.

October 16, 2017 1:15 pm

Mark S Johnson: The only person on this thread discussing measures of a population mean is you, and it’s almost certain the only training in statistics you’ve ever had involved SPSS.

Error in a measure is assumed to be normally distributed, not the measure itself. You need to meditate on that. The accuracy of a measure has nothing to do with the uncertainty of the estimate. The “law of large numbers” doesn’t improve accuracy, it improves precision. You’re wrong to argue otherwise.

Clyde Spencer
October 15, 2017 12:02 pm

Bartleby,
That is particularly true if there is a systematic error in the accuracy. If you have a roomful of instruments, all out of calibration because over time they have drifted in the same direction, using them to try to obtain an average will, at best, give you an estimate of what the average error is, but it will not eliminate the error. The only way that you are going to get the true value of the thing you are measuring is to use a high-precision, well-calibrated instrument.

October 16, 2017 1:40 pm

Certainly true if there is systemic error, which really means the measure is somehow biased (part of an abnormal distribution); unless the error of estimate is normal, the law of large numbers can’t be used at all. It can never be used to increase accuracy.

The whole idea of averaging multiple measures of the same thing to improve precision is based on something we call a “normal error distribution” as you point out. We assume the instrument is true within the stated accuracy, but that each individual observation may include some additional error, and that error is normally distributed.

So, by repeatedly measuring and averaging the result, the error (which is assumed normal) can be arithmetically reduced, increasing the precision of the estimate by a factor defined by the number of measures. This is the “Students T” model.

But accuracy isn’t increased, only precision. 100 measures using a device accurate +/- 2cm will result in a more precise estimate that’s accurate to +/- 2cm.

Accuracy and Precision are two very different things.

Bartemis
October 16, 2017 4:51 pm

‘The whole idea of averaging multiple measures of the same thing to improve precision is based on something we call a “normal error distribution”…’

Normal (or Gaussian) distributions are not required, though a great many measurement error sets do tend to a Normal distribution due to the Central Limit Theorem.

All that is required is that the error be equally distributed in + and – directions. Averaging them all together then means they will tend to cancel one another out, and the result will, indeed, be more accurate. Accuracy means that the estimate is closer to the truth. Precision means… well, a picture is worth a thousand words. These arrows are precise:

October 16, 2017 9:48 pm

Bartemis illustrates very effectively, the difference between accuracy and precision.

Nick Stokes
October 16, 2017 10:11 pm

Bartleby,
“100 measures using a device accurate +/- 2cm will result in a more precise estimate that’s accurate to +/- 2cm.

Accuracy and Precision are two very different things.”
Yes, if you are talking about a metrology problem, which is the wrong problem here. No-one has ever shown where someone in climate is making 100 measures of the same thing with a device. But there is one big difference between accuracy and precision, which is in the BIPM vocabulary of metrology, much cited here, but apparently not read. It says, Sec 2.13 (their bold):
“NOTE 1 The concept ‘measurement accuracy’ is not a quantity and is not given a numerical quantity value. “

Which makes sense. Accuracy is the difference between the measue and the true value. If you knew the true value, you wouldn’t be worrying about measurement accuracy. So that is the difference. If it has numbers, it isn’t accuracy.

October 17, 2017 4:13 pm

Nick Stokes (perhaps tongue in cheek) writes: “So that is the difference. If it has numbers, it isn’t accuracy.”

Nick, if it doesn’t have numbers, it isn’t science. 🙂

Nick Stokes
October 17, 2017 4:26 pm

Bartleby,
“isn’t science”
Well, it’s in the BIPM vocabulary of metrology.

October 17, 2017 10:32 pm

Nick, there’s an old, old saying in the sciences that goes like this:

If you didn’t measure it, it didn’t happen.”

I sincerely believe that. So any “discipline” that spurns “numbers” isn’t a science. QED.

Nick Stokes
October 17, 2017 11:39 pm

Bartleby,
I’m not the local enthusiast for use of metrology (or BIPM) here. I simply point out what they say about the “concept ‘measurement accuracy’”.

October 17, 2017 11:51 pm

Nick Stokes writes: “I’m not the local enthusiast for use of metrology (or BIPM) here. I simply point out what they say about the “concept ‘measurement accuracy’”

OK. I don’t think that changes my assertion, that science is measurement based and so requires the use of numbers.

I’m not sure if you’re trying to make an argument from authority here? Id so it really doesn’t matter what the “BIPM” defines; accuracy is a numerical concept and it requires use of numbers. There’s no alternative.

If, in the terms of “metrology”, numbers are not required, then the field is no different from phrenology or astrology, neither of which is a science. Excuse me if you’ve missed that up until now. Numbers are required.

Clyde Spencer
October 15, 2017 11:45 am

Mark S Johnson,

We have a very different take on what Kip has written. My understanding is that the tide gauges can be read to a precision of 1mm, which implies that there is a precision uncertainty of +/- 0.5mm. HOWEVER, it appears that the builders of the instrumentation and site installation acknowledge that each and all of the sites may have a systematic bias, which they warrant to be no greater than 2 cm in either direction from the true value of the water outside the stilling well. We don’t know whether the inaccuracy is a result of miscalibration, or drift, of the instrument over time. We don’t know if the stilling well introduces a time-delay that is different for different topographic sites or wave conditions, or if the character of the tides has an impact on the nature of the inaccuracy. If barnacles or other organisms take up residence in the inlet to the stilling well, they could affect the operation and change the time delay.

The Standard Error of the Mean, which you are invoking, requires the errors be random (NOT systematic!). Until such time as you can demonstrate, or at least make a compelling argument, that the sources of error are random, your insistence on using the Standard Error of the Mean is “WRONG!”

I think that you also have to explain why the claimed accuracy is more than an order of magnitude less than the precision.

October 15, 2017 12:00 pm

Clyde, a single well cannot measure global average sea level. It does not sample with respect to the geographic dimension. Again there is confusion here with the precision/accuracy of an individual instrument, and the measurement of an average parameter of a population. Apples and oranges over and over and over.

Clyde Spencer
October 15, 2017 12:10 pm

Mark S Johnson,

I never said that a single well measured the average global sea level, and I specifically referred to the referenced inaccuracy for multiple instruments.

You did not respond to my challenge to demonstrate that the probable errors are randomly distributed, nor did you explain why there is an order of magnitude difference between the accuracy and precision.

You seem to be stuck on the idea that the Standard Error of the Mean can always be used, despite many people pointing out that its use has to be reserved for special circumstances. You also haven’t presented any compelling arguments as to why you are correct. Repeating the mantra won’t convince this group when they have good reason to doubt your claim.

October 15, 2017 12:16 pm

Clyde the reason it’s called Standard Error of the Mean is because I’m talking about measuring the mean and am not talking about an individual measurement.

This is not about measuring the same block of metal 1000 times to improve the measurement. It’s about measuring 1000 blocks coming off the assembly line to determine the mean value of the block’s you are making.

Clyde Spencer
October 15, 2017 12:46 pm

Mark S Johnson,

You said, “…I’m talking about measuring the mean.” Do you own a ‘meanometer?” Means of a population are estimated through multiple samples, not measured.

You also said, “This is not about measuring the same block of metal 1000 times to improve the measurement. It’s about measuring 1000 blocks coming off the assembly line to determine the mean value of the block’s you are making.”

In the first case, you are primarily concerned about the accuracy and precision of the measuring instrument. Assuming the measuring instrument is accurate, and has a small error of precision, the Standard Error of the Mean can improve the precision. However, no amount of measuring will correct for the inaccuracy, which introduces a systematic bias. Although, if the electronic measuring instrument is wandering, multiple measurements may compensate for that if the deviations are equal or random at each event. But, if you have such an instrument, you’d be advised to replace it rather than try to compensate after the fact.

In the second case, you have the same problems as case one, but you are also confronted with blocks that are varying in their dimensions. Again, if the measuring instrument is inaccurate, you cannot eliminate a systematic bias. While the blocks are varying, you can come up with a computed mean and standard deviation. However, what good is that? You may have several blocks that are out of tolerance and large-sample measurements won’t tell you that unless the SD gets very large; the mean may move very little if any. What’s worse, if the blocks are varying systematically over time, for example as a result of premature wear in the dies stamping them, neither your mean or SD is going to be very informative with respect to your actual rejection rate. They may provide a hint that there is a problem in the production line, but it won’t tell you exactly what the problem is or which items are out of tolerance. In any event, even if you can justify using the Standard Error of the Mean to provide you with a more precise estimate of the mean, what good does it do you in this scenario?

Paul Blase
October 15, 2017 12:52 pm

“In the second case, you have the same problems as case one, but you are also confronted with blocks that are varying in their dimensions. In this case you shouldn’t be worrying about your instrument, your concern is your manufacturing process!

October 16, 2017 1:11 pm

Clyde –

You’re playing into the hands of someone ignorant. It’s a common fault on public boards like this.

Both of you (by that I mean Johnson too) are freely exchanging the terms “accuracy” and “uncertainty”; they are not the same. Until you both work that out you’re going to argue in circles for the rest of eternity.

October 17, 2017 11:35 pm

Nick Stokes ==> Said: October 16, 2017 at 10:11 pm

But there is one big difference between accuracy and precision, which is in the BIPM vocabulary of metrology, much cited here, but apparently not read. It says, Sec 2.13 (their bold): “NOTE 1 The concept ‘measurement accuracy’ is not a quantity and is not given a numerical quantity value. “

And the rest of the note? The very next sentence….. is!

A measurement is said to be more accurate when it offers a smaller measurement error*.

This is exactly what Kip Hansen has argued all along and exactly what Bartleby just wrote** and yet you have just gone out of your way to cherry pick the quote and completely butcher the context of the very definition you are referring to!

*And measurement error is defined at 2.16 (3.10) thusly: “measured quantity value minus a reference quantity value”
**Bartleby wrote: “100 measures using a device accurate +/- 2cm will result in a more precise estimate that’s accurate to +/- 2c.”

Nick Stokes
October 18, 2017 12:58 am

SWB,
“The very next sentence…”
The section I quoted was complete in itself, and set in bold the relevant fact: “is not given a numerical quantity value. Nothing that follows changes that very explicit statement. And it’s relevant to what Bartleby wrote: “a more precise estimate that’s accurate to +/- 2cm”. BIPM says that you can’t use a figure for accuracy in that way.

Crispin in Waterloo but really in Beijing
October 16, 2017 6:25 am

Well said, Kip.

Mark S: “the +/- 2cm is the standard deviation of the measurement”

No, it is not the SD. The SD can only be calculated after a set of readings has been made. The 2cm uncertainty is a characteristic of the instrument, determined by some calibration exercise. It is not an ‘error bar’, it is an inherent characteristic of the apparatus. Being inherent, replicating measurements or duplicating the procedure will not reduce the uncertainty of each measurement.

Were this not so, we would not strive to create better instruments.

You make an additional error I am afraid: each measurement stands alone, all of them. They are not repeat measurements of ‘the same thing’ for it is well known in advance that the level will have changed after the passage of second. The concept you articulate relates to making multiple measurements of the same thing with the same instrument. An example of this is taking the temperature of a pot of water by moving a thermocouple to 100 different positions within the bulk of the water. The uncertainty of the temperature is affected by the uncertainty of each reading, again, inherent to the instrument and the SD of the data. One can get a better picture of the temperature of the water by making additional measurements, but the readings are no more accurate than before, and the average is not more accurate just because the number of readings is increased. Making additional measurements tells us more precisely where the middle of the range is, but does not reduce the range of uncertainty. This example is not analogous to measuring sea level 86,400 times a day as it rises and falls.

Whatever is done using the 1-second measurements, however processed, the final answer is no more accurate than the accuracy of the apparatus, which is plus minus 20mm.

October 17, 2017 11:39 pm

Help admin or mod or mods. A close block quote went astray just above. Please, thank you 😉

October 18, 2017 7:56 pm

Nick Stokes==> October 18, 2017 at 12:58 am:

The section I quoted was complete in itself, and set in bold the relevant fact: “is not given a numerical quantity value“. Nothing that follows changes that very explicit statement. And it’s relevant to what Bartleby wrote: “a more precise estimate that’s accurate to +/- 2cm”. BIPM says that you can’t use a figure for accuracy in that way.

Talk about perversity – I can’t imagine it would be anything else – if you really are being intellectually honest!

Here is the whole reference (Their bold):

2.13 (3.5)
measurement accuracy
accuracy of measurement
accuracy

closeness of agreement between a measured quantity value and a true quantity value of a measurand

NOTE 1 The concept ‘measurement accuracy’ is not a quantity and is not given a numerical quantity value. A measurement is said to be more accurate when it offers a smaller measurement error.

NOTE 2 The term “measurement accuracy” should not be used for measurement trueness and the term “measurement precision” should not be used for ‘measurement accuracy’, which, however, is related to both these concepts.

NOTE 3 ‘Measurement accuracy’ is sometimes understood as closeness of agreement between measured quantity values that are being attributed to the measured.

How could you completely miss the definition of Accuracy?

It is defined as the “closeness of agreement between a measured quantity value and a true quantity value of a measurand.”

It is very clear that the term is not numeric but ordinal and of course, ordinal quantities have mathematical meaning as you would well know!

Nick Stokes
October 18, 2017 9:26 pm

“It is very clear that the term is not numeric but ordinal and of course, ordinal quantities have mathematical meaning as you would well know!”
Yes. And what I said, no more or less, is that it doesn’t have a number. And despite all your huffing, that remains exactly true, and is the relevant fact. I didn’t say it was meaningless.

Auto
October 15, 2017 1:38 pm

Don’t feed the troll.

Auto

skorrent1
October 15, 2017 7:04 pm

When I first considered the “law of large numbers” years ago, I applied an engineer’s mental test for myself. If I have a machine part that needs to be milled to an accuracy of .001 in, and a ruler that I can read to an accuracy of 1/16 in, could I just measure the part with a ruler 1000 times, average the result, and discard my micrometer? I decided that I would not like to fly in an aircraft assembled that way.

Alan Rakes
October 16, 2017 5:04 am

Mark, I am far from an expert but do remember a little of what I leaned in my classes on stochastic processes. If I were able to assume that the distribution from which I was measuring was a stationary or at least wide sense stationary, then the process of multiple measurements as you imply could in fact increase the accuracy. This is actually how some old style analog to digital converters worked by using a simple comparator and counting the level crossings in time you can get extra bits of accuracy. This is similar to your assertion here.

The main flaw here is that you must make the stationarity assumption. Sorry, but temperature measurements and tidal gauge measurements are far from stationary. In fact, the pdf is a continuing varying parameter over time so I have a hard time agreeing with your assertion about the improvement in accuracy.

RW
October 16, 2017 8:11 pm

This is essentially about significant digits. Not the standard deviation of a sample of sample means. These two things are different. Ok? You cannot manufacture significant digits by taking samples. Period.

Peter Morgenroth
October 14, 2017 11:17 pm

It may be worth remembering – no calculated figure is entitled to more significant figures (accuracy) than the data used in the calculation.

MarkW
October 15, 2017 7:29 am

In fact, the further your calculations get from the original measured number, the greater the uncertainty gets.

October 15, 2017 8:44 am

Three measurements, each with one digit of significance: 0.2, 0.3 and 0.5

The calculated average is what?

Is it 0?
is it .33?
or is it .33333 ?
In fact the more digits you add, the closer you come to the real value, namely one third.

October 15, 2017 10:31 am

Mark, what you illustrate in your example is the reduction of uncertainty and convergence on the true value that can be accomplished when averaging multiple observations of the same thing using the same instrument (or instruments calibrated to the same accuracy). It assumes several things, the one thing not mentioned in Kip’s article or your example is that all measures come from a quantity that’s normally distributed. So there are at least three assumptions made when averaging a quantity and using the “law of large numbers” to reduce uncertainty in the measure;

– That all measures are of the same thing.
– That all measures have the same accuracy.
– That the measures are drawn from an underlying normal distribution.

All three assumptions must be met for the mean to have “meaning” 🙂

Briefly, if you average the length of 100 tuna, and the length of 100 whale sharks, you won’t have a meaningful number that represents the average length of a fish. In fact, if you were to plot your 200 observations, you’d likely find two very distinct populations in your data, one for whale sharks and another for tuna. The data don’t come from a normal distribution. In this case, any measure of uncertainty is useless since it depends on the observations coming from a normal distribution. No increase in instrument accuracy can improve precision in this case.

I’ll get to this again in my comment on Kip’s essay below.

October 15, 2017 10:55 am

Bartleby, I believe this is the crux of the wealth of misunderstanding here: “That all measures are of the same thing.”
….
A population mean is not a “thing” in your analysis of measurement.

You can’t measure a population mean with a single measure, you need to do random sampling of the population to obtain an estimator of the mean.

This is not at all like weighing a beaker full of chemicals on a scale.

You don’t conduct an opinion poll by going to the local bar and questioning a single patron….you need a much larger SAMPLE to get an idea of what the larger population’s opinion is. In the extreme case where N(number of obs) = population size, your measure of the average has zero error.

Tom Halla
October 15, 2017 1:11 pm

The “average” temperature is not of any real value, it is the change in temperature, and then, as a change in the equator-polar gradient that seems to matter in climate. Purporting to find changes to the nearest thousandth of a degree with instruments with a granularity of a whole degree appears to be an act of faith by the warmist community. Credo quia absurdiam?

Jim Gorman
October 15, 2017 1:15 pm

Mark S; You miss the point. What is the mean of 0.2+- 0.5, 0.3+- 0.5, and 0.5+- 0.5. Where the +- is uncertainty. Is it 0.3+- 0.5? How will even an infinite number of measurement reduce the uncertainty?

The range is going to be 0.8 to -0.5. You can say the mean is 0.3333, but I can say it is 0.565656 and be just as correct. Basically, just the mean without the uncertainty limits is useless.

October 16, 2017 9:05 pm

“Bartleby, I believe this is the crux of the wealth of misunderstanding here: “That all measures are of the same thing.”
….
A population mean is not a “thing” in your analysis of measurement.”

Mark, you’ve been beaten enough. Go in peace.

Clyde Spencer
October 15, 2017 12:17 pm

Peter,

The actual rule is that no calculated result is entitled to more significant figures than the LEAST precise multiplier in the calculation.

I suspect that some mathematicians and statisticians unconsciously assume that all the numbers they are working with have the precision of Pi. Indeed, that might be an interesting test. Calculate PI many times using only measurements with one significant figure and see how close the result comes to what is known.

Nick Stokes
October 15, 2017 3:44 pm

Clyde,
“Calculate PI many times using only measurements with one significant figure”
Something like this was done, by Buffon, in about 1733. Toss needles on floorboards. How often do they lie across a line. That is equivalent to a coarse measure. And sure enough, you do get an estimate of π.

RW
October 16, 2017 8:52 pm

Omg. Look, the example with needles just bakes perfect accuracy into the pie. Now let’s try marking needles as over a line or not with effing cataracts or something…good lord. I don’t understand why the idea of “your observations are fundamentally effing limited man!” is so hard to understand here. Nothing to do with minimizing random sampling error.

Peter Sable
October 14, 2017 11:23 pm

Kip is correct if the temperature never deviates from 72degF +/- 0.5degF. You will just write down 72 degF and the error will indeed be has he indicates.

Fortunately the temperature varies far more than that. One day, the temperature high/ow is 72/45 from 71.5 true and 45.6 true, the next day it is 73/43 from 72.3 true and 44.8 true, the next day it is 79/48 from 79.4 true and 47.9 true, and so on. The noise that is the difference between the true and recorded measurement has an even distribution as he notes, but can be averaged as long as the underlying signal swings bigger than the resolution of 1degF.

The Central Limit is a real thing. You average together a bunch of data with rectangular distribution you get a normal distribution. Go ahead and look at the distribution of a 6 sided dice. With one dice it’s rectangular. With two dice it’s a triangle. Add more and more dice and it’s a normal distribution.

Fortunately the signal varies by more than the 1 bit comparator window for the sigma-delta A/D and D/A converters in your audio and video systems, which operate on similar principles. It would be quite obvious to your ears if they failed to work. (yes, they do some fancy feedback stuff to make it better, but you can get a poor man’s version by simple averaging. I’ve actually designed and built the circuits and software to do so)

Peter

Jim Gorman
October 15, 2017 1:44 pm

You assume you know the “true” temperature. Lets change that to all that you know is 72/45 +- 0.5, 73/43 +- 0.5, and 79/48 +- 0.5. Where the +- is uncertainty. Does the mean also have an uncertainty of +- 0.5. If not why not. Will 1000 measurements change the fact that each individual measurements has a specific uncertainty and you won’t really know the “true” measurement?

Peter Sable
October 15, 2017 9:29 pm

for 1,000 measurements the *difference* between the true and the measured will form a rectangular distribution. If that distribution is averaged the average forms normal distribution, per the central limit theorem. The mean of that distribution will be zero, and thus the mean of the written-down measurements will be the ‘true’ measurement.

Try performing the numerical experiment yourself. It’s relatively easy to do in a spreadsheet.

Or go listen to some music from a digital source. The same thing is happening.

Jim Gorman
October 16, 2017 5:53 am

Peter; The problem is that you don’t know the true value? It lies somewhere between +- 0.5 but where is unknown.

Peter Sable
October 16, 2017 7:01 am

Peter; The problem is that you don’t know the true value? It lies somewhere between +- 0.5 but where is unknown.

How odd that your digital sound system appears to know.

You do know the true value for some period (integrating between t0 and t1) as long as the input signal varies by much greater than the resolution of your instrument. You do not know the temperature precisely at t0 or any time in between t0 and t1. But for the entire period you do know at a precision greater than that of your instrument. This is how most modern Analog to Digital measurement systems work.

Whether a temperature average is a useful concept by itself is not for debate here (I happen to think it’s relatively useless). But it does have more precision than a single measurement.

Nick Stokes posted an example above. Try running an example for yourself. It just requires a spreadsheet.

Jim Gorman
October 16, 2017 9:17 am

Peter; consider what you are integrating. Is it the recorded value or the maximum of the range or the minimum of the range or some variations of maximum, minimum, and recorded range?

And I’m sorry but integrating from t0 to t1 still won’t give the ‘true’ value. It can even give you a value to a multitude of decimal places. But you still can’t get rid of the uncertainty of the initial measurement.

Consider your analog to digital conversion. You have a signal that varies from +- 10.0 volts. However, your conversion apparatus is only accurate to +- 0.5 volts. How accurate will your conversion back to analog be?

Peter Sable
October 17, 2017 8:56 am

Consider your analog to digital conversion. You have a signal that varies from +- 10.0 volts. However, your conversion apparatus is only accurate to +- 0.5 volts. How accurate will your conversion back to analog be?

Do you mean accuracy or precision? I’ll try to answer both.

If you mean precision:

It depends on the frequency and input signal characteristics. In the worst case of a DC signal with no noise at any other frequency, the precision is +/- 0.5 volts.

If however I’m sampling a 1Khz signal at 1Mhz and there is other random noise at different frequencies in the signal, then my precision is 0.5V/sqrt(1000) = 0.016 volts @ 1khz. I can distinguish 0.016V changes in the 1Khz signal amplitude by oversampling and filtering (averaging). I’m trading off time precision for voltage precision.

if you mean accuracy

If you mean accuracy AT DC, do you mean the accuracy of the slope or the offset? A linear calibration metric is typically expressed in terms of y=mx+b, I don’t know if you are talking about m or b… Likely ‘b’, or you would have used a different metric than volts (you would use a relative metric, like percentage). e.g. “accuracy = 1% +/- 0.5V” is what you might see in a calibration specification.

Assuming you are talking about b, then since amplitude is typically a delta measurement, then the b is irrelevant (cancels out), same answer as above. You know the amplitude of the 1Khz signal within 0.016V.

Getting back to climate, as long as ‘b’ does not vary, you get the same answer for the temperature trend, since it is also a delta measurement. IMHO ‘b’ does vary quite a bit over time, more than the BE or other folks are taking into account (see Anthony’s work), but that’s not Kip’s argument.

Peter

Jer0me
October 14, 2017 11:27 pm

I’m also somewhat surprised that they do not use ‘banker’s rounding’ (google it). Not using BR adds an upwards bias with a large amount of data, which is why banks do use it.

Earthling2
October 15, 2017 1:54 am

Banker’s Rounding would sure explain a .5 degree increase in global temperature the last 150 years. Given that thermometers then were hardly accurate to even 1 degree reading the scale on the glass 50 years ago, and then depending what your eye level to the thermometer was reading the scale in what were fairly crude weather stations. The 1 decree C global temperature increase the last 150 years claimed by Science must also fall “randomly” within the +/- 0.5 deviation, especially if there is upward bias to do so. So half of all global warming might just be banker’s rounding.

Nick Stokes
October 15, 2017 2:08 am

“Not using BR adds an upwards bias with a large amount of data”
It’s one way of avoiding bias. Any pseudo random tie-break would also do, and that’s probably what they do use if rounding is an issue. But it’s not clear that it is an issue.

Geoff Sherrington
October 15, 2017 2:29 am

Nick,
Here is a BOM comment on rounding and metrication.
http://cawcr.gov.au/technical-reports/CTR_049.pdf
“The broad conclusion is that a breakpoint in the order of 0.1 °C in Australian mean temperatures appears to exist in 1972, but that it cannot be determined with any certainty the extent to which this is attributable to metrication, as opposed to broader anomalies in the climate system in the years following the change. As a result, no adjustment was carried out for this change”
When we are looking at a 20th century official warming figure of 0.9 deg C, the 0.1 degree errors should become an issue. Geoff

Nick Stokes
October 15, 2017 3:00 am

Geoff,
“the 0.1 degree errors”
They aren’t saying that there is such an error. They are saying that there seems to be a small climate shift of that order, and they can’t rule out metrication as a cause, even though they have no evidence that it caused changes.

An awful lot of numbers were converted with variable skill, but those authors have no special knowledge to offer (and say so). I remember my first passport post-metrication; my height was 1.853412 m! At one stage I looked at old news readings in F to check against GHCN (in C); I never found a conversion error.

Greg
October 15, 2017 4:18 am

BR is symmetrical since half of the .5 values get rounded up , the other half get rounded down.

What will introduce a bias is when temperatures were marked in whole degrees by truncation. When and where this was used and stopped being used will introduce a 0.5 F shift if not correctly known from meta data and corrected for.

October 15, 2017 5:20 am

A broader quotation from the BoM document cited by Geoff is:

“All three comparisons showed mean Australian temperatures in the 1973-77 period were from 0.07 to 0.13°C warmer, relative to the reference series, than those in 1967-71. However, interpretation of these results is complicated by the fact that the temperature relationships involved (especially those between land and sea surface temperatures) are influenced by the El Niño-Southern Oscillation (ENSO), and the 1973-77 period was one of highly anomalous ENSO behaviour, with major La Niña events in 1973-74 and 1975-76. It was also the wettest five-year period on record for Australia, and 1973, 1974 and1975 were the three cloudiest years on record for Australia between 1957 and 2008 (Jovanovic et al., 2011).

The broad conclusion is that a breakpoint in the order of 0.1 °C in Australian mean temperatures appears to exist in 1972, but that it cannot be determined with any certainty the extent to which this is attributable to metrication, as opposed to broader anomalies in the climate system in the years following the change. As a result, no adjustment was carried out for this change”

So several years of the wettest, cloudiest weather on record in Australia, linked to two major La Nina events, caused the mean temperature to increase by about 0.1C? And unworthy of adjustment?

Really?

More than 50% of Australian Fahrenheit temperatures recorded before 1972 metrication were rounded .0F. Analysis of the rounding influence suggests it was somewhere between 0.2C and 0.3C, which sits quite comfortably with an average 0.1C warming amid rainy, cloudy climate conditions you’d normally expect to cool by 0.1C.

Corruption of the climate record continued with the 1990s introduction of Automatic Weather Stations. The US uses five minute running averages from its AWS network in the ASOS system to provide some measure of compatibility with older mercury thermometers. Australia’s average AWS durations are something of a mystery, anywhere from one to 80 seconds (see Ken Stewart’s ongoing analysis starting at https://kenskingdom.wordpress.com/2017/09/14/australian-temperature-data-are-garbage/).

Comparing historic and modern temps in Australia is like comparing apples with oranges, both riddled with brown rot.

Clyde Spencer
October 15, 2017 12:59 pm

Jer0me,
There are several rounding schemes that have been invented and many are still in use in specialized areas. However, the argument that makes the most sense to me is that in a decimal system of numbers the sets of {0 1 2 3 4} {5 6 7 8 9} are composed of 5 digits each, and exactly subdivide the interval before repeating. Thus, when rounding, one should round ‘down’ (retain the digit) if any of the digits in the position of uncertainty are in the first set, and one should round ‘up’ (increment the digit) if any of the digits are in the second set.

TimTheToolMan
October 15, 2017 3:55 pm

Not so because you aren’t actually rounding down the zero, its already zero… and so there are actually 4 elements that are rounded downward and 5 elements that are rounded upward so the scheme is asymmetrical and upward biased.

Clyde Spencer
October 15, 2017 8:03 pm

Tim,
No, the digit in the uncertain position has been estimated as being closer to zero than it is to 1 or nine. The zero has a meaning, unlike the absence of a number.

TimTheToolMan
October 15, 2017 11:13 pm

Clyde

The zero has a meaning, unlike the absence of a number.

And the meaning is the number you’re rounding to. Think of it this way…out of the set {0,1,2,3,4} in 4 of the 5 cases cases the rounding will produce a downward adjustment. Out of the set {5,6,7,8,9} all 5 of the cases produce an upward adjustment. That cant be a symmetrical adjustment if each of the outcomes is equally probable.

Don K
October 14, 2017 11:51 pm

“In scientific literature, we might see this in the notation: 72 +/- 0.5 °F. This then is often misunderstood to be some sort of “confidence interval”, “error bar”, or standard deviation”

The confusion is understandable? It’s been sixty years, but I’m quite sure they taught me at UCLA in 1960 or so that the 72 +/- notation is used for both precision based estimates and for cases where the real error limits are somehow known. It’s up to the reader to determine which from context or a priori knowledge?

I’d go of and research that, but by the time I got an answer — if I got an answer — this thread would be long since dead. Beside which, I’d rather spend my “How things work time” this week trying to understand FFTs.

Anyway — thanks as usual for publishing these thought provoking essays.

Nick Stokes
October 14, 2017 11:51 pm

Kip,
You do have over a century of scientific understanding against you. And you give almost no quantitative argument. And you are just wrong. Simple experiments disprove it.

In the spirit of rounding, I took a century of Melbourne daily maxima (to 2012, a file I have on hand). They are given to 0.1°C. That might be optimistic, but it doesn’t matter for the demo. For each month, I calculated the average of the days. Then I rounded each daily max to the nearest °C, and again calculated the average. Here are the results:

```Month To 1 dp   To 0 dp   Diff
Jan   26.0478   26.0545   -0.0067
Feb   26.0595   26.0535   0.006
Mar   24.0706   24.0652   0.0054
Apr   20.3757   20.3803   -0.0046
May   16.9282   16.9242   0.004
Jun   14.2591   14.2597   -0.0006
Jul   13.7578   13.7416   0.0162
Aug   15.0923   15.0832   0.0091
Sep   17.4591   17.4493   0.0098
Oct   19.8232   19.8177   0.0055
Nov   22.0112   22.0087   0.0025
Dec   24.2994   24.2966   0.0028
```

As you’ll seen despite the loss of accuracy in rounding (To 0 dp), the averages of those 100 years, about 3000 days, does not have an error of order 1. In fact, the theoretical error is about 0.28/sqrt(3000)= 0.0054°C, and the sd of the differences shown is indeed 0.0062. 0.28 is the approx sd of the unit uniform distribution.

October 15, 2017 12:10 am

Brilliant example Nick.
This diproves Kip’s claim
Jan

Greg
October 15, 2017 12:52 am

What Nick’s example shows is that rounding error is approximately gaussian ( normally ) distributed , contrary to Kip’s assertion.

That is only one very small part of the range of problems in assessing the uncertainty in global means. Sadly even this simple part Kip gets wrong from the start. The article is not much help.

Nick Stokes
October 15, 2017 1:15 am

“that rounding error is approximately gaussian”
Actually, there’s no requirement of gaussian. It just comes from the additivity of variance Bienayme. If you add n variables, same variance, the sd of sum is σ*sqrt(n), and when you divide by n to get the average, you get the 1/sqrt(n) attenuation.

Greg
October 15, 2017 4:35 am

Thanks Nick. That article refers to “random” variables, how is that different to normally distributed?

“of the same variance” is also key problem in global temps since SST in different regions do not have the save variance. That is without even talking about about the illegitimate mixing with land temps which vary about twice a quickly due to lesser specific heat capacity and is why you can not even add them to sea temps, let alone the rest of the data mangling.

You can not play with physical variables a freely as you can with stock market data.

Nick Stokes
October 15, 2017 9:51 am

Greg,
“That article refers to “random” variables, how is that different to normally distributed?”
Random variables can have all manner of distributions. Gaussian (normal), Poisson, uniform etc.

” is also key problem”
Same variance here just simplifies the arithmetic. The variances still add, equal or not.

AndyG55
October 15, 2017 8:12 pm

Well done Nick.

You have also highlighted your lack of comprehension of basic maths 🙂

“n” readings of +/- 0.5 uniformly distributed between 0 and 1.

Standard deviation is INDEPENDENT of “n”

“n” readings +/- 0.5 uniformly distributed from any 1 unit group eg (between 17.5 & 18.5)

And suddenly you think the standard deviation becomes dependent on “n”? Really ?????

Do you want to think about that…………… just once?

No probably not. Just keep trotting out your statistical gibberish.

Nick Stokes
October 15, 2017 8:53 pm

“And suddenly you think the standard deviation becomes dependent on “n”? “
Where did I say that? The argument here is about standard error of the mean. Which is also related to the standard deviation of a set of realisations of the mean.

I think you’re out of your depth here, Andy.

Don K
October 15, 2017 1:09 am

Nick. I’m sure you’re right. But, Kip has a point also. If I take a cheap Chinese surveying instrument that measures to the nearest 10cm and measure the height of the Washington Monument (169.046 m), I’m probably going to get an answer of 169.0m and averaging a million measurements isn’t going to improve whatever answer I get. (As long as the monument refrains from moving? Can I improve my measurement by jiggling my measuring instrument a bit while making a lot of observations?)

I’m not quite clear on the what the difference is between the two situations. Or even whether there is a difference.

Nick Stokes
October 15, 2017 1:25 am

Don K,
“I’m not quite clear on the what the difference is between the two situations.”
Mark Johnson has it right below. The difference is that one is sampling, and sampling error is what matters. In any of these geophysical situations, there aren’t repeated measures of the same thing. There are single measures of different things, from which you want to estimate a population mean.

So why do measurement errors attenuate? It is because for any of those measures, the error may go either way, and when you add different samples, they tend to cancel. In Kip’s 72F example, yes, it’s possible that the three readings could all be down by 0.5, and so would be the average. But it’s increasingly unlikely as the number of samples increases, and extremely unlikely if you have, say, 10.

Don K
October 15, 2017 7:08 am

Thanks for trying Nick. As I say, I’m sure you are correct. But I also think Kip is probably correct for some situations. What I’m having trouble with is that it appears to me there are not two fundamentally different situations, but rather two situations connected by a continuous spectrum of intermediate situations. So, I’m struggling with what goes on in the transition region (if there is one) between the two situations. And how about things like quantization error? As usual, I’m going to have to go off and think about this.

TimTheToolMan
October 15, 2017 4:02 pm

Don K writes

But I also think Kip is probably correct for some situations.

Situations where there was a bias involved in the measurements for example…

Nick Stokes
October 15, 2017 4:08 pm

“Situations where there was a bias involved in the measurements”
No, Kip’s examples have nothing about bias. He said so here. You don’t see examples like this involving bias. They aren’t interesting, because once stated, the solution is obvious; remove or correct for the bias. There’s nothing else.

TimTheToolMan
October 15, 2017 7:16 pm

Nick writes

They aren’t interesting, because once stated, the solution is obvious; remove or correct for the bias.

Fair enough from Kip’s later comment but practically speaking you cant easily say you have no bias in your measurements especially in measuring something as complex at GMST or GMSL.

Peter Sable
October 15, 2017 9:40 pm

But I also think Kip is probably correct for some situations.

He’s correct for the situation which he carefully prepares above. If the signal you are sampling never deviates beyond the resolution of the instrument, you are stuck with the resolution of the instrument.

Fortunately for your sound system and for temperature averages, the signal does deviate over time by more than the resolution, and thus you can get an accuracy greater than that of the resolution of the measurement instrument by averaging together multiple measurements.

Your sound system in your stereo (unless you are an analog nut) samples at 10s of Mhz frequencies using a 1-bit D/A (or A/D) and then “averages” down the signal to 192Khz giving you nice 24 bit sound at 20Khz. At least, that’s how the Burr-Brown converter in my expensive pre-amp works. I also helped design such systems…

Peter

(I put “averages” in quotes because it’s more sophisticated than that. In fact they purposefully introduce noise to force the signal to deviate by more than the resolution. The “averages” the climate folks use are boxcar averages which is probably the worst choice for a time series…

Peter Sable
October 17, 2017 9:02 am

Peter ==> If only they were finding the means for “water level at the Battery at 11 am 12 Sept 2017” they would get wonderfully precise and accurate means for that place and time with a thousand measurements. Digitizing music doesn’t attempt to reduce the entire piece of music to one single precise note.

That’s an argument that the average sea level over some long period of time is not physically meaningful.

That’s a different argument than what you discuss in the above article.

As far as music, the single precise note is sampled thousands of times at low resolution and then averaged in a way that is physically meaningful to your ear. That was my point. If you want to argue that averaging the entire musical piece is not meaningful, well, I would agree with you. But I wouldn’t argue about the precision of that average, I would just argue that it’s not meaningful…

Peter

Peter Sable
October 18, 2017 3:24 pm

The attempt to use thousands of six-minute means to arrive at a very precise monthly mean is like reducing an entire piece of music to a single precise note — it is only the precision claimed that is meaningless — it is possible to get a very nice useful average mean sea level within +/- 2cm or maybe double that +/-4 cm with all other variables and source of uncertainty added in.

It’s not quite so black and white. Consider music. If I averaged out the 10-20Khz part of the signal I would certainly lose musical quality (although someone with hearing loss might not notice), but I would improve the precision at 100Hz). I would still be able to hear and calculate the beats per minute of the music, for example.

The same issue if I was trying to detect tides. If I average over 48 hours or monthly I’m not going to see the tides in my signal since the tides are ~6 hours peak-trough.

If I’m interested in how the sea level is changing from decade to decade, however, averaging to a yearly level is perfectly reasonable, and you actually gain precision in doing so, since all the small perturbations are averaged out and additionally you trade decreased time precision for increased sea level precision. This is where we seem to disagree, and I’ll stand on 25 years of engineering experience (including as an engineer designing calibration equipment for electronics), plus can provide textbook references if you want. The Atmel data sheet I provided in a post above is one example.

I think however that small long term changes in the average the surface temperature over the planet is not physically relevant. For the global average, I can change the time axis for an X-Y axis (making this a 3-D problem) and the above analysis about averaging and trading time precision for temperature precision applies – it’s just not physically relevant. The average global temperature in combination with time is not really physically relevant (as opposed to the monthly average temperature in the El Nino region IS physically relevant). I’d refine that argument and say 1degC change for global temperatures is not physically relevant, but 10degC likely is. (-10degC is an ice age).

I also believe there’s an issue with measuring long term temperature trends that only a few have addressed. From Nyquist we know that we cannot see a signal with a period greater than sample rate / 2, but few people realize Nyquist is symmetrical. We cannot see signals with a frequency LOWER than the window length / 2.
So for example in a 120 year temperature record we cannot resolve anything longer than 60 year cycles. And it’s actually worse than this if you have multiple overlapping long cycles like say for example PDO and multiple friends out of phase with each other… (Numerical analysis suggests 5 cycles required, which also corresponds to the normal oversampling rate on digital oscilloscopes for similar reasons, based on professional experience). I’d like to see a temperature record of 350 years before drawing strong conclusions about long term climate trends….

Peter

Phil
October 15, 2017 2:06 am

For a time series, an “average” is not an average. It is a smooth or a filter. When you “average” 30 days of temperature readings to obtain a monthly “average,” you are applying a 30-day smooth to the data by filtering out all wavelengths shorter than 30 days. It is a filter, not an average. Dividing by the square root of n does not apply to smooths. You know better. You are very knowledgeable. What you are doing in your chart is comparing two different ways to do a smooth. Again, it is not an average. The only way that you can apply the square root of n to claim an improvement in measurement uncertainty is if each measurement were of the same thing. However, every day when you take a temperature reading, you are measuring a property that has changed. You can take an infinite number of readings and the smooth of such readings will have the same uncertainty as the most uncertain of the readings. You do not get the benefit of claiming a statistical miracle. The problem arises by treating a time series as if it consisted of a collection of discrete measurements of the same thing. The average temperature of January 1 is not an estimate of the “average temperature” of the month of January. Same goes for each day of January. You do not have 30 measurements of the “average temperature” of January!

Nick Stokes
October 15, 2017 2:15 am

“You do not have 30 measurements of the “average temperature” of January!”
No. I have 100. Each year’s 31-day average is a sample of a population of January averages. And they are literally averages; they do have filter properties too, though that is more awkward. But filtering also attenuates noise like measurement error or rounding.

Phil
October 15, 2017 2:22 am

Phil
October 15, 2017 2:29 am

Each year’s 31-day average is a sample of a population of January averages.

NO. Each January is a smooth of something different. You are not taking one hundred measurements of a single hole’s diameter, so that you can divide by the square root of 100 and claim that you have an improved uncertainty of the diameter of that single hole. You are taking 100 measurements of the diameter of 100 different holes, because each January is different, so you do not get the benefit of dividing by the square root of 100.

Nick Stokes
October 15, 2017 2:45 am

Phil
“When you remove terms of an equation, you are discarding information. So, in effect, you are claiming improved uncertainty by discarding information!”

Of course averaging discards information. You end up with a single number. Anyone who has lived in Melbourne will tell you that the average Jan max of 26°C is not a comprehensive description of a Melbourne summer. It estimates an underlying constant that is common to January days. In Fourier terms, it is the frequency zero value of a spectrum. But by reducing a whole lot of information to a single summary statistic, we can at least say that we know that one statistic well.

Phil
October 15, 2017 2:47 am

Let me put it another way. You have a hole whose diameter is changing continuously. Measuring the diameter 100 times does not improve your uncertainty as to the diameter of the hole, because each time you measured it, the diameter had changed. When you apply a 30-day smooth to the series of diameter measurements, you are simply reducing the resolution of your time series data. This may be helpful in determining if the hole is getting bigger or smaller, but it does not improve the uncertainty of each diameter measurement, because each time you measure you are only sampling it once, so you have 100 measurements of sample size n, where n=1. You can only divide by the square root of 1. You cannot claim that your uncertainty is improved. You need to treat the series of measurements as a time series and only use statistical theorems appropriate for time series. Using statistical theorems applicable to non-time series data on time-series data will provide (respectfully) spurious results.

Nick Stokes
October 15, 2017 3:07 am

Phil
“You have a hole whose diameter is changing continuously.”
Well, an example is the ozone hole. We can check its maximum once a year. And as years accumulate, we have a better idea of the average. There it is complicated by the fact that we think there may be secular variation. But even so, our estimate of expected diameter improves.

Phil
October 15, 2017 3:23 am

(the average Jan max) estimates an underlying constant that is common to January days.

Again, most respectfully, no. The average of Jan max is not an underlying constant. You may claim that the average of Jan max is a constant, but, in reality, the temperature is continuously changing. You may claim that the filtered data that you call “the average of Jan max” is not significantly different from zero from year to year based on certain statistical tests, but you cannot pretend that “the average of Jan max” is a constant. Temperature is changing continuously.

Of course averaging discards information. You end up with a single number.

Please do not confuse issues. Averaging (dividing the sum of 100 measurements by 100) 100 distinct measurements of a hole whose size does not change does not discard any information. In that instance, you can claim that you can improve on the uncertainty of just measuring it once, by dividing by the square root of 100. “Averaging” (dividing the sum of 100 sequential data points by 100) 100 measurements of a hole whose size is changing continuously is a mathematical operation on a time series called smoothing. The result is not the mean of a population. It is a filter which removes certain wavelengths and thus discards information. Although, the computational steps bear great similarity, the two operations are quite distinct mathematically and I think you know that.

…by reducing a whole lot of information to a single summary statistic, we can at least say that we know that one statistic well

Once again, I respectfully disagree. How well you know that “single summary statistic” depends not only on how you reduce the information but also on the nature of the information that you are reducing. When the “whole lot of information” consists of time-series data, and what you are measuring is changing from measurement to measurement, then you cannot claim that you “know” the “single summary statistic” any better than you know the least certain data point in the series of data points that mathematical operations are being performed on, because each time you measure this continuously changing thing, you are only measuring it once. The only exception I can think of is in certain high quality weather stations where three sensors are installed and temperature is measured simultaneously by all three. At those particular weather stations and ONLY at those particular weather stations can it be claimed that the sample size, n, is greater than 1. At those stations and ONLY at those stations is it appropriate to divide the uncertainty of the sensor by the square root of 3 to obtain an improved uncertainty of each temperature measurement by the system of three sensors at each particular time of measurement.

Phil
October 15, 2017 3:43 am

Well, an example is the ozone hole. We can check its maximum once a year. And as years accumulate, we have a better idea of the average. There it is complicated by the fact that we think there may be secular variation. But even so, our estimate of expected diameter improves.

Let’s assume that each time the ozone hole is measured, the uncertainty of that measurement is, for the sake of argument, plus-or-minus one square mile. You cannot “average” the historical maximum ozone hole measurements and claim that you know the size of the ozone hole with an uncertainty less than the hypothetical plus-or-minus one square mile. You do not have a better idea of the average maximum ozone hole size as the years “accumulate.” As the years accumulate, the characteristics of the filter that you are using change so that for 10 years of history, you may reduce that to one statistic that would be a 10 year smooth, discarding all wavelengths shorter than 10 years in length. When you have 20 years of history, you may reduce that to a different statistic that would be a 20 year smooth, discarding all wavelengths shorter than 20 years in length, but the uncertainty of each smooth would remain the same at the hypothetical one square mile.

Clyde Spencer
October 15, 2017 1:12 pm

Phil,
You said, “When you remove terms of an equation, you are discarding information.” I totally agree. An easy way to demonstrate this is to plot the daily temperatures and also plot the monthly temperatures and compare them. If one calculates the standard deviation of the annual data, I would expect that the standard deviation would be larger for the daily data than for the monthly data. Also, I would expect the daily data to have a larger range.

NeedleFactory
October 15, 2017 2:26 am

I set about to disprove Kip’s assertion, using Mathematica, and found a satisfying (to me) proof.
Then I read the comments, and found the above comment by Nick Stokes.
Although I am a warming skeptic, and Nick (I think) is not, I must concur with Nick.
Since he said it well, I’ll not bother to discuss my simulation — it’s quite trivial.

tty
October 15, 2017 4:15 am

Did you check the source code in Mathematica first? Did you even read (and understand) the manual thoroughly? Statistics/mathematics packages embody a whole lot of assumptions that the average user is almost never aware of. A lot of the bad statistics around these days are due to the fact that most people never actually learn the underlying theory any longer. They just follow the recipe without knowing if they have the right ingredients.

Peter Sable
October 15, 2017 9:45 pm

I did the same 2 years ago using Matlab. And since I’ve saved companies \$millions by using statistics, I’m quite confident in the source code..

(I was actually checking to see what the result of auto-correlation was for space-based averaging, such as what Berkeley Earth uses. They underestimate the std deviation by about 2.5x because they don’t take this into account… there’s also other issues with BE (their algorithm for determining whether to infill is likely too sensitive) but I digress)

tty
October 16, 2017 6:30 am

You would be surprised how many people have not the slightest idea what autocorrelation is, though it is hard to think of any kind of climate data that are not autocorrelated.

Old England
October 15, 2017 2:38 am

Nick, what you need to explain to me is how any treatment of data removes the original uncertainty – because whatever number you come up with it is still bound (caveatted) by the original +/- 0.1 deg or whatever the original uncertainty is; i.e. in your example 0.2 deg C.

And remember in the series you have used that most of the numbers had a +/- 1 deg F before BOM played with them to reach temperatures to 4 decimal places from a 2 deg F range that must still apply.

Geoff Sherrington
October 15, 2017 3:00 am

Nick,

Remember that a disproportionate number of original temperature readings were taken to the nearest whole degree F. If they later got some added figure after the decimal because of conversion from F to C, by dropping these off again for your exercise you are merely taking the data back to closer to where it started. Even post-decimal, If you think of a month when all the original observations were in whole degrees, you are merely going in a loop to no effect. It is unsurprising that you find small differences.
To do the job properly, you need to examine the original distribution of digits after the decimal.
………
But you are missing a big point from Kip’s essay. He postulates that observations of temperature need not follow a bell-shaped distribution about the mean/median or whatever, but are more often a rectangular distribution to which a lot of customary statistics are inapplicable. I have long argued that too much emphasis has been put on statistical treatments that do more or less follow normal distributions, with too little attention to bias errors in a lot of climate science.

Early on, I owned an analytical chemistry lab, a place that lives or dies on its ability to handle bias errors. The most common approach to bias detection is by the conduct of analyses using other equipment, other methods with different physics, like X-ray fluorescence compared with atomic absorption spectrometry compared with wet chemistry with gravimetric finish. In whole rock analysis the aim is to control bias so that the sum of components of the rock specimen under test is 100%. Another way to test accuracy is to buy standard materials, prepared by experts and analysed by many labs and methods, to see if your lab gives the same answer. Another way is it be registered with a quality assurance group such as NATA which requires a path to be traced from your lab to a universal standard. Your balance reports a weight that can be compared with the standard kilogram in Paris.
Having seen very little quality work in climate science aimed at minimising of bias error and showing the trace to primary standards, one might presume that the task is not routinely performed. There are some climate authors who are well aware of the bias problem and its treatment, but I do wish that they would teach the big residual of their colleagues to get the act right.
It will be a happy future day when climate authors routinely quote a metrology measurement authority like BIPM (Bureau of Weights and Measures, Paris) in their lists of authors. Then a lot of crap that now masquerades as science would be rejected before publication and save us all a lot of time wading through sus-standard literature to see if any good material is there.
Don’t you agree? Geoff.

Nick Stokes
October 15, 2017 3:18 am

Geoff,
The history of the data here doesn’t matter. It’s about the arithmetic. It’s a data set with a typical variability. If the original figures were accurate, adding error in the form of rounding makes little difference to the mean. If they had been F-C conversion errors, measurement errors or whatever, they would have attenuated in the same way. The exception is if the errors had a bias. That’s what you need to study.

That is the deal with homogenisation, btw. People focus on uncertainties that it may create. But it is an adjunct to massive averaging, and seeks to reduce bias, even at the cost of noise. As this example shows, that is a good trade.

re BIPM – no, that misses the point. As Mark Johnson says elsewhere, it’s about sampling, not metrology.

LdB
October 15, 2017 5:39 am

Nick the question that is being asked badly and you have not answered so I will ask you directly. Can you always homogenize data, and lets fire a warning shot to make you think, both Measured Sea Level and Global temperature are proxies. I have no issue with your statistics but your group has a problem they are missing.

October 15, 2017 9:39 am

+10 you said “metrology”.

Nick Stokes
October 15, 2017 1:46 pm

LdB
“Can you always homogenize data”
The question is, can you identify and remove bias, without creating excessive noise? That depends partly on scope of averaging, which will damp noise and improve the prospects. As to identifying bias, that is just something you need to test (and also to make sure you are not introducing any).

LdB
October 15, 2017 7:12 pm

So basically you have a rather large gap in your science knowledge that you can’t homogenize everything.

Nick Stokes
October 15, 2017 9:24 pm

It simply means that as with any numerical procedure, you have to check if it is working. With temperature homogenisation, that is done extensively, eg Menne and Williams.

LdB
October 19, 2017 7:15 am

I am less worried about the temperature readings than the Tidal gauges. Having seen many situations in which Central Limit Theory fails in signal processing the tidal guage situation does have my alarm bells ringing do you know if anyone has tested it?

Nick Stokes
October 15, 2017 9:40 am

Kip
“Do the simple experiment”
Here is the fallacy in your first case. You can take it that the distribution of each range is uniform, range +-0.5. So the first reading looks like this:

The variance is 1/12. But when you take the sum of 71 and 72, the probabilities are convolved:

The range is +-1, but the variance is 1/12+1/12=1/6. When you sum all of them, the distribution is convolved again (with the running mean) and is

The range is now +-1.5, and the variance 1/4. To get the average, you divide the x-axis by 3. That brings the range back to +-0.5, but the variance is now 1/36. The range is theoretically 1, but with very small probabilities at the ends.

You can see that the distribution is already looking gaussian. This is the central limit theorem at work. The distribution gets narrower, and the “possible” range that you focus on becomes extreme outliers.

October 15, 2017 9:41 am

Kip – I think we have to keep pulling people back to considering the REALITY of what the ORIGINAL measurement purports to quantify.

Bartemis
October 15, 2017 11:43 am

Nick is correct. It is well established statistical theory that averaging of quantized signals can improve accuracy. The usual model is, however, based upon several assumptions. Rather than flat out rejection of the efficacy, which is well established, a counterargument should focus on the assumptions, and whether they are satisfied.

Quantization Noise Assumptions:

1) the data are homogeneous
2) the measurements are unbiased
3) the underlying signal is traversing quantization levels rapidly and independently

Under these assumptions, one can model quantization as independent, zero mean, additive noise uniformly distributed between -Q/2 to +Q/2, where Q is the quantization interval. The RMS of the error is then Q/sqrt(12). Averaging N samples then reduces the RMS to Q/sqrt(12N), and the averages are reasonably close to being normally distributed for large N. Such averaging is routine in a wide variety of practical applications, and the results in those applications do generally adhere to the model.

To what degree are these assumptions satisfied for the temperature series? Well, the data are not homogeneous, because temperature is an intensive variable. Its physical significance varies with the local heat capacity. And, the likelihood that the measurements are unbiased is vanishingly small, due to the sparse sampling, the ad hoc methods employed to merge them together, and the issue of homogeneity referenced above.

Assumption #3 is, in fact, the only assumption that likely does hold. Thus, I do not see the line of reasoning of this article as being particularly fruitful. It is attacking one of the stronger links in the chain, while the other links are barely holding together.

Greg
October 15, 2017 2:22 pm

Thanks Nick. I like convolutions. Reminds me of a project is did on a scanning spectrometer once. Fine slits in the collimator give a sharp spectral resolution but sometimes light levels are too weak an you need to open up to wider slits. This convolutes the spectral peaks ( scanning wavelength ) with your first graph and causes broadening, losing resolution. In fact both inlet and outlet slits are finite leading to something like your second graph.

If the slits are not equal this leads to an isosceles trapezoid form convoluted with the scan signal. The fun bit is to try to deconvolute to recover the original resolution. 😉

The third one is quite a surprise. It’s obviously distorted but remarkably bell shaped. This implies that a three pole running mean would be a fairly well behaved filter, even with same window length each time, as opposed to the asymmetric triple RM I show here:

https://climategrog.wordpress.com/gauss_r3m_freq_resp/

Nick Stokes
October 15, 2017 6:57 pm

Kip,
“You may have as precise a MEAN as you wish, but you may not ignore the original measurement accuracy.”
“The question and answer verify that both the individual 1-second measurements and the 6-minute data value represents a range of water level 4 cm wide, 2 cm plus or minus of the value recorded.”</i
That wasn't what they said at all. They spelled it out:
“Sigma is the standard deviation, essential the statistical variance, between these (181 1-second) samples.”
That is statistical. It says that the probability of being in that range is 66%. You can’t get away from probability in that definition. And the probability reduces with averaging.

And you have tthe wrong measurement accuracy. The only thing here that could be called that is the 1mm resolution of the instrument. The +-20mm is a statistical association between the water inside and that outside. It is a correlation, and certainly is capable of being improved by better sampling.

Tom Halla
October 15, 2017 7:03 pm

i don’t think one can determine sea level to closer than 20mm, as at least with the Pacific, there is always more chop than that.

Nick Stokes
October 15, 2017 9:52 pm

Kip,
Yes, it’s clear in that expanded string that their reference to sigma was to the variation of six-minute readings, not the 2cm estiamte. Sorry to have misunderstood that. But I still think the use of the 2cm (or 5mm) measures have to be considered as standard deviations. That is normal notation and practice. I see that the NOAA guy didn’t explicitly contradict you on that, but I don’t think he was confirming. Here is what NIST says about theexpression of measurement uncertainty:

Standard Uncertainty
Each component of uncertainty, however evaluated, is represented by an estimated standard deviation, termed standard uncertainty with suggested symbol ui, and equal to the positive square root of the estimated variance”

RW
October 16, 2017 9:11 pm

Nick’s examples are exercises in the world of sampling statistics where his probabilities are fixed to some theoretical distribution (he’s using the normal distribution a lot) the parameters of which depend massively on the size of each sample. That should the first clue into how ‘magic’ the error reduction gets when he increases the size of the sample. Also enlightening is that these kinds of exercises assume you take the same-sized sample each time. The next step in his lecture series should now be on how one comes up with the parameters for a distribution of sample means when the sizes of the samples differ. At least that will make this more applicable to the tidal gauge / temp measure issue. But one still cannot overcome the limits of the observation. Kip doesn’t care about sampling error.

Nick Stokes
October 16, 2017 9:32 pm

“he’s using the normal distribution a lot”
No, I’m not at all. The only theoretical number I used was the sd of the uniform distribution (sqrt(1/12)). It’s true that the central limit theorem works, but I’m not relying on it. It’s just using additivity of variance, and that will work equally for different sized samples, and for non-normal distributions.

RW
October 16, 2017 11:46 pm

Nick. When one invokes the central limit theorem one invokes the Normal distribution, because the latter is used to approximate the distribution of sample means. The standard deviation of that distribution gets smaller and smaller as you increase the size of each sample, and the shape of that distribution will look more and more Normal as you increase the number of samples. In a single sample, the standard error of the mean (SEM) is a sample estimate of the standard deviation described above. It is different from the sample standard deviation (SD). By itself, the SEM is a long-run estimate of the precision of the means of lots of samples. Both the SD and the SEM vary from sample to sample due to, at the very least, random sampling error. Under ideal circumstances, the sample mean is an unbiased estimator of the population mean. Under those circumstances, the sample mean will still not hit the population mean (because random sampling error), snd the SEM provides an expectarion of how closely the sample.meams should cluster together if you took a pile of additional samples each the same size as the first. Again. Precision. The mean of a sampling distribution of means will equal the population mean if both are distributed Normal. The central tendency theorem is invoked to assume that the distribution of sample means is Normal, even though the samples are drawn from a population that is non-Normal. This sets up valid null hypothesis tests that concern the means of sampling distributions of means and, say, a single sample.mean. It does not necessarily allow for unbiased estimation of the population mean using the mean of the sampling distribution of means, let alone our lonely single sample mean. So you are invoking the Normal distribution, a lot, when you refer tonthebcentral limit theorem. You’re dealing with sampling distributions.

Nick Stokes
October 17, 2017 12:14 am

“Nick. When one invokes the central limit theorem”
But I didn’t. I observed its effect. All my example did was to take means of various samples of daily maxima, and compare with the identically calculated means of data that had been rounded to integer values. No assumptions. The differences were small. I compared them to what is expected by additive variance (it matched) but that is not essential to the conclusion. I showed that the difference in means was nothing like the 0.29°C effect of rounding on individual data.

But in all this, I haven’t heard you support Kip’s view that measurement error passes through to the sample mean without reduction. You do say that it somehow adds differently, but don’t say how. How would you calculate the effect of observation uncertainty on the mean?

Darkwing
October 15, 2017 7:04 am

But this confuses the issue completely. The posting is not about removing the error from rounding, but from uncertainty in measurement. Your argument is utterly irrelevant to the question at hand.The post is addressing the physical fact that using a ruler that only measures accurately in millimetres twice won’t make it give you a measurement in picometers. You can’t use a high school ruler a million times to measure the size of an atom. Measurement accuracy does not improve with repeated samples.

October 15, 2017 9:44 am

o/t.personal comment.
you are not the _08 guy are you?

Probably not, just coincidence, but if you are you will know who I am (7).

Darkwing
October 15, 2017 10:42 am

If that’s directed at me I’m a frayed knot.

Jim Gorman
October 15, 2017 2:13 pm

+1

RW
October 16, 2017 9:19 pm

Darkwing gets it. Nick is just obfuscating.

Ray in SC
October 15, 2017 7:22 am

Nick,

You are missing Kip’s point. His assertion is that your January reading should be 26.0478 +/- 0.1.

Nick Stokes
October 15, 2017 9:43 am

+-0.05, I think. And he would assert that after rounding it should be +-0.5. But it clearly isn’t. I actually did it, for 12 different months. And nothing like that error is present in the means.

Darkwing
October 15, 2017 10:40 am

Stokes

You’re still missing the point. Why would the error be present in the means? There is no there there to begin with, in the means or otherwise. How can you say something is or isn’t present if it was never measured in the first place?

We are not discussing errors in means, we are discussing errors in measurement.

ATheoK
October 15, 2017 9:52 am

“Nick Stokes October 14, 2017 at 11:51 pm
Kip,
You do have over a century of scientific understanding against you. And you give almost no quantitative argument. And you are just wrong. Simple experiments disprove ”

Pure hand waving, Nick.

Explain how century old temperatures, eyeball read from mounted shaded thermometers can be added to modern, never certified or recertified for accuracy, temperature thermistors?

Then an alleged average calculated out to four decimal places? Which by sheer absurdity only appears accurate.
e.g. Jan maxima average is 26°C, period.

Calculation of an alleged four decimal place version and/or difference does not represent greater accuracy than January’s 26°C.
It is all pretense, not reality.

Then you want everyone to accept that mishandling a Century of data accurately represent the entire and all potential weather cycles?

Hand waving, Nick.

Nick Stokes
October 15, 2017 10:09 am

“Hand waving”
No, it’s an introduction to a concrete example with real data.

ATheoK
October 15, 2017 7:03 pm

“Nick Stokes October 15, 2017 at 10:09 am
“Hand waving”
No, it’s an introduction to a concrete example with real data.”

Real data!?
You call four decimal place numbers from “0.n” maximum 1 decimal place physical measurements, “real data”?

That claim is a mathematical shell game using an imaginary pea.
Yes, you are hand waving.

Nick Stokes
October 16, 2017 1:10 am

“You call four decimal place numbers from…”
No, I call them calculated results. I need the decimals to show what the difference is. But the robustness of the calculation. To at least two decimals, you get the same result if you reduce data from 1 dp to 0dp.

ATheoK
October 16, 2017 9:05 pm

“Nick Stokes October 16, 2017 at 1:10 am
“You call four decimal place numbers from…”
No, I call them calculated results. I need the decimals to show what the difference is. But the robustness of the calculation. To at least two decimals, you get the same result if you reduce data from 1 dp to 0dp.”

You claim false value for your imaginary four decimal places.
Nor can you prove four decimal place value when using integers and single decimal place recorded numbers as data.

You use “robustness” just as the climate team does when they’re skating bad research or bad mathematics past people.

October 15, 2017 12:42 pm

Get an eight foot pole that has markings at 1,2,3….8 feet.
..
Use this pole to measure 10,000 adult American males randomly selected. Each measurement is to the nearest foot.

When you sum all the measurements it will be roughly 58300 to 58400.

When you divide the sum by 10,000, you’ll get 5.83 to 5.84

Congratulations, you just measured the average height of an American male to less than the nearest inch. Pretty amazing considering your pole only has markings at one foot intervals!!!

Clyde Spencer
October 15, 2017 1:38 pm

Mark S Johnson,

Well if you have any stock in companies that manufacture highly accurate and highly precise measuring instruments you had better sell it. You have just let the cat out of the bag that anyone can get by with much cheaper, crude instrumentation if they just measure 10,000 samples.

Based on your remarks, I don’t believe that you have read my article that preceded the one Kip cited. Let me then share a quote from it:
“Furthermore, Smirnoff (1961) cautions, ‘… at a low order of precision no increase in accuracy will result from repeated measurements.’ He expands on this with the remark, ‘…the prerequisite condition for improving the accuracy is that measurements must be of such an order of precision that there will be some variations in recorded values.’” But, most importantly, you must be measuring the same thing!

October 15, 2017 1:56 pm

Again Clyde, you post: “you must be measuring the same thing”

I posted: ” 10,000 adult American males”

See the difference?…….

October 15, 2017 9:04 pm

Clyde Spencer: “Well if you have any stock in companies that manufacture highly accurate and highly precise measuring instruments you had better sell it. You have just let the cat out of the bag that anyone can get by with much cheaper, crude instrumentation if they just measure 10,000 samples.” When you are measuring the height of only one person, 10,000 samples are going to agree, and be up to 6 inches off with 95% chance of being up to 5.7 inches off when done with Mark S. Johnson’s 8-foot pole with perfect calibration and resolution of 1 foot. But if you are looking for an average height among 10,000 persons, Mark S. Johnson’s measuring pole can determine that with a much smaller +/- with 95% confidence. And if Mark S. Johnson’s pole has all of its markings being incorrect by the same amount or the same percentage, it can still be used to track growth or shrinkage of a large random population to the nearest inch if that changes by more than an inch, with high confidence.

Greg
October 15, 2017 11:25 pm

Is your claim that measurement accuracy means nothing if you just have enough numbers to churn?

It is a question of quantisation or resolution, ie precision, not accuracy. You should not use the two terms interchangeably. They have precise and different meanings.

It is not that the precision “means nothing” but less precision can be compensated by more readings.

TimTheToolMan
October 16, 2017 4:52 am

Mark S Johnson writes

Congratulations, you just measured the average height of an American male to less than the nearest inch.

Except you’re an inch out on the true average and you couldn’t do it at all if the markings were at 3 foot intervals. You seem to want to ignore the measurements themselves when arguing how accurate you can be. Its a fatal mistake.

Slacko
October 16, 2017 10:29 pm

Mark S Johnson October 15, 2017 at 12:42 pm
“Congratulations, you just measured the average height of an American male to less than the nearest inch. Pretty amazing considering your pole only has markings at one foot intervals!!!”

What’s even more amazing is that you also got the height of Australian males to the nearest inch. I’m really impressed.

Peter Sable
October 18, 2017 6:08 pm

Well if you have any stock in companies that manufacture highly accurate and highly precise measuring instruments you had better sell it.

Too late. I helped developed such a system in 1995 at an electronics test and measurement company. The technique was developed many decades before that but only became economically viable in the 1990s due to the newer CMOS manufacturing capabilities.

Currently I have a Burr-Brown 24-bit ADC (59 ppb precision) with a 1 bit (+/- 50%) sampler in my stereo pre-amp. It sounds so good I run my analog record player through it. In 1995 we were happy to get 18 bits using the same technique for a digital multi-meter.

Your 1-foot interval for the American male population won’t work because the signal (actual heights) doesn’t vary by more than a foot. However, if you want 1/10th of an inch precision then measuring each male to 1-2 inches precision is quite sufficient. Just make sure when you calibrate your stick you calibrate your 1 inch tickmarks to 1/10th of an inch precision.

Peter

October 18, 2017 6:26 pm

Peter Sable says: “Your 1-foot interval for the American male population won’t work because the signal (actual heights) doesn’t vary by more than a foot. ”

Nope, it will work because there are 6foot 3 inch males in the population, and there are 5 foot 2 inch males in the population. There are even some 4 foot 4 inch males and some 7 foot inch ones.

The key fact you don’t understand is that some males will be smaller than 5 foot 6 inches ,and some will be larger. It’s the relative proportion of each that determines the average.

Nick Stokes
October 18, 2017 6:43 pm

I agree with Peter there. Calculating the average is trying to estimate ∫hP(h) dh where h is height, P is the pdf. The coarse ruler is like trying to evaluate the integral with quantiles. You can get a good approx with 1″ intervals, which is less than 1/10th of he range. But when you get intervals close to the range, the integration is likely inaccurate.

Peter Sable
October 18, 2017 6:50 pm

Nope, it will work because there are 6foot 3 inch males in the population, and there are 5 foot 2 inch males in the population. There are even some 4 foot 4 inch males and some 7 foot inch ones.

There aren’t enough in the population sample to span the range of 1 foot. you are right if you happen to know the exact mean of the population you could use a “are you taller or shorter” measurement and estimate the mean from that.

For an analog input signal to a 1-bit DAC it’s possible to know (or rather calibrate) the true mean of the population and then the proportion gives you sample average as you indicate I don’t think you know that mean a-priori with a population. Also, your population had better have an even distribution. I suspect there are more 6’6″ males in the population than 4’6″ males.

When the variance of the signal approaches the precision of the instrument, then the devil is in the details. We’re talking about 1degC precision with a 10degC diurnal variation, so not apples-apples to your yardstick example.

October 18, 2017 7:09 pm

Nick & Peter….

Sorry to inform both of you, but, the numerical PROPORTION of 5 foot measures to 6 foot measures will contribute the most to determine the average when the sum of the measures is divided by 10,000. There will be some 4-foot measurements, and there will be some 7 foot measurements, but their numbers will be relatively small.

What makes any argument against my “8 foot pole” example fail, is that we know prior to executing my procedure, what the average is. Also known is how height is distributed. With these two facts, you will have a hard time showing my exampple failing.

October 18, 2017 7:14 pm

Peter, the analogy of DAC is inappropriate. DAC sampling does not measure a population mean. It approximates an instantaneous value which is the antithesis of a population value.

Nick Stokes
October 18, 2017 8:47 pm

OK, I tried it, and Mark’s method did still do well, with 1′ intervals. I assumed heights normally distributed, mean 5.83, sd 0.4. Centered, the expected numbers were

```4.5    5.5    6.5    7.5
190    6456   3337    17
```

Weighted average is 5.818, so it is within nearest inch.

Jim Gorman
October 15, 2017 2:01 pm

You are missing the point. What is the uncertainty of each of the daily maxima? Run your averages where the measurements are all at the top of range of uncertainty and then again when they are all at the bottom of the range. Now tell us what the “real” value is. If there are uncertainties, you just can’t assume the middle of the range is the correct reading.

Crispin in Waterloo but really in Beijing
October 16, 2017 9:01 am

Nick, we already went through this once and you haven’t learned how this works.

“As you’ll seen despite the loss of accuracy in rounding (To 0 dp), the averages of those 100 years, about 3000 days, does not have an error of order 1. In fact, the theoretical error is about 0.28/sqrt(3000)= 0.0054°C, and the sd of the differences shown is indeed 0.0062. 0.28 is the approx sd of the unit uniform distribution.”

You are making the same mistake as last time – you are leaving out the uncertainty of the readings, and treating them as if they are gold. You have calculated the centre of the range of uncertainty and called your construct the ‘theoretical error’. The uncertainty of each reading is 20mm up or down and you have shown nothing that reduces it.

You have provided an SD based on the data, but forgot to add the uncertainty for each reading, for which a different formula applies. You are trying to sell the idea that 3000 readings makes the result ‘more accurate’. The accuracy of the result is determined (only) by the instrument, which is why we rate the accuracy of instruments so we can pick one appropriate for the task at hand. You can’t just leave out the instrumental uncertainty because you have 3000 readings. They are 3000 uncertain readings and that uncertainty propagates.

It is a surprise to me that so many contributors do not understand this. Kip wrote it out in plan bold letters: measuring 1000 things once each with an inaccurate instrument does not provide a less-inaccurate result. That is the property of measurement systems – uncertainties propagate through all formulae including the one you show.

Measuring with a plus-minus 20mm tide gauge 1000 times over a 4000mm range does not provide an average that is known to better than plus-minus 20mm because that is the accuracy of the readings. Any claim for a more accurate result is false.

If you used the same equipment to measure the water level in a lake with waves on it, knowing that the level does not change, is a different matter in terms of how stats can be applied because that is taking multiple measures of the same thing with the same instrument. That still wouldn’t increase the accuracy, but the stats that can be applied are different. It certainly wouldn’t make the result more precise either because the precision remains 1mm. Your formula estimates quite precisely where the centre of the error range is located. Nothing more. The ‘real answer’ lies somewhere within that range, not necessarily in the middle as you imply. That is why it is called a “range”.

Nick Stokes
Reply to  Crispin in Waterloo but really in Beijing
October 16, 2017 7:50 pm

Crispin
“You have calculated the centre of the range of uncertainty and called your construct the ‘theoretical error’. The uncertainty of each reading is 20mm up or down and you have shown nothing that reduces it.

You have provided an SD based on the data, but forgot to add the uncertainty for each reading, for which a different formula applies.”
My example was of temperatures in Melbourne. But how do you “add the uncertainty”? What different arithmetic would be done? There seems to be a view that numbers are somehow endowed with original sin, which cannot be erased and has to be carried in the calculation. But how?.

In fact all my example did was to take a set of readings with high nominal precision, sacrifice that with rounding, and show that the average so calculated is different to a small and predictable extent. Any “original sin” derived from measurement uncertainty would surely be swamped by the rounding to 1C, or if not, I could round to 2C, still with little change. If the exact readings could have been optained, they would be a very similar series before rounding, and would change in the same way.

One test of these nonsense claims about irreducible error is to actually calculate a result (protagonists never do) and show the error bars. They will extend far beyond the range of the central values calculated. That does not make nonsense of the calculation. It makes nonsense of the error bars. If they claim to show a range over which the thing calculated could allegedly vary, and it never does, then they are wrong.

RW
Reply to  Crispin in Waterloo but really in Beijing
October 16, 2017 9:27 pm

Nick, the errors at the different levels (observation vs. random sampling) will sum to give you the true estimate of error. If the errors are correlated (unlikely) then they sum but are also influenced by the direction and magnitude of the correlation between them. It is like Kip said, this isn’t typical undergrad stats, unfortunately (which is more a dig at oversimplified undergrad stats).

Nick Stokes
Reply to  Crispin in Waterloo but really in Beijing
October 16, 2017 9:39 pm

“Nick, the errors at the different levels (observation vs. random sampling) will sum to give you the true estimate of error. “
So how would you sum the observation errors? Say they amount to 0.5C per observation. Why would that sum differently than, say, 0.5C of rounding?
Kip wants to say that 0.5C observation error means 0.5C error in mean of 1000 observations. Do you believe that?

Tom Halla
October 16, 2017 9:44 pm

No, Nick, Kip Hansen is stating that the average does not mean anything without an error band of .5C., if the data going into the average had that error band.

RW
Reply to  Crispin in Waterloo but really in Beijing
October 17, 2017 12:09 am

Nick. Kip already mentioned it. The errors are essentially fixed, the observations finite and known. Therefore the SD will be +/- 0.5. (was it cm?) Var=(n/n)E{0.5^2}. SD = Var^0.5. This is your first level variance. Sum it with variance from each additional level of estimation. With all the different sites of measuring water level, each probably exposed to different factors which probably overlap sometimes from site to site, I would guess that sea level would be considered a random effect if this were a meta analysis. Variability (precision) within each site and variability in sea level betweem sites would need to be taken into account as well in order to get the ‘true’ unceetainty in the uber avergage.

Nick Stokes
Reply to  Crispin in Waterloo but really in Beijing
October 17, 2017 12:22 am

RW,
“Var=(n/n)E{0.5^2}”
Do you mean 1/n? I can’t figure the second term, but it sounds a lot like you’re agreeing with Mark Johnson and me that the std error of the mean drops as sqrt(1/n). What you’re saying doesn’t sound at all like Kip’s
” the means must be denoted with the same +/- 0.5°F”

And what do you make of Kip’s insistence that ranges, not moments, are what we should be dealing with?

RW
Reply to  Crispin in Waterloo but really in Beijing
October 17, 2017 9:27 am

Nick. Yes 1/n like you are thinking but because the error is 0.5 for each observation the equation becomes n/n …0.5^2 ‘n’ times…i just pulled the n out of the summation (‘E’) per summation rules to make it easier for you to see thay it has no effect at that level. We are back to what Kip said originally. We have also established that the 0.5 +/- is a standard deviation as i think was said by someone already (you?).

The SEM is not SD/(n-1)^0.5 as someone else wrote, it is simply SD/n^0.5 . The n-1 only comes with the calculation of sample variance. Here, we use n for variance because we have the population of observations. We are not generalizing to a population of observations.

Nick Stokes
Reply to  Crispin in Waterloo but really in Beijing
October 17, 2017 9:44 am

“because the error is 0.5 for each observation the equation becomes n/n …0.5^2 ‘n’ times…i just pulled the n out of the summation (‘E’) per summation rules to make it easier for you to see thay it has no effect at that level. “
You’ll need to spell that out in more detail. If you are summing n variances, the summands are, after scaling by the 1/n factor of the average, (0.5/n)^2. So the thing in front should be (n/n^2).

As for “We are back to what Kip said originally.”, no, Kip is very emphatic that 0.5 is not a sd, and we should not think of probability (what else?):
“In scientific literature, we might see this in the notation: 72 +/- 0.5 °F. This then is often misunderstood to be some sort of “confidence interval”, “error bar”, or standard deviation.”

Nick Stokes
Reply to  Crispin in Waterloo but really in Beijing
October 17, 2017 4:16 pm

Kip,
I think no agreement is possible because you reject probability as a basis for quantifying uncertainty, and I insist there is nothing else. People here like quoting the JCGM guide; here is one thing it says:

3.3.4 The purpose of the Type A and Type B classification is to indicate the two different ways of evaluating uncertainty components and is for convenience of discussion only; the classification is not meant to indicate that there is any difference in the nature of the components resulting from the two types of evaluation. Both types of evaluation are based on probability distributions (C.2.3), and the uncertainty components resulting from either type are quantified by variances or standard deviations.

You like intervals. But
1) meaningful intervals rarely exist in science. Numbers lie within a range as a matter of probability; extremes of any order can’t be ruled out absolutely. If an interval is expressed, it is a confidence interval, perhaps implying that the probability of going beyond can be ignored. But not zero, and the bounds are arbitrary, depending on what you think can be ignored, which may differ for various purposes, and may be a matter of taste.
2) Intervals do not combine in the way you like to think. Science or Fiction set out some of the arithmetic, as did I and others. When you combine in an average, the only way the ends of an interval can stay populated is if all the measures are at that end. So it is one-sided, and takes an extraordinary coincidence.

You don’t have absolutes in science. Heissenberg insists that you might be on Mars. All the oxygen molecules in your room might by chance absent themselves. One does not think about these things because the probabilities are extremely low. But you can’t get away from probability.

The practical problem with your musings is that they describe a notion of uncertainty which is not that of a scientific audience, as the JCGM note shows. So it doesn’t communicate. I also believe that it just isn’t one that you could quantify or use systematically. That is what StatsFolk have learnt to do.

RW
October 16, 2017 8:38 pm

This shows nothing aside from how the number of significant digits you use has little influence on the standard deviation of a sample of sample means (i.e. the standard error of the mean). You are talking inferential sample statistics. All the gains you are referring to combat random sampling error. The post concerns uncertainty in the measurements themselves. These are different things. The former is hugely helped by taking more samples and/it increasing the n in each sample, whereas the latter is not overcome by this.

Nick Stokes
October 16, 2017 9:12 pm

“You are talking inferential sample statistics. All the gains you are referring to combat random sampling error. The post concerns uncertainty in the measurements themselves. These are different things. “
They are. And the post is talking about the wrong one. In climate, many different kinds of measurement are combined. The post imagines that somehow the measurement uncertainty of each aligns, and can be added with no effect of cancellation. It doesn’t explain how.

There may indeed be some alignment; that would create a bias. An example is TOBS. People make great efforts to adjust for changes in that.

TimTheToolMan
October 17, 2017 5:16 am

Nick writes

There may indeed be some alignment; that would create a bias. An example is TOBS.

Another might be how the satellite chases the tidal bulge around the earth when doing sea level measurements such that month averages have biases.

garymount
October 14, 2017 11:54 pm

Is temperature truly infinite in the continuum like time is, or does it have quanta associated with it like radiation?
I found a sample document to read, but I haven’t extensively studied quantum mechanics yet :
TEMPERATURE IN QUANTUM DYNAMICS
ALESSANDRO
SERGI

ABSTRACT
. What is the meaning of the thermodynamical temperature in quantum mechanics? What is its role in the classical limit? What can we say about the interplay between quantum and thermal fluctuations? Can we impose a constant-temperature constraint within dynamical simulations on quantum systems as we do in simulations of classical systems?

https://www.scribd.com/mobile/document/40884849/Temperature-in-quantum-mechanics

LdB