The ‘Trick’ of Anomalous Temperature Anomalies

By Kip Hansen  Re-Blogged From WUWT

It seems that every time  we turn around, we are presented with a new Science Fact that such-and-so metric — Sea Level Rise, Global Average Surface Temperature, Ocean Heat Content, Polar Bear populations, Puffin populations — has changed dramatically — “It’s unprecedented!” — and these statements are often backed by a graph illustrating the sharp rise (or, in other cases, sharp fall) as the anomaly of the metric from some baseline.  In most cases, the anomaly is actually very small and the change is magnified by cranking up the y-axis to make this very small change appear to be a steep rise (or fall).  Adding power to these statements and their graphs is the claimed precision of the anomaly — in Global Average Surface Temperature, it is often shown in tenths or even hundredths of a Centigrade degree.  Compounding the situation, the anomaly is shown with no (or very small) “error” or “uncertainty” bars, which are, even when shown,  not error bars or uncertainty bars  but actually statistical Standard Deviations (and only sometimes so marked or labelled).


I wrote about this several weeks ago in an essay here titled “Almost Earth-like, We’re Certain”.   In that essay, which the Science and Environmental Policy Project’s Weekly News Roundup characterized as “light reading”,  I stated my opinion that “they use anomalies and pretend that the uncertainty has been reduced.   It is nothing other than a pretense.  It is a trick to cover-up known large uncertainty.”

Admitting first that my opinion has not changed, I thought it would be good to explain more fully why I say such a thing — which is rather insulting to a broad swath of the climate science world.   There are two things we have to look at:

  1. Why I call it a “trick”, and 2.  Who is being tricked. 



What exactly is “finding the anomaly”?  Well, it is not what it is generally thought.  The simplified explanation is that one takes the annual averaged surface temperature and subtracts from that the 30-year climatic average and what you have left is “The Anomaly”.

That’s the idea, but that is not exactly what they do in practice.  They start finding anomalies at a lower level and work their way up to the Global Anomaly.  Even when Gavin Schmidt is explaining the use of anomalies, careful readers see that he has to work backwards to Absolute Global Averages in Degrees — by adding the agreed upon anomaly to the 30-year mean.

“…when we try and estimate the absolute global mean temperature for, say, 2016. The climatology for 1981-2010 is 287.4±0.5K, and the anomaly for 2016 is (from GISTEMP w.r.t. that baseline) 0.56±0.05ºC. So our estimate for the absolute value is (using the first rule shown above) is 287.96±0.502K, and then using the second, that reduces to 288.0±0.5K.”

But for our purposes, let’s just consider that the anomaly is just the 30-year mean subtracted from the calculated GAST in degrees.

As Schmidt kindly points out, the correct notation for a GAST in degrees is something along the lines of 288.0±0.5K — that is a number of degrees to tenths of a degree and the uncertainty range ±0.5K.  When a number is expressed in that manner, with that notation, it means that the actual value is not known exactly, but is known to be within the range expressed by the plus/minus amount.



This illustration shows this in actual practice with temperature records….the measured temperatures are rounded to full degrees Fahrenheit — a notation that represents ANY of the infinite number of continuous values between 71.5 and 72.4999999…

It is not a measurement error, it is the measured temperature represented as a range of values 72 +/- 0.5.  It is an uncertainty range, we are totally in the dark as to the actual temperature — we know only the range.

Well, for the normal purposes of human beings, the one-degree-wide range is quite enough information.  It gets tricky for some purposes when the temperature approaches freezing — above or below frost/freezing temperatures being Climatically Important for farmers, road maintenance crews and airport airplane maintenance people.

No matter what we do to temperature records, we have to deal with the fact that the actual temperatures were not recorded — we only recorded ranges within which the actual temperature occurred.

This means that when these recorded temperatures are used in calculations, they must remain as ranges and be treated as such.    What cannot be discarded is the range of the value.  Averaging (finding the mean or the median) does not eliminate the range — the average still has the same range.  (see Durable Original Measurement Uncertainty ).

As an aside:  when Climate Science and meteorology present us with the Daily Average temperature from any weather station, they are not giving us what you would think of as the “average”, which in plain language refers to the arithmetic mean — rather we are given the median temperature — the number that is exactly halfway between the Daily High and the Daily Low.   So, rather than finding the mean by adding the hourly temperatures and dividing by 24, we get the result of Daily High plus Daily Low divided by 2.  These “Daily Averages” are then used in all subsequent calculations of weekly, monthly, seasonal, and annual averages.  These Daily Averages have the same 1-degree wide uncertainty range.

On  the basis of simple logic then, when we finally arrive at a Global Average Surface Temperature, it still has the original uncertainty attached — as Dr. Schmidt correctly illustrates when he gives Absolute Temperature for 2016 (link far above) as   288.0±0.5K.  [Strictly speaking, this is not exactly why he does so — as the GAST is a “mean of means of medians” — a mathematical/statistical abomination of sorts.] As William Briggs would point out  “These results are not statements about actual past temperatures, which we already knew, up to measurement error.” (which measurement error or uncertainty is at least +/- 0.5).

The trick comes in  where the actual calculated absolute temperature value is converted to an anomaly of means. When one calculates a mean (an arithmetical average — total of all the values divided by the number of values), one gets a very precise answer.  When one takes the average of values that are ranges, such as 71 +/- 0.5, the result is a very precise number with a high probability that the mean is close to this precise number.   So, while the mean is quite precise, the actual past temperatures are still uncertain to +/-0.5.

Expressing the mean with the customary ”+/- 2 Standard Deviations” tells us ONLY what we can expect the mean to be — we can be pretty sure the mean is within that range.  The actual temperatures, if we were to honestly express them in degrees as is done in the following graph, are still subject to the uncertainty of measurement:  +/- 0.5 degrees.


[ The original graph shown here was included in error — showing the wrong Photoshop layers.  Thanks to “BoyfromTottenham” for pointing it out. — kh ]

The illustration was used (without my annotations) by Dr. Schmidt in his essay on anomalies.  I have added the requisite I-bars for +/- 0.5 degrees.  Note that the results of the various re-analyses themselves have a spread of 0.4 degrees  — one could make an argument for using the additive figure of 0.9 degrees as the uncertainty for the Global Mean Temperature based  on the uncertainties above (see the two greenish uncertainty bars, one atop the other.)

This illustrates the true uncertainty of Global Mean Surface Temperature — Schmidt’s acknowledged +/- 0.5 and the uncertainty range between reanalysis products.

In the real world sense, the uncertainty presented above should be considered the minimum uncertainty — the original measurement uncertainty plus the uncertainty of reanalysis.   There are many other uncertainties that would properly be additive — such as those brought in by infilling of temperature data.

The trick is to present the same data set as anomalies and claim the uncertainty is thus reduced to 0.1 degrees (when admitted at all) — BEST doubles down and claims 0.05 degrees!




Reducing the data set to a statistical product called anomaly of the mean does not inform us of the true uncertainty in the actual metric itself — the Global Average Surface Temperature  — any more than looking at a mountain range backwards through a set of binoculars makes the mountains smaller, however much it might trick the eye.

Here’s a sample from the data that makes up the featured image graph at the very beginning of the essay.  The columns are:  Year — GAST Anomaly — Lowess Smoothed

2010  0.7    0.62
2011  0.57  0.63
2012  0.61  0.67
2013  0.64  0.71
2014  0.73  0.77
2015  0.86  0.83
2016  0.99  0.89
2017  0.9    0.95

The blow-up of the 2000-2017 portion of the graph:


We see global anomalies given to a precision of hundredths of a degree Centigrade.  No uncertainty is shown — none is mentioned on the NASA web page displaying the graph (it is actually a little app, that allows zooming).   This NASA web page, found in NASA’s Vital Signs – Global Climate Change section, goes on to say that “This research is broadly consistent with similar constructions prepared by the Climatic Research Unit and the National Oceanic and Atmospheric Administration.”   So, let’s see:

From the CRU:


Here we see the CRU Global Temp (base period 1961-90) — annoyingly a different base period than NASA which used 1951-1980.  The difference offers us some insight into the huge differences that Base Periods make in the results.

2010   0.56 0.512
2011 0.425 0.528
2012   0.47 0.547
2013 0.514 0.569
2014   0.579  0.59
2015 0.763 0.608
2016   0.797 0.62
2017 0.675 0.625

The official CRU anomaly for 2017 is 0.675 °C — precise to thousandths of a degree.  They then graph it at 0.68°C.  [Lest we think that CR anomalies are really only precise to “half a tenth”, see 2014, which is 0.579 °C. ]   CRU manages to have the same precision in their smoothed values — 2015 = 0.608.

And, not to discriminate, NOAA offers these values, precise to hundredths of a degree:

2010,   0.70
2011,   0.58
2012,   0.62
2013,   0.67
2014,  0.74
2015,  0.91
2016,  0.95
2017,  0.85

[Another graph won’t help…]

What we notice is that, unlike absolute global surface temperatures such as those quoted by Gavin Schmidt at RealClimate, these anomalies are offered without any uncertainty measure at all.  No SDs, no 95% CIs, no error bars, nothing.  And precisely to the 100th of a degree C (or K if you prefer).

Let’s review then:   The major climate agencies around the world inform us about the state of the climate through offering us graphs of the anomalies of the Global Average Surface Temperature showing a steady alarmingly sharp rise since about 1980.  This alarming rise consists of a global change of about 0.6°C.  Only GISS offers any type of uncertainty estimate and that only in the graph with the lime green 0.1 degree CI bar used above. Let’s do a simple example: we will follow the lead of Gavin Schmidt in this August 2017 post and use GAST absolute values in degrees C with  his suggested uncertainty of 0.5°C.  [In the following, remember that all values have °C after them – I will use just the numerals from now on.]

What is the mean of two GAST values, one for Northern Hemisphere  and one for Southern Hemisphere?  To make a real simple example, we will assign each hemisphere the same value of 20 +/- 0.5 (remembering that these are both °C). So, our calculation:   20 +/- 0.5 + 20 +/- 0.5 divided by 2 equals ….. The Mean is an exact 20.  (now, that’s precision…)

What about the Range?  The range is +/- 0.5.  A range 1 wide.  So, the Mean with the Range is 20 +/- 0.5.

But what about the uncertainty?     Well the range states the uncertainty — or the certainty if you prefer — we are certain that the mean is between 20.5 and 19.5.

Let’s see about the probabilities  — this is where we slide over to “statistics”.

Here are some of the values for the Northern and Southern  Hemispheres, out of the infinite possibilities inferred by 20 +/- 0.5:  [we note that 20.5 is really 20.49999999999…rounded to 20.5 for illustrative purposes.]  When we take equal values, the mean is the same, of course.  But we want probabilities — so how many ways can the result be  20.5 or 19.5?  Just one way each.

NH           SH
20.5 —— 20.5 = 20.5 only one possible combination
20.4         20.4
20.3         20.3
20.2         20.2
20.1         20.1
20.0         20.0
19.9         19.9
19.8         19.8
19.7         19.7
19.6         19.6
19.5 —— 19.5 = 19.5 only one possible combination

But how about 20.4 ?  We could have 20.4-20.4, or 20.5-20.3, or 20.3-20.5 — three possible combinations. 20.3?  5 ways    20.2?  7 ways   20.1?  9 ways   20.0?  11 ways .  Now we are over the hump and 19.9? 9  ways  19.8? 7 ways  19.7? 5 ways  19.6? 3 ways  and 19.5? 1 way.

You will recognize the shape of the distribution:


As we’ve only used eleven values for each of the temperatures being averaged, we get a little pointed curve.   There are two little graphs….the second (below) shows what would happen if we found the mean of two identical numbers, each with an uncertainty range of +/- 0.5, if they had been rounded to the nearest half degree instead of the usual whole degree.  The result is intuitive — the mean always has the highest probability of being the central value.


Now, that may seem so obvious as to be silly.  After all, that’s that a mean is — the central value (mathematically).  The point is that with our evenly spread values across the range — and, remember, when we see a temperature record give as XX +/- 0.5 we are talking about a range of evenly spread possible values,  the mean will always be the central value, whether we are finding the mean of a single temperature or a thousand temperatures of the same value.  The uncertainty range, however, is always the same.  Well, of course it is!   Yes, has to be.

Therein lies the trick — when they take the anomaly of the mean, they drop the uncertainty range altogether and concentrate only on the central number, the mean, which is always precise and statistically close to  that central number.   When any uncertainty is expressed at all, it is expressed as the probability of the mean being close to the central number — and is disassociated from the actual uncertainty range of the original data.

As William Briggs tells us:  “These results are not statements about actual past temperatures, which we already knew, up to measurement error.”   

We already know the calculated GAST (see the re-analyses above).  But we only know it being somewhere within its known uncertainty range,  which is as stated by Dr. Schmidt to be +/- 0.5 degrees.   Calculations of the anomalies of the various means do not tell us about the actual temperature of the past — we already knew that — and we knew how uncertain it was.

It is a TRICK to claim that by altering the annual Global Average Surface Temperatures to anomalies we can UNKNOW the known uncertainty.



As Dick Feynman might sayThey are fooling themselves.  They already know the GAST as close as they are able to calculate it using their current methods.  They know the uncertainty involved — Dr. Schmidt readily admits it is around 0.5 K.    Thus, their use of  anomalies (or the means of anomalies…) is simply a way of fooling themselves that somehow, magically, that the known uncertainty will simply go away utilizing the statistical equivalent of “if we squint our eyes like this and tilt our heads to one side….”.

Good luck with that.


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