Guest essay by C.R. Dickson
Most people have no trouble relating to temperature, because they use it every day when they set the thermostat in their homes, adjust the temperature dial on an oven, or watch a weather report on TV. On the other hand, practically no one recognizes a temperature anomaly, the yardstick for measuring man-made global warming. That’s because outside of climate studies, no one uses it.
A temperature anomaly is the difference obtained by subtracting an average temperature from real temperature data. Climate studies work with anomalies instead of real temperatures because anomalies are assumed to be more accurate over large geographical areas (see note 1). The rapidly rising graphs of temperature anomalies also conveniently dramatize catastrophic global warming.
So it’s easy to see why a few journalists made a big fuss over a very flat looking graph of average global temperatures posted in a tweet from the National Review. The graph in Figure 1 below (see notes 2 and 3) is like the one displayed in the tweet.
This graph supposedly hides global warming because the small increases in temperatures aren’t obvious. An online article in The Huffington Post stated it was an improper visualization that makes “just about anything seem stagnant,” and The Fix at The Washington Post complained that “it is misleading” because it “hides the actual change in temperatures.” Also online, Business Insider said the graph zooms “out so much that it makes it seem like global average temperatures haven’t changed at all.”
Of course, the journalists decided the temperature graph was up to no good, and they countered with their own graphs of the national debt and the Dow Jones Industrial Averages. It was graph vs. graph on the way to the world’s end.
With a bit of an effort, it’s easy to discover that the temperature changes are identical for both global temperature anomalies and for global temperatures (see note 4). The difference is that the graph of the anomalies is a magnified view, not a normal one.
Magnification doesn’t change the object you are viewing; it just lets you see more details. A blood cell or a microbe doesn’t get any bigger when it’s magnified; it only looks larger.
For example, the normal view of a piece of glass shown in Figure 2 appears to be very smooth. As can be seen in Figure 3, the magnified view has numerous peaks and valleys making the surface look rough, not smooth. Although the imperfections seem larger in the magnified view, they are the same size as in the normal view.

The same thing happens with reconstructed temperatures and temperature anomalies. When you magnify the average global temperatures in Figure 4, the unseen changes become visible, as Figure 5 clearly shows.

Fortunately, people normally do not use a magnified version of the world to proceed with their daily lives. That’s why no one drives down a highway guided by a microscope magnifying the road’s surface. For the same reason, weather forecasters use the real temperatures instead of magnified temperature anomalies.
Because it’s so difficult to observe man-made global warming, some experts at NASA GISS believe the accuracy of climate models requires a one hundredfold increase in order to see the small amount of warming.
“A doubling in atmospheric carbon dioxide (CO2), predicted to take place in the next 50 to 100 years, is expected to change the radiation balance at the surface by only about 2 percent. If a 2 percent change is that important, then a climate model to be useful must be accurate to something like 0.25%. Thus today’s models must be improved by about a hundredfold in accuracy, a very challenging task.”
A paper by Graeme Stephens et al. in Nature Geoscience also shows how hard it is to find global warming. They reported the uncertainty in the earth’s warming imbalance as 0.6 watts per m2 ± 17 watts per m2. The enormously large uncertainty in this very small number means that it is difficult, if not impossible, to observe. Just like NASA said it was!

But how small is this imbalance? It’s only 0.06 percent of the 1,000 watts per m2 of sunlight falling on the earth’s surface at noon. Another interesting comparison is that 0.6 watts per m2 is like a small AA battery discharging over a few hours (see Figure 5). Consider this: Little batteries that turn on televisions do not power hurricanes.
Small numbers with large error bars, combined with excessive averaging, is a recipe for ambiguous results. The reaction to the temperature graph is a perfect example of how political motivations can twist ambiguities into disagreements. Confusion is created by using temperature as if it were the same as an anomaly, but somehow the temperature graph is misleading while the anomaly graph is not. What is hidden is the fact that both graphs display no real temperature data.
Fortunately, unambiguous data is the cornerstone of scientific research. If independent researchers cannot obtain the same answer, then there is something wrong with the data, the experiment, or both. Speculations, theories, and hypotheses come and go in science, but good data lasts forever. That is why catastrophic man-made global warming, like all consensus “science,” will eventually go the way of phlogiston, spontaneous generation, and luminiferous ether.
NOTES:
1. Hansen et al. discusses using anomalies instead of actual temperatures, and there is some limited information on errors. Hansen also complains about talk shows, politics, public perception, and the news media on pages 20-23. Real Climate talks about temperature and anomalies and for additional discussions go here, here, and here.
2. The graph in the tweet showed up in a WUWT comment here. Additional comments led to this site . The graphs in Figures 1, 4, and 5 are in degrees Fahrenheit because that’s what the National Review graph used.
3. The NASA GISS tabulated values were updated in the process of making the above graphs. A large number of historical values were changed without explanation making the tabulated values a moving target.
4. To create temperature anomalies NASA GISS takes real-world temperatures and subtracts a subjective “best estimate for the global mean for 1951-1980,” which is calculated to be 14 degrees Celsius, or 52.7 degrees Fahrenheit. The temperature changes (ΔT) for both graphs are the same because one graph is offset from the other by a constant 52.7 degrees F.
5. The solar irradiance is for AM 1.5 (approximately 48.2 degrees zenith). A value of 3.9 watt hour (14 kilojoules maximum energy) is typical for 1.5 volt AA battery discharging at a 50 mA drain. (0.6 watts / m2) x (6.5 hour) = 3.9 watt hour / m2.
ABOUT:
C. R. Dickson is a retired chemist and physicist with a Ph.D. from Columbia University. He has worked for Polaroid, Allied Chemical, RCA, and the Solarex Thin Film Division, a solar cell company formed as an RCA technology spinoff. He also served as a scientific advisor to the United Nations Industrial Development Organization in Vienna, Austria.
Discover more from Watts Up With That?
Subscribe to get the latest posts sent to your email.

0.6 watts per m^2 ± 17 watts per m^2.
Nobody can see a signal of 0.6 within noise of 17, they just can’t. To pretend otherwise is a farce.
It should be 0.6 plus/minus 0.4 w/m2 (not 17). See figure T10 in the attached link.
http://wattsupwiththat.com/2016/01/10/el-nio-shortens-the-pause-by-just-one-month/
Duncan, incorrect. The number you cite is TOA. The number the post cites is surface. The uncertainty difference is all the stuff in the middle, called Earths atmosphere. Both values from the same Stephens et. al. 2012 paper, main summary figure. Itself a revision of Trenberth and Fasullo’s estimate of 2009.
Not only that, ristvan, if you look again at the Stephens paper, you’ll see that they have misplaced the decimal in the TOA error. It should be ±3.9 W/m^2, not ±0.39 W/m^2.
I sent emails twice to Graeme Stephens asking about that, but he never replied.
PF, I should have caught that myself after all the mileage gotten out of that paper. Yup. TY.
Kelvin based at 0 K is the only correct way to graph Figure 4. The difference between the putative 19th century global mean temperature and the supposed 1 °C warming of the 21st century is about 0.3% on that scale.
Pat Frank,
He explains that in his paper:
“Fluxes leaving Earth at the TOA are also well documented, although inherently less accurate with an uncertainty of ±4 Wm–2 on the net TOA flux that mostly stems from calibration errors on measurements of the outgoing fluxes12,15. This uncertainty is almost an order of magnitude larger than the imbalance of 0.58 ±0.4 Wm–2 inferred from OHC information. The outgoing TOA fluxes presented in Fig. B1 are the TOA CERES fluxes adjusted within the measurement uncertainty to match this OHC inferred imbalance”
Walter,
Yes, degrees K is the appropriate scale, especially when talking about the sensitivity. Look at this plot of sensitivity plotted to scale with power density and temperature along with the SB Law (emissivity = 1 and emissivity = 0.62).
http://www.palisad.com/co2/tp/fig1.png
Two ‘SB’ sensitivities are shown based on average surface temperature (287K) and average planet emissions (255K). Measurements from 3 decades of weather satellite data are shown as little red dots and align almost perfectly with the emissivity = 0.62 version of the SB law. Each little dot is the monthly average surface temperature vs. planet emissions for constant latitude slices of the planet. Notice that the SB sensitivities nearly bound the skeptics estimates of the sensitivity.
The IPCC nominal sensitivity of 0.8C per W/m^2 is also shown. Why have so many people missed the fact that the presumed sensitivity is a linearization error! In the AR’s, they justify the metric as approximately linear, which of course it is, except that the nominal slope is that of SB and not a slope passing through zero that completely ignores the immutable consequences of the T^4 relationship between power density and temperature,
Nick Stokes, the total uncertainties in the diagrammatic report are the rms of the individual uncertainties. That for TOA is miscalculated.
Ristvan, I disagree. The post references [quote] “uncertainty in the earth’s (not surface) warming imbalance as 0.6 watts per m2 ± 17 watts per m2”. I still think it is misleading or just incorrectly referenced.
From the paper itself-
“The net energy balance is the sum of individual fluxes. The current uncertainty in this net surface energy balance is large, and amounts to approximately 17 Wm–2”
It’s talking about the “net energy balance” AT THE SURFACE Duncan. NOT the TOA.
That you think it is misleading is YOUR OPINION. It is the opinion of others here, that it is NOT misleading. Anyone who is confused can merely read the actual paper, which the author liked to IN THE ARTICLE, to clarify this issue for themselves.
.
I think that is one of the author’s POINTS KTM….this article is a PERFECT rebuttal to almost every single argument we see on WUWT!!! It’s ALL MADE UP-exaggerated, inferred, speculated, adjusted, MAGNIFIED!!! And when you view it all from the proper perspective-instead of in the microscope-you realize how INSANE and STUPID and utterly WRONG the argument for AGW is, at it’s very foundation!
Thank you C. R. Dickson!
Exactly Aphan.
When I tell warmists that those tiny changes with those massive error bars are really nothing to worry about they go ballistic.
Can’t beat winding-up a warmist 😉
Yes. Thank you C. R. Dickson.
In relation to the problem of controlling the climate, the signal power is not 0.6 watts per m^2 but rather is nil. The noise power is not 17 watts per m^2 but rather is nil. Control differs from telecommunication in the respect of requiring information from the future. Energy from the future is prohibited by relativity theory. Information from the future is not prohibited by this theory.
Not sure I follow, did I miss the sarc tag?
Paul:
I write to give you a warning.
Oldberg did not omit a sarc tag. This is not the first time he has claimed he can obtain “Information from the future”. He was demolished by davidmhoffer when he previously raised this nonsense on WUWT. And despite my repeated requests he has yet to tell me the winning lottery numbers for next week.
Pressing Oldberg on this will result in you being dragged down Alice’s rabbit hole. Please don’t do it.
Richard
Thanks for the tip Richard. I was curious how Terry could resolve difference between information and energy. I could think of dozens of ways to convey energy that might appear as information. Meaning both, or more likely neither, are possible.
HA, it appears that the movie “Back to the Future” has had a really profound effect on what Terry Oldberg believes is scientific fact and reality.
Richard
Try to get the us powerball lottery numbers from him, it is up to $1.5 BILLION (yes Billion with a B) the drawing is 1/13 at 8:00 est.
“Energy from the future is prohibited by relativity theory. Information from the future is not …”
Sorry Terry but I choked up coffee on this one.
“I smoke two joints in time of peace
And two in time of war
I smoke two joints before I smoke two joints,
And then I smoke two more”
— Sublime.
Well it is just like the gasoline price anomaly at my local gas station.
For the last 25 years I have lived at this place, the gas at my local Arco station has always cost $20.
Never changes !!
g
Even without challenging the science, the whole exercise is ridiculous. We are expected to believe that a 2 Celsius degree change in the global average temperature will lead to catastrophic results. If the planet’s average temperature is 14 deg C (or 287 deg. A), a 2 degree change represents an energy change of 0.7%. If the planet were than sensitive, we wouldn’t be here.
I’d never thought about it this way. I like it. Instead of basing a 2ºC on the Celsius scale, it should be based on the Kelvin scale. The difference is the same, but relating it to 0º would make for a very small percentage.
How about Gore scale that made the earth’s sub-surface temperature million of degrees ?
This is actually the most correct way to view the change as it displays the energy differences proportionally.
A colour scale proportionally changing by Kelvin to on all those maps of temperature differentials would end up a single colour to the naked eye.
Now how do we get the IPCC to recast their assessments in Kelvin?
Francisco- now apply what you just learned…that view…to EVERYTHING about climate science. Everything is MAGNIFIED…stretched…..expanded….until it looks HUGE and SCARY and UNDENIABLY dangerous.
Look at the glass! How smooth and shiny and perfect it appears when you look at it with the naked eye-but how jagged and ugly and raw and dangerous it appears once you MAGNIFY IT.
C.R. Dickson this is a freaking GENIUS analogy to use!
This is EXACTLY what SOME “climate scientists” are doing over and over and over again! They are MAGNIFYING everything to insane, absurd proportions! Mike Mann-MAGNIFIED what a handful of freaking TREES said into MANN MADE GLOBAL WARMING. Cook et al 2013 MAGNIFIED their own personal ASSUMPTIONS into a 97% CONSENSUS!
Trenberth MAGNIFIED a miscalculation into HEAT HIDING IN THE DEEP OCEANS.
Hansen MAGNIFIED his paranoia into CATASTROPHIC PREDICTIONS
Lewandowsky MAGNIFIES his own flawed assumptions and stupidity into MENTAL ILLNESS ON EVERYONE!!!
Josh…where is Josh? I need a cartoon about this—!!! Let’s start calling them MAG MEN!!! I’ll get a tshirt company ready! We need SWAG about the MAGS!!!
Excellent article. I wonder who down-voted it? (Anthony knows these things.) I like graphs, because they show at a glance things that might take a while to explain in text.
Next, @Francisco,
Here’s your ºK chart:
http://4.bp.blogspot.com/-lPGChYUUeuc/VLhzJqwRhtI/AAAAAAAAAS4/ehDtihKNKIw/s1600/GISTemp%2BKelvin%2B01.png
(click in charts to embiggen)
If those “political journalists” don’t like anomaly charts, they need to complain to NASA/GISS (and for the record, “political journalist” is an oxymoron, like “giant shrimp”, etc. A better term is “propagandist”).
Speaking of anomalies, this interesting chart shows that the high temperatures are not affected. It’s the low temps that change most:
http://www.science20.com/files/images/global.png
THIS is the kinds of chart the ‘political journalists’ want people to see:
http://www.realclimate.org/images//Marcott.png
Could they have made it any scarier? But of course, it’s bogus.
We can out-chart those chumps, easy-peasy. This is what’s really happening:
http://i1.wp.com/www.powerlineblog.com/ed-assets/2015/10/Global-2-copy.jpg
There’s nothing either unusual or unprecedented happening. The whole climate scare is a false alarm; a hoax intended to get carbon taxes passed, among other things. But the public is starting to come around. Remember the little boy who cried “Wolf!”? In the end, the wolf ate him.
DBSTEALEY why not plot CO2 on the same graph of absolute temperature that should look pretty decorrelated.
Trebla nailed it. The CAGW counter is that:
1) the rate of change is unprecedented
2) the rate of change is accelerating
Since it is really hard to measure the “global” temperature with contemporary technology, it’s even harder to ascertain the rate of change…much harder to that to get the acceleration of the rate of change.
3) The Earth is always either warming or cooling. Always has been. Always will.
The CAGW counter is that:
1) the rate of change is unprecedented
2) the rate of change is accelerating
Except that none of that is true:
http://jonova.s3.amazonaws.com/graphs/hadley/Hadley-global-temps-1850-2010-web.jpg
What data did you use for that graph?
http://www.science20.com/files/images/global.png
sfx2020,
The chart was posted on a blog called ‘science 2.0 dot com’.
The global mean temperature during the LGM was only about 5 or 6 degrees below what it is to-day. I’d say that could be pretty catastrophic.
To be fair, the warmers are not saying 2 deg would be catastrophic.
Really? What are they saying? And why the urgency? Why the push to “act now”? What are they tearing their hair out for?
“To be fair, the warmers are not saying 2 deg would be catastrophic.”
True enough; in Paris they were saying 1.5 deg.
(and you didn’t answer A.D.E’s question)
“To be fair, the warmers are not saying 2 deg would be catastrophic.”
How do you define “catastrophic” John Finn?
Catastrophic-“involving or causing sudden great damage or suffering” “extremely unfortunate” “dreadful, tragic, disastrous”
Bold below mine:
http://insideclimatenews.org/news/21072015/new-study-says-even-2-degrees-warming-highly-dangerous
“But a new draft study being published this week by a team of 17 leading international climate scientistswarns that even 2 degrees of warming is “highly dangerous” and could cause sea level rise of “at least several meters” this century, leaving most of the world’s coastal cities uninhabitable.
“The economic and social cost of losing functionality of all coastal cities is practically incalculable,” the authors write. “It is not difficult to imagine that conflicts arising from forced migrations and economic collapse might make the planet ungovernable, threatening the fabric of civilization.”
John Finn, now go ahead and try to convince us all that what those 17 leading international climate scientists said could happen, would not be catastrophic!
Trebla, energy tracks t^4, so 289K is 2.8% higher than 287K, not 0.7%
Six degrees cooler, and New York City is buried beneath 100m of ice. Six degrees warmer and New York City is buried beneath 100m of water. Yes, it really is that sensitive.
Richard Lindzen once said that the theory of CAGW essentially boiled down to a philosophical debate. Does one believe that a long running natural system amplifies perturbations; or minimizes them?
Perhaps your scenario is correct. But, no, I don’t believe the system is so sensitive that it would take a truly miniscule perturbation and run away into Armageddon with it.
When I envisioned the time it would take for the people of New York City to be submerged under a hundred feet of rising sea water, I thought of this scene from Austin Powers:
FM,
Your post led me to this pleasurable (& O/T) waste of my time:
The best of Austin Powers’ Fat Bastard.
Not for everyone, I’m afraid. But I LOL’d! ☺
It used to be a standard for graphs to have a double slash near the bottom on the left edge to clearly highlight any break from an assumed Y axis starting point of “zero”. Now I don’t see any published graphs that do that.
I still have a paperback copy of a wonderful little book, “How to Lie with Statistics”, from when I was in college. Here’s a .pdf link for it:
http://www.horace.org/blog/wp-content/uploads/2012/05/How-to-Lie-With-Statistics-1954-Huff.pdf
While I loathe defending anything from the AGW side, you’re misunderstanding the graph. It’s not a simple zoom of the temperature graph; rather, the Y axis is a plot of the anomaly (temperature difference from mean) vs. time. There is no large Y-axis component not being shown… it’s not there.
Dave Waller-
WHICH graph are you referring to SPECIFICALLY? There are a LOT of graphs posted above your quote, some showing anomalies, some showing actual temperatures, or averages. The Y axis on the two charts side by side actually do show the “blow up” or “zoomed in” You have to look at how the increments of temperature on the zoomed in chart ALSO changed. The chart on the left shows the markers in 20 degree increments. The one on the right shows the markers at HALF degree increments. So it IS a perfect “zoom in”, you either just don’t see it, or won’t admit it.
Nice, thanks.
I think it was graphs like that in Daryl Huffs ‘How to lie with Statistics’ that changed me all those years ago.
Just like the CO2 concentration graph… if you don’t zoom it up to ludicrous amounts, it’s mostly a straight line.
Global temperatures as calculated by the climate shamans is still stubbornly stable. Particularly on a y-axis that recognizes the 90C spread in temperatures on a daily basis.
Per this,
Could someone tell me that when CO2 goes from 350 ppm to 400 ppm, what other gas(es) dropped by 50 ppm?
jmarshs,
It’s not necessary for another gas to offset the rise. It’s only parts per million; there’s room for more.
We sold some of the nitrogen molecules to Mars….:) To make room, jmarshs.
The atmosphere expands j. It really is NOT a greenhouse with a solid glass ceiling that can only “hold” so many molecules of “atmosphere”.
Oxygen
@Aphan,
And if the volume expands (which I believe it does) then the temperature drops, right? Like a DX (direct expansion) system? And as the surface area increases due to a volume increase, then the greater the cooling? This would be true irregardless of the composition of the atmosphere.
@dbstealey, Parts Per Million means that if CO2 is 400 ppm, then the combined total of other gases is 999,600 ppm. If CO2 rises to 500 ppm, then the other gasses are equal to 999,5000 ppm. If CO2 were 1,000,000 ppm, then there would be no other gases.
jmarshs,
Yes, but it doesn’t really matter since we’re talking about only 50 parts per million. The problem in this whole debate is all the wild-eyed arm waving and running around in circles over something that does not matter at all. It wouldn’t matter if CO2 was 500 ppm. We couldn’t even tell, without using very sensitive instruments to measure it.
So yes, you and Werner Brozek are both right. But it just doesn’t matter.
You haven’t noticed the obesity epidemic? That’s where the other gases went.
When one gas goes up the others drop proportionately- a million molecules. CO2 goes up 50 million to 400, nitrogen(70%) drops 700,000 to 699,997.55 molecules.
jmarshs
You only get temperature changes for fixed volumes; earth’s atmosphere is not constrained by a huge box. Also, ppm can change by adding more mass to the system. Same tonnage of O2 and N2, etc, just more tonnage of CO2.
That is only true if you have the dry air ppm and then add water vapour. But in our case, hydrocarbons are burned so only oxygen is reduced. And since CO2 went up by 0.01% over the last 200 years, O2 went down by 0.01% from 20.96% to 20.95%. And as dbstealey says, that does not matter.
See:
http://www.worldgreen.org/images/stories/KeelingOxygenCurve.jpg
C’mon people, nothing has to reduce, ppm is a proportion, like per cent. Doesn’t matter if it’s a glass or a swimming pool.
More interestingly, look at CO2 levels at 1M years ago and older. Life flourished on earth when CO2 levels were much higher than today’s or even IPCC projections of catastrophe. Including when our ancestors walked around on the hottest part of the planet.
I have a similar difficulty with even beginning to believe those who preach a theory of cataclysmic man-made average global warming, aka Climate Change, which says a CO2 induced temperature rise of 2 degrees centigrade in 85 years has to be avoided, literally at all costs, and then telling us we are on track of achieving this goal because they are measuring and have average global temperature rises to prove it. Assuming the present average global temperature is roughly 15 degrees centigrade, that is 2 degrees rise on 15 degrees in 85 years or 0.15% per year or very roughly 0.02 degrees centigrade per year. Then I consider the very many independent variables that affect average global temperatures over and above CO2 and the very many factors that affect the accuracy and reliability of each separate temperature measurement in all areas of the world that goes to make up the annual average global temperature! And then I give up with the warmists, whoever they are and however eminent they are. I gave up believing in Fairy Tales very many years ago, and particularly Fairy Tales which is and will cost us a fortune!!
Graph in Kelvins starting at absolute zero.
The average person is familiar with temperature in everyday life.
And when you show true graphs, the idea that a 1 degree change visible in figure 5 is risible to average people when changes of 10-20 degrees daily are common and much wider over the seasons.
So graph 4 seems scaled appropriately to every day experience, no matter what the alarmists say or blog.
This must be why the latest strategy is to say that there is no such thing as normal weather anymore.
If they can convince people that the 10-20 degree daily swings are themselves “unnatural”, suddenly they sense Climate Change every time the sun comes up or goes down. They’re trying to turn people into climate hypochondriacs.
Just like the advocates of constant, obsessive medical surveillance have turned us into “risk factor” hypochondriacs. I don’t believe anything that no one can prove is a “problem” qualified for “problem” status.
Why did the author use a graph based on GISS rather than UAH? While it is the strongest argument on each side, it doeas get more than a little deceptive overall.
Phil Jones misplaced the data once again?
My bad. Thinking CRU…..
The author of the article is not the creator of the graph. He wrote the article about the graph that is causing AGW aneurysms . Take it up with the creator of the graph.
Excellent article. Perhaps someone could create a video of a person dipping their toe in the bath, trying to detect a one to two degree difference in temperature, or turning the house thermostat up or down by a degree and forecasting catastrophes.
Two important things to keep in mind when viewing graphs. Scale and anomaly base periods.
I have never been a fan of the anomaly graph as its look and feel depends upon the dates of the 30 year moving average you select. it also hides that many countries were actually hotter last century and the one before.
In absolute terms it was hotter in the 30’s in the US, and it was hotter in Australia in the mid 1800’s than it was today.
I remember when they were so disappointed that the 1930’s peak was higher than the 1998 peak and then set about adjusting the past and the anomaly graph came to the fore. take absolute temperatures and that rise at the end of the anomaly graph fades into insignificance.
The global mean temperature anomaly is an application of a Central Limit Theorem variant in probability theory to the analysis meteorological temperature statistics. That is all very well, provided the data underlying those statistics, the random variables, complies with the conditions associated with the particular CLT.
However there are a number of controversial peculiarities of climate data that suggest the CLT is not valid in this application. The most obvious is the use of so-called homogenisation to bias data, but there are also issues of changes in variance over time, the very large differences in variance by latitude and by land/sea measurement site, and the relatively small number of observation sites for weather station measurements and their poor geographical distribution.
Do any commenters share my concerns?
Leo G, you’re dead on correct. Misuse of the CLT is rife throughout the surface temperature community. They have consistently and universally assumed all measurement error is random, both land surface and SST. After subtraction of some estimated error mean, they assume all remaining uncertainty declines as 1/sqrtN; effectively to zero.
It’s complete nonsense. The assumption is as non-verified as it is universal. A recent paper includes discussion of this scientific grotesquerie here.
There is a reason beyond graphical deception that anomalies are useful to warmunists. In the CMIP5 climate models, the hindcast ‘actual’ temperatures vary by 4C from lowest to highest. Mauritsen, J. Adv. Modelling Earth Systems 2013. Essay Models all the way Down discusses it and some of the reasons for such a big fail. So, the climate models cannot get evaporation, condensation, snow versus rain…right. Any water phase change is in model to model disagreement. Solution, take anomalies and hide the truth about how bad they all are.
You might like to ask why prolific WUWT posters like Lord M and Bob T graph anomalies extensively. Closet warmunists?
Nick, you would complain no matter what they used.
They use anomalies so simple warmists, like yourself, can understand them.
I have used the above GISS graph above (fig. 4) on several blogs/forums and FB and the warmists get all confused and flustered. Some even haven’t a clue what the are looking at.
If a graph isn’t on a 45 degree angle (or more), or doesn’t look like a hockey stick, it confuses them.
They use them because you do. Hard to explain how ridiculous your graphs are without using the same measurement system.. Stop being obtuse and start thinking for yourself.
Nick Stokes:
You write
I don’t need to ask because I know. They like to use the same units as warmunists when assessing the data used by warmunists: this is called consistency.
Their use of proper consistency does NOT imply they are “Closet warmunists”, and as recently as yesterday on WUWT “Lord M” stated that he recognises the use of global temperature anomalies hides important information about global temperature variations.
Richard
C.R. Dickson:
You say
Sorry, but that is NOT true of temperature anomalies.
The use of global temperature anomalies hides important information.
For example, alarmists often say ‘global temperature rise must be kept below 2.0°C’ and they are shocked to discover that global temperature rises by nearly double that during each year while nobody notices. In reality, it is global temperature anomaly that UNFCCC claims (for no scientific reason) should be constrained to not rise by 2.0°C.
In each year the GLOBAL average temperature rises by 3.8°C during 6 months (January to June) and falls by 3.8°C during the other 6 months (June to January).
And global temperature is highest when the Earth is most distant from the Sun (i.e. when radiative forcing is lowest) during each year: which suggests that global temperature variation is determined more by internal variability of the climate system than by radiative forcing variations.
The seasonal effect is because water is a better heat sink than land so oceans vary temperature less than land with the seasons. There is more land in the Northern Hemisphere (NH) than the Southern Hemisphere (SH)
Therefore, NH summer average temperatures are hotter than SH summer average temperatures.
And
NH winter average temperatures are colder than SH winter average temperatures.
But global temperature is the average of NH and SH average temperatures.
Also, the absolute global temperatures indicated by climate models differ between models and differs from calculated global temperature by more than the rise of global temperature claimed (e.g. by IPCC) to have happened since the industrial revolution. These discrepancies are hidden by use of global temperature anomalies: any difference between modeled and observed global temperature is zero difference between modeled and observed global temperature anomaly.
Richard
Could you provide us some back up on this? ie a link to monthly average temps (vs average monthly anomaly plots we usually see) – basically showing what is subtracted out to get the anomaly. I think most perceive this to be a constant each & every month – ie the same base is taken out each month, which isn’t the case according to your post. If you have a link, would be a great addition to the “global temperature ” reference page here at WUWT.
I think I just found the answer to my own question:
See link:
Bob – I know you are a regular here – do you have a link here for the raw data? It would be interesting to plot this annual variation data along side “worst case scenarios ” to put it in perspective – could be a real nice post by someone here at WUWT
Jeff here is both hadSST an anomaly and ERSSTv3 a true SST data set . you can get both via http://climexp.knmi.nl
http://climategrog.files.wordpress.com/2016/01/ace_amo_2015.png
http://judithcurry.com/2016/01/11/ace-in-the-hole/#comment-757592
See:
http://theinconvenientskeptic.com/2013/03/misunderstanding-of-the-global-temperature-anomaly/
Richard-
RSC said to CR-“Sorry, but that is NOT true of temperature anomalies.The use of global temperature anomalies hides important information.”
Richard, he’s saying that same thing, but hes saying that “it takes a bit of effort” to see what the anomaly charts are HIDING. People view the “magnified” anomaly charts and mentally think “temperatures are rising” because people don’t “think” in “anomalies, they think in temperatures.
“any difference between modeled and observed global temperature is zero difference between modeled and observed global temperature anomaly.”
His point exactly. The changes are the same, there’s no difference between them. But AGWers are freaking out because they don’t understand that. They’ve been shown the anomalies all along but THOUGHT they were seeing temperature changes. They were taught “rising fast” through anomaly use. When you correct their thinking by showing the actual temperature plotted-they think the charts are WRONG because it does not show “rising fast” any more.
Aphan:
Sorry, both your points are wrong when you write
C.R. Dickson did NOT say the same as me.
C.R. Dickson says anomalies “magnify” changes, but I point out that anomalies DELETE information of changes.
It takes more than “a bit of effort” to see what has been deleted.
As example, I cited that the seasonal variation is deleted from anomaly charts. I explained that the seasonal variation provides 3.8°C rise in global temperature during each year, but that seasonal is 0.0°C change in global temperature anomally. And I thank Werner Brozek for his link that provides additional explanation of it. I suggest you would benefit from reading Werner’s link – which I note is titled “Misunderstanding of the global temperature anomaly”- because your post says you misunderstand the matter.
I stated two significant indications that are deleted by the deletion of the seasonal variation.
And as additional example, I pointed out that use of anomalies deletes information on the failure(s) of climate models to emulate global temperature(s).
Deletion of important information is NOT “the same” as “magnification” of the information.
Richard
“The difference is that the graph of the anomalies is a magnified view, not a normal one.”
richard, I just quoted him. He SAID it magnifies THE VIEW, not the changes. Its like zooming in on something with a camera. The resulting picture focuses on only a small part of a bigger view, thus it hides or cuts out part of the picture that gives more context.
A chart of anomalies in ONE THING of course does NOT have to represent anything else- like seasonal variations etc. Its not dishonest to do, if ALL you want to highlight is one particular variable! You can’t say that information was intentionally ommitted or concealed without proof of that. It’s irrational.
Aphan:
I know you “quoted him” and – as I explained – he is wrong. You claiming he is right means you are wrong.
Cleary, you don’t understand my explanation of the matter so I again ask you to read the link provided by Werner Brozek. Use of anomalies deletes all seasonal variation in global and hemispheric temperatures: it does NOT “magnify” the variation.
Richard
RichardSCourtney-
Um…did you read ALL of Werner’s article? And his comments in the comment section? Because someone in the comment section asked-
“I am confused about the logic used in the first part of this article. In the opening, you state that “this one misunderstanding is what allows warmists to get away with…” I assume that the term “warmists” refers to people who accept AGW. You then show that Spencer used a deceptive headline in the article. It seems to me that you have just presented an example of how AGW skeptics (Watts & Spencer) mis-represent climate data to make their argument. If my understanding of your argument is correct, it is not the “warmists” who are getting away with something, it is the “skeptics”. – See more at: http://theinconvenientskeptic.com/2013/03/misunderstanding-of-the-global-temperature-anomaly/#sthash.T5JNoCAf.dpuf”
Werner responded:(bold mine)
“The line was not from Spencer and nor is it deceptive Much like your comment on absorption, it is simplification that is incorrect. The problem is that most people accept anomaly as the same thing as temperature. Which is why I wrote what I did. Watts put my article on his page which shows that he accepts this clarification that I put together.
No one got away with anything except the people that push the inaccurate use of anomaly to the point that it is incorrectly accepted by most people as the Earth’s temperature. My article should clarify the issue for everyone.- See more at: http://theinconvenientskeptic.com/2013/03/misunderstanding-of-the-global-temperature-anomaly/#sthash.T5JNoCAf.dpuf”
The article DOES clarify things Richard, just not the way you THINK it does.
Fact- C.R.’s article is about WHY a simple chart (Figure 1) posted by someone else online, one that just shows temperatures INSTEAD of anomalies, caused such raging hissy fits among warmist “climate scientists”. HE (C.R.) explains in his article pretty much the same thing that Werner does in Werner’s article-“Climate scientists hate it when people show real temperature because it is impossible to see much warming when you look at the seasonal changes in the actual temperature.”
Fact-THEN C.R. explains WHY posting charts with JUST anomalies on them is misleading by COMPARING it to “magnifying” something. Glass-to magnified glass. And then posts two charts (fig 4 and fig 5) to ILLUSTRATE his ANALOGY/POINT.
WERNER explains that it is NOT inaccurate to plot charts using anomalies. It’s CORRECT to do so when you ONLY want to show ANOMALIES. It is not the charting of anomalies that is INACCURATE. It is “push(ing) the use of anomaly to the point that it is incorrectly accepted by most people as the Earth’s temperature” that is “inaccurate”. It is the BEHAVIOR that is inaccurate, NOT the charts.
C.R. DOES NOT SAY that the behavior is accurate either! He AGREES with Werner! It is YOU that thinks that C.R. DOES NOT…and you are WRONG.
Aphan:
It is clear that you do not read what is written.
Werner’s link begins saying
bolding to provide emphasis added by RSC.
It is distortion – n.b. NOT “magnification – when warming is transformed to cooling.
Use of anomalies does NOT magnify the “view”: it distorts the “view”by deleting information.
As I said, the above essay is wrong to say that use of anomalies magnifies the “view”and you are wrong when you assert that the essay is not wrong.
Richard
richardscourtney-
“As I said, the above essay is wrong to say that use of anomalies magnifies the “view”and you are wrong when you assert that the essay is not wrong.”
Richard, your opinions about the essay is your own. But you aren’t just stating your opinion, you are asserting that the author meant something that cannot be proven based upon a careful examination of what he actually said! It is WRONG, and slightly nuts, for you to insinuate that what CR said about “magnifying the view” in his own illustration, of his own point, in his own article, about a specific graph, was meant, by CR, to be extrapolated across all graphs ever created by anyone in scientific history!
Because CR did not indicate anything of the sort to be his intention anywherein his article, YOU seem to be guilty of creating a strawman argument against him!
You are also interpreting Werner’s article selectively, and incorrectly, in order to bolster your illogical and irrational claims about CR DICKSON!
It seems like you aren’t capable of honestly and accurately evaluating the actual evidence presented to you by rebuttal from anyone these days. You just stubbornly deny it and continue to embrace whatever position you originally came to, no matter how irrational it is. I hope you are well.
“They reported the uncertainty in the earth’s warming imbalance as 0.6 watts per m2 ± 17 watts per m2. “
No, they reported it as 0.6 watts per m2 ± 0.4 watts per m2. The figure you cited is of surface imbalance, which is not an observed quantity, but derived from a budget of the very large energy exchanges there.
As to
“Magnification doesn’t change the object you are viewing; it just lets you see more details. “
the term “magnification” is wrong. There is no “unmagnified” scale for a graph. The proper scale is one that communicates the information . I don’t think you would be impressed if in hospital you find they were monitoring your temperature with a graph scaled in Kelvin (from zero).
Your Fig 4 claims to be a “normal” view of global warming. Its scale is from -10F to 100F. Who said that was normal? A proper scale is one that shows the range of global average measurements. If you do that, you’ll actually communicate the information in the data, whether using anomalies or not. If you don’t, it’s just bad graphing.
CR-“Magnification doesn’t change the object you are viewing; it just lets you see more details. “
Nick- “The term “magnification” is wrong.”
Your opinion.
“There is no “unmagnified” scale for a graph. The proper scale is one that communicates the information.”
Um, he’s not talking about the GRAPH being magnified, he’s talking about the “object” of the graph being magnified….the data. Like the “glass” images. It’s the exact same OBJECT being viewed in both images. But magnifying it’s properties brings out the “anomalies” in the glass. It makes the data look jagged and rough and chaotic, rather than smooth and consistent and level.
Using temperature anomalies (the jagged, rough, chaotic variations) does the same thing. It communicates a totally different message than the one communicated from the proper perspective. People don’t view the climate in anomalies. They view the jagged ups and downs of the anomalies and think “temperature” changes. Posting a chart of temperature changes IN THE PROPER SCALE to communicate TEMPERATURE CHANGES made the AGWer’s heads explode. THAT is the point. (Note he CHANGES the scale between Fig 4 and Fig 5 too…magnifying the SCALE for the graphs.)
“I don’t think you would be impressed if in hospital you find they were monitoring your temperature with a graph scaled in Kelvin (from zero).”
ROFL…I’d be HIGHLY suspicious if they monitored my temperature with a GRAPH in the first place! Even more so if they graphed it using “anomalies” rather than actual temperatures!
“Your Fig 4 claims to be a “normal” view of global warming. Its scale is from -10F to 100F. Who said that was normal? A proper scale is one that shows the range of global average measurements. If you do that, you’ll actually communicate the information in the data, whether using anomalies or not. If you don’t, it’s just bad graphing.”
It’s a graph based on the GRAPH that inspired the article, so it uses the same scale as that one. You want to discuss the scale of the graphs, take it up with the creator of the original graph. That graph’s data points were “AVERAGE GLOBAL TEMPERATURE” and the scale is -10 to 110F (you missed the extra 10). If the “average global temperature” is 60 degrees, then you CAN logically graph that as HALFWAY BETWEEN -10 and 110 F. It’s perfectly normal and balanced graphing.
“I’d be HIGHLY suspicious if they monitored my temperature with a GRAPH in the first place!”
Huh? Have you ever seen a hospital temperature chart for a patient? It has observations added on a temp scale with time of obs along the base.
What possible name could you use for it except “graph”?
Why are -10 and 110 chosen as the end points?
@chris
Probably because 99.99+% of the world population lives in climatic conditions bounded by those two temperatures.
DJ Hawkins – No, that is incorrect. Phoenix, for example, regularly is above 110. On average, 11 times per year. The Middle East, home to 200M people, is regularly above 110, as are many parts of India, home to 1.5B people. Roughly 1/3 of the planet’s population sees temperatures above 110 on a yearly basis during the summer months. So it’s false that 99.99% of the population lives in climactic conditions bounded by these two temperatures.
steveta_uk-“I’d be HIGHLY suspicious if they monitored my temperature with a GRAPH in the first place!”
“What possible name could you use for it except “graph”?”
A THERMOMETER??
“an instrument for measuring and indicating temperature, typically one consisting of a narrow, hermetically sealed glass tube marked with graduations and having at one end a bulb containing mercury or alcohol that expands and contracts in the tube with heating and cooling.”
A “graph”is “a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.”
“Nick Stokes
January 11, 2016 at 2:21 pm
A proper scale is one that shows the range of global average measurements.”
Averages are *NOT* measurements. It’s a mathematical result *MADE UP* using a dataset of numbers. Understand that the datasets do not have to be “adjusted” to see why a global average is meaningless with regards to temperature.
Not “made up”, calculated. By your logic, the Dow Jones Average is “made up” since it is an average of 30 stocks weighted by market cap.
“Not “made up”, calculated. By your logic, the Dow Jones Average is “made up” since it is an average of 30 stocks weighted by market cap.”
Exactly! You’re catching on!
When i used to be able to present talks on climate changes to the uninformed masses, this was one of my arguments – tearing apart the alarmists preponderance for providing temperature graphs with either no scale or a grossly exaggerated scale – and then I’d provide a few example from media and the beloved alarmist scientists… My other favourite was the selected data graphs. Reduced one student of Santer’s to tears once with that (which didn’t improve my employability!)
Sorry..meant show graphs with selected time ranges that show time periods where the temperature was rising – even if the time period for the temperature rise was getting rather long in the tooth and falling temps had ensued for several years after the cut off for the graph, but not the publication date of the paper
Ben Santer makes me cry all the time and I don’t even have to be one of his students! Sounds like you helped one of his students “make a breakthrough”. Good for you!
These graphs still understate the insignificance of the effect of CO2 on our estimated surface temperature . Learning why 0 based scales are essential in dealing with ratios is something which is virtually 8th grade pre-algebra math and is the reason why only Kelvin temperature is useful in computations . The use of truncated blown up scales non 0 based scales is one of the main examples in the classic How to Lie with Statistics .
A true picture of the effect of CO2 on temperature is given by this 0 based graph of both CO2 and temperature . In any magnified view which blows up the near noise level variations in our estimated temperature , the CO2 plot will be near vertical .
http://cosy.com/Science/CO2vTkelvin.jpg .
That is the best visual representation of this issue. Thanks for sharing.
The article was inspired by a graph that only had one variable on it-temperature change. All of the other graphs show the insignificance of any temperature change over the time period. Introducing a second variable-CO2 could be viewed as “manipulative” and “deceptive” since the graph in question did not include that variable.
Bob Armstrong,
It doesn’t make the line much flatter …
http://4.bp.blogspot.com/-lPGChYUUeuc/VLhzJqwRhtI/AAAAAAAAAS4/ehDtihKNKIw/s1600/GISTemp%2BKelvin%2B01.png
… but I’ve still got it covered. And while I’d agree that Kelvin is a brilliant temperature scale for use in thermodynamics calculations, it sucks out loud for climate ratios because mean surface temperature of the planet is pretty friggen far above absolute zero [1] …
http://www.giss.nasa.gov/research/briefs/hansen_15/fig1.gif
… and has been for at least the past 5 million years. Upon even a cursory inspection, the truly numerate will note that over the past 800 kyrs of glaciation cycles, average surface temperature has only varied on the order of 6 K, and quickly realize that the 2 K warming “limit” is 33% of that range.
Ice sheets on this rock are not tricked by silly y-axis scaling games, for — somewhat ironically — ability to reason is a requirement for being fooled.
—————
[1] Mentally add 14 to the values in this plot to get absolute temperature in Celsius, and an additional 273.15 to get them in Kelvin.
It doesn’t make the line much flatter
Ha ha ha ha ha ha ha ha
Stephen Richards- “It doesn’t make the line much flatter Ha ha ha ha ha ha ha ha”
So it’s not just ME right? You can SEE how hilarious Brandon’s reasoning is?
Brandon JUST posted two charts and claimed “average surface temperature has only varied on the order of 6 K,and quickly realize that the 2 K warming “limit” is 33% of that range.”
He THEN demonstrates that he knows how to convert Celsius into Kelvin.
Now, if we move the ZERO POINT line on Brandon’s 2nd and 3rd graphs down to the bottom of the chart, like is done on the Kelvin chart, it changes absolutely nothing about the data on the chart. What it WOULD demonstrate is that for the past 2 million years, Earth’s “average” temperature has risen ABOVE that line 6 times. All by itself. All without human intervention.
Now…Brandon’s charts OBVIOUSLY can only be “anomaly charts” because there is no friggin way we could have recorded and “averaged” Earth’s actual daily, or monthly or yearly temperatures over the past 5 million years. Those just cannot be anything other than based on anomalies. Brandon has been programmed to view “anomaly changes” as “temperature changes” which is BAD…BAD BAD according to this article.
So Brandon Gates….when scientists talk about a 2C warming limit….are they talking about a 2 C TEMPERATURE increase?….because actual temperature charts SHOW NO INCREASE-which is what we keep pointing out to you over and over and over again-and that means NO WORRIES right? And if the chart above yours, the Kelvin chart, is ALSO made from “anomalies” it shows there is NO INCREASE in anomalies either! So no worries!
I particularly like the graph in figure 4. compared to figure 5. I believe it is a GISS graph (or based on their figures) I use it a lot and think the “skeptics” and media/politicians should use it more…updated to 2016 if possible. I once had a large version of fig. 4., but can’t seem to find it…
Even the high priced (> $200 USD) microprocessor controlled, anticipating, intelligent “Smart Response” home thermostats do NOT keep the average temperature of your entire home within the 2 degrees F over any 24 hour period many days of the year. I have two recording thermometers and the temperature can vary as much as 4 degrees on a typical day, and mine has an outdoor temperature sensor to “help” minimize excursions or variations and help the furnace make the right amount of heat or the AC use an efficient number of stages. Most days the temperature varies by more than 3 degrees.
150 years and they are worried about less than two degrees change, get serious!
I just love y-axis scaling games …
http://2.bp.blogspot.com/-RqcviJ-LBFo/VTWzaE5AFNI/AAAAAAAAAbk/J2BX64yHG9A/s1600/GISS%2BGlobal%2Bvs%2BCMIP5%2BTemps%2BF.png
… because it’s a great way to make CMIP5 look a lot better than they get credit for in some circles.
Billions of dollars to predict a flat line does not make CMIP5 look better. It raises suspicion on the value derived from the expense.
How quickly the argument shifts with the context.
FTOP_T-
I think what you are trying to point out is that “IF” CIMP5 is accurate in the chart BG just posted, then BG just proved that when you chart both CIMP5 and GISS data in the proper manner, they AGREE AND PROVE that there has just been the SLIGHTEST rise in temperatures F since 1880, even with all that added human CO2! If THAT is true, then as you pointed out, it ALSO proves that we have wasted BILLIONS of dollars to monitor NOTHING. Right? Perfectly logical conclusion to arrive at from the evidence BG posted.
Now, I wonder if he’ll SEE that both the GISS and the CIMP5 data, presented properly, proves that he has NOTHING to worry about because temps are NOT rising in any abnormal way even though CO2 is? If not, I can only logically assume he’s either blind, or some kind of loony that will have to attempt to “quickly shift his argument” or “change the context” in some way to disprove his own, just now, proof.
I just love bar graph color scheme games.
The warmunists have not been particularly clever about the data processing. It’s just a shame that, as with all other aspects of the debate, the public just hasn’t been informed.
I had an enjoyable episode very recently with somebody who sought to mock my views, not least when I tried to explain how the pause-buster “dataset” was made up. He had seen the term ERSSTv4, but his face was a picture when I told him that ER meant “Extended Reconstructed”. The argument, at least for a while, was over.
Plus many. A great sound bite.
That the public is uninformed is a sad, but true fact. That they PREY upon the public’s lack of information is an evil, but true fact.
How to lie (or not) with statistics
I like to keep an open mind. I liked the energy and environmental policies of POTUS Bush based on what he did, not what those with Bush hatred syndrome said he did.
After a few attempts at discovering the Obama plan, I gave up because it amounted to throwing TARP money at wind and solar. So when war on coal was decreed by executive action based on overwhelming scientific evidence, I went to the White House web page to learn for myself the science.
Did you know that the old and young are more sustainable to the hazards of hot weather?
Of course this true. The first step in telling a lie, is to state a well known truth.
So, back to the graphs. A few ago were vacationing on our sail boat. A weather advisory was issued for a heat advisory. Plot 110 degree F on the graph. You do not need a phd to figure out there is a problem and it is today not 100 years from now.
Air conditioning on our sail boat is a bucket of river water in the face. Since I like to take my wife sailing in the future, I asked if she would like to go to a hotel with A/C rather than the bucket.
Took my wife to the emergency room instead and she got two stents put in.
Obama’s war a coal is a war on cheap electricity to mitigate hot weather.
See this graph: USHCN surface temperatures – before and after “adjustments”:
http://realclimatescience.com/wp-content/uploads/2015/12/2015-12-18-12-36-03.png
Note temperatures cooled from ~1940-1975 as atmospheric CO2 increased, not only in the USA but globally. This indicates that climate sensitivity to atmospheric CO2 is near-zero, not 1C, 3C or 6C.
That graph ought to be displayed every day. It tells a story of deception.
TA
Kudos to James for being the first to do it…..
https://suyts.wordpress.com/2014/05/27/how-global-warming-looks-on-your-thermometer/
Thanks, that’s what I was looking for. I will save this – somehow somewhere…
Oops! This …
http://www.macrotrends.net/1319/dow-jones-100-year-historical-chart
Do that with stock price and see how long you last
You mean like this? What’s your point?
Again … oops. This!
http://www.macrotrends.net/1319/dow-jones-100-year-historical-chart
teapartygeezer January 11, 2016 at 7:30 pm
“You mean like this? What’s your point?”
Mosher doesn’t have a point. He is throwing a straw man into the thread. The stock market is governed by two emotions, greed and fear which have nothing to do with actual temperatures.
“Do that with stock price and see how long you last”
I do and have lasted a long time with a good long profitable view, if we looked at the stock market the other way we would be day traders and risk being broke every day.
Did Mosher just post the equivalent of “Well bad evil stupid people do it with stock prices then it’s ok for scientists to do it too!”????? Bravo….very….very….bold reasoning that.
Define “last”.