Daily Averages? Not So Fast…

Guest Essay by Kip Hansen (with graphic data supplied by William Ward)


featured_imagefeatured_imageOne of the advantages of publishing essays here at WUWT is that one’s essays get read by an enormous number of people — many of them professionals in science and engineering.

In the comment section of my most recent essay concerning GAST (Global Average Surface Temperature) anomalies (and why it is a method  for Climate Science to trick itself) — it was brought up [again] that what Climate Science uses for the Daily Average temperature from any weather station is not, as we would have thought, the average of the temperatures recorded for the day (all recorded temperatures added to one another divided by the number of measurements) but are, instead, the Daily Maximum Temperature (Tmax) plus the Daily Low Temperature (Tmin) added and divided by two.  It can be written out as (Tmax + Tmin)/2.

Anyone versed in the various forms of averages will recognize the latter is actually the median of  Tmax and Tmin — the midpoint between the two.  This is obviously also equal to the mean of the two — but since we are only dealing with a Daily Max and Daily Min  for a record in which there are, in modern times,  many measurements in the daily set, when we align all the measurements by magnitude and find the midpoint between the largest and the smallest we are finding a median (we do this , however,  by ignoring all the other measurements altogether, and find the median of a two number set consisting of only Tmax and Tmin. )

This certainly is no secret and is the result of the historical fact that temperature records in the somewhat distant past, before the advent of automated weather stations,  were kept using Min-Max recording thermometers — something like this one:


Each day at an approximately set time, the meteorologist would go out to her Stevenson screen weather station, open it up, and look in at a thermometer similar to this.  She would record the Minimum and Maximum temperatures shown by the markers, often she would also record the temperature at the time of observation, and then press the reset button (seen in the middle) which would return the Min/Max markers to the tops of the mercury columns on either side.  The motion of the mercury columns over the next 24 hours would move the markers to their respective new Minimums and Maximums for that period.

With only these measurements recorded, the closest to a Daily Average temperature that could be computed was the median of the two.  To be able to compare modern temperatures to past temperatures, it has been necessary to use the same method to compute Daily Averages today, even though we have recorded measurements from automated weather stations every six minutes.

Nick Stokes discussed (in this linked essay) the use and problems of Min-Max thermometers as it relates to the Time of Observation Adjustments.  In that same essay, he writes

Every now and then a post like this appears, in which someone discovers that the measure of daily temperature commonly used (Tmax+Tmin)/2 is not exactly what you’d get from integrating the temperature over time. It’s not. But so what? They are both just measures, and you can estimate trends with them.

And Nick Stokes is absolutely correct — one can take any time series of anything, find all sorts of averages — means, medians, modes —  and find their trends over different periods  of time.

In this case, we have to ask the question:  What Are They Really Counting?  I find myself having to refer back to this essay over and over again when writing about modern science research which seems to have somehow lost an important  thread of true science — that we must take extreme care with defining what we are researching — what measurements of what property of what physical thing will tell us what we want to know?

Stokes maintains that any data of measurements of any temperature averages  are apparently just as good as any other — that the median of (Tmax+Tmin)/2 is just as useful to Climate Science as a true average of more frequent temperature measurements, such as today’s six-minute records.      What he has missed is that if science is to be exact and correct, it must first define its goals and metrics — exactly and carefully.

So, we have raised at least three questions:

1. What are we trying to measure with temperature records? What do we hope the calculations of monthly and annual means and their trends, and the trends of their anomalies [anomalies here always refers to anomalies from some climatic mean], will tell us?
2. What does (Tmax+Tmin)/2 really measure? Is it quantitatively different from averaging all the six-minute (or hourly) temperatures for the day? Are the two qualitatively different?
3. Does the currently-in-use (Tmax+Tmin)/2 method fulfill the purposes of any of the answers to question #1?

I will take a turn at answering these question, and readers can suggest their answers in comments.

What are we trying to measure?

The answers to question #1 depends on who you are or what field of science you are practicing.

Meteorologists measure temperature because it is one of the key metrics of their field.  Their job is to know past temperatures and use them to predict future temperatures on a short term basis — tomorrow’s Hi and Lo, weekend weather conditions and seasonal predictions useful for agriculture.  Temperature predictions of extremes are an important part of their job — freezing on roadways and airport runways, frost and freeze warning to agriculture, high temperatures that can affect human health and a raft of other important meteorological forecasts.

Climatologists are concerned with long-term averages of ever changing weather conditions for regions, continents and the planet as a whole.  Climatologists concern themselves with the long-range averages that allow them to divide various regions into the 21 Koppen Climate Classifications and watch for changes within those regions.   The Wiki explains why this field of study is difficult:

“Climate research is made difficult by the large scale, long time periods, and complex processes which govern climate. Climate is governed by physical laws which can be expressed as differential equations. These equations are coupled and nonlinear, so that approximate solutions are obtained by using numerical methods to create global climate models. Climate is sometimes modeled as a stochastic [random] processbut this is generally accepted as an approximation to processes that are otherwise too complicated to analyze.”     [emphasis mine — kh]

The temperatures of the oceans and the various levels of the atmosphere, and the differences between regions and atmospheric  levels, are, along with a long list of other factors,  drivers of weather and the long-term differences in temperature are thus of interest to climatology.  The momentary equilibrium state of the planet in regards to incoming and outgoing energy from the Sun is currently one of the focuses of climatology and temperatures are part of that study.

Anthropogenic Global Warming scientists (IPCC scientists)  are concerned with proving that human emissions of CO2 are causing the Earth climate system to retain increasing amounts of incoming energy from the Sun and calculate global temperatures and their changes in support of that objective.  Thus, AGW scientists focus on regional and global temperature trends and the trends of temperature anomalies and other climatic factors that might support their position.

What do we hope the calculations of monthly and annual means and their trends will tell us? 

Meteorologists are interested in temperature changes for their predictions, and use “means” of past temperatures to set an expected range to know and predict when things are out of these normally expected ranges.  Temperature differences between localities and regions drive weather which makes these records important for their craft.  Multi-year comparisons help them to make useful predictions for agriculturalists.

Climatologists want to know how the longer-term picture is changing — Is this region generally warming up, cooling off, getting more or less rain?  — all of these looked at in decadal or 30-year time periods.  They need trends for this. [Note:  not silly auto-generated ‘trend lines’ on graphs that depend on start-and-end points — they wish to discover real changes of  conditions over time.]

AGW scientists need to be able to show that the Earth is getting warmer and use temperature trends — regional and global, absolute and anomalies — in the effort to prove the AGW hypothesis  that the Earth climate system is retaining more energy from the Sun due to increasing CO2 in the atmosphere.

What does (Tmax+Tmin)/2 really measure? 

(Tmax+Tmin)/2, meteorology’s daily Tavg, is the median of the Daily High (Tmax) and the Daily Low (Tmin) (please see the link if you are unsure why it is the median and not the mean).   The monthly TAVG is in fact the median of the Monthly Mean of Daily Maxes and the Monthly Mean of the Daily Mins.  The Monthly TAVG, which is the basic input value for all of the subsequent regional, statewide, national, continental, and global calculations of average temperature (2-meter air over land), is calculated by finding the median of the means of the Tmaxs and the Tmins for the month for the station, arrived at by adding all the daily Tmaxs for the month and finding their mean (arithmetical average) and adding all the Tmins for the month, and finding their mean, and then finding the median of those two values.    (This is not by a definition that is easy to find — I had to go to original GHCN records and email NCEI Customer Support for clarification).

So now that we know what the number called monthly TAVG is made of,  we can take a stab at what it is a measure of.

Is it a measure of the average of temperatures for the month?  Clearly  not.   That would be calculated by adding up the Tavg for each day and dividing by the number of days in the month.  Doing that might very well give us a number surprising close to the recorded monthly TAVG — unfortunately, we have already noted that the daily Tavgs are not the average temperatures for their days but at the medians of the daily Tmaxs and Tmins.

The featured image of this essay illustrates the problem, here it is blown up:


This illustration is from an article defining Means and Medians, we see that if the purple traces were the temperature during a day, the median would be identical for wildly different temperature profiles, but the true average, the mean, would be very different.  [Note: the right hand edge of the graph is cut off, but both traces end at the same point on the right — the equivalent of a  Hi for the day.]  If the profile is fairly close to a “normal distribution” the Median and the Mean are close together — if not, they are quite different.

Is it quantitatively different from averaging all the six-minute (or hourly) temperatures for the day?  Are the two qualitatively different?

We need to return to the Daily Tavgs to find our answer.  What changes Daily Tavg?   Any change in either the daily Tmax or the Tmin.  If we have a daily Tavg of 72, can we know the Tmax and Tmin?  No, we cannot.   The Tavg for the day tells us very little about the high temperature for the day or the low temperature for the day.  Tavg does not tell us much about how temperatures evolved and changed during the day.

Tmax 73, Tmin 71 = Tavg 72
Tmax 93, Tmin 51 = Tavg 72
Tmax 103, Tmin 41= Tavg 72

The first day would be a mild day and a very warm night, the second a hot day and an average sort of night.  The second could have been a cloudy warmish day, with one hour of bright direct sunshine raising the high to a momentary 93 or a bright clear day that warmed to 93 by 11 am and stayed above 90 until sunset with only a short period of 51 degree temps in the very early morning.    Our third example, typical of the high desert in the American Southwest, a very hot day with a cold night.  (I have personally experienced 90+ degree days and frost the following night.) (Tmax+Tmin)/2 tells us only the median between two extremes of temperature, each of which could have lasted for hours or merely for minutes.

Daily Tavg, the median of Tmax and Tmin,  does not tell us about the “heat content” or the temperature profile of the day.  If daily Tmaxs and Tmins and Tavgs don’t tell us the temperature profile and “heat content” of their days, then the Monthly TAVG has the same fault — being the median of the mean of Tmaxs and Tmins —  cannot tell us either.

Maybe a graph will help illuminate this problem.


This graph show the difference between daily Tavg  (by (Tmax+Tmin)/2 method) and the true mean of daily temperatures, Tmean.  We see that there are days when the difference is three or more degrees with an eye-ball average of a degree or so, with rather a lot of days in the one to two degree range.  We could punch out a similar graph for  Monthly TAVG and real monthly means, either of the actual daily means or from averaging (finding the mean) of all temperature records for the month).

The currently-in-use Tavg and TAVG (daily and monthly) are not the same as actual means of the temperatures during the day or the month, they are both quantitatively different and qualitatively different — they tells us different things.

So,  YES, the data are qualitatively different and  quantitatively different.

Does the currently-in-use (Tmax+Tmin)/2 method fulfill the purposes of any of the answers to question #1?

 Let’s check by field of study:

Meteorologists measure temperatures because it is one of the key metrics of their field.   The weather guys were happy with temperatures measured to the nearest full degree.  One degree one way or the other was not big deal (except at near freezing).  Average weather can also withstand an uncertainty of a degree or two.  So, my opinion would be that (Tmax+Tmin)/2 is adequate for the weatherman, it is fit for purpose in regards to the weather and weather prediction.  For weather, the weatherperson knows  the temperature will vary naturally by a degree or two across his area of concern, so a prediction of  “with highs in the mid-70s” is as precise as he needs to be.

Climatologists are concerned with long-term ever changing weather conditions for regions, continents and the planet as a whole.   Climatologists know that past weather metrics have been less-than-precise — they accept that (Tmax+Tmin)/2 is not a measure of the energy in the climate system but it gives them an idea of temperatures on a station, region, and continental basis, close enough to judge changing climates —  one degree up or down in the average summer or the winter temperature for a region is probably not a climatically important change — it is just annual or multi-annual weather.  For the most part, climatologists know that only very recent temperature records get anywhere near one or two degree precision.  (See my essay about Alaska for why this matters).

Anthropogenic Global Warming scientists (IPCC scientists)  are concerned with proving that human emissions of CO2 are causing the Earth climate system to retain increasing amounts of incoming energy from the Sun.  Here is where the differences in quantitative values, and the qualitative differences, between (Tmax+Tmin)/2 and a true Daily/Montly mean temperature comes into play.

There are those who will (correctly) argue that temperature averages (certainly the metric called GAST) are not accurate indicators of energy retention in the climate system.  But before we can approach that question, we have to have correct quantitative and qualitative measures of temperature reflecting changing heat energy at weather stations. (Tmax+Tmin)/2 does not tell us whether we have had a hot day and a cool night, or a cool day and a warmish night.  Temperature is an intensive property (of air and water, in this case) and not properly subject to addition and subtraction and averaging in the normal sense — temperature of an air sample (such as in an Automatic Weather Station – ASOS) — is related to but not the same as the energy (E) in the air at that location and is related to but not the same as the energy in the local climate system.  Using (Tmax+Tmin)/2 and TMAX and TMIN (monthly mean values) to arrive at monthly TAVG does not even accurately reflect what the temperatures were and therefore will not, and cannot, inform us properly (accurately and precisely) about the energy in the locally measured climate system and therefore  when combined across regions and continents, cannot inform us properly (accurately and precisely) about the energy in regional, continental or the global climate system — not quantitatively in absolute terms and not in the form of changes, trends, or trends of anomalies.

AGW science is about energy retention in the climate system — and the currently used mathematical methods —  all the way down to the daily average level — despite the fact that, for much of the climate historical record, they are all we have — are not fit for the purpose of determining changing energy retention by the climate system to any degree of quantitative or qualitative accuracy or precision.

Weathermen and women are probably well enough served by the flawed metric as being “close enough for weather prediction”.  Hurricane prediction is probably happy with temperatures within a degree or two – as long as all are comparable.

Even climate scientists, those disinterested in the Climate Wars, are happy to settle for temperatures within a degree or so — as there are a large number of other factors, most which are more important than “average temperature”, that combine to make up the climate of any region.  (see again the Koppen Climate Classifications).

Only AGW activists insist that the miniscule changes wrested from the long-term climate record of the wrong metrics are truly significant for the world climate.


Bottom Line:

The methods currently used to determine both Global Temperature and Global Temperature Anomalies rely on a metric, used for historical reasons, that is unfit in many ways for the purpose of determining with accuracy or precision whether or not the Earth climate system is warming due to additional energy from the Sun being retained in the Earth’s climate system and is unfit in many ways for the purpose of determining the size of any such change and,  possibly,  not even fit for determining the sign of that change.   The current method does not properly measure a physical property that would allow that determination.

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Author’s Comment Policy:

The basis of this essay is much simpler than it seems.  The measurements used to form GAST(anomaly) and GAST(absolute) — specifically (Tmax+Tmin)/2, whether daily or monthly) are not fit for the purpose of determining those global metrics as they are presented to the world by AGW activist scientists.     They are most often used to indicate that the climate system is retaining more energy and thus warming up….but the tiny changes seen in this unfit metric over climatically significant periods of time cannot tell us that, since they do not actually measure the average temperature, even as experienced at a single weather station.  The additional uncertainty from this factor increases the overall uncertainty about GAST and its anomalies to the point that the uncertainty exceeds the entire increase since the mid-20th century.  This uncertainty is not eliminated through repeated smoothing and averaging of either absolute values or their anomalies.

I urge readers to reject the ever-present assertion that “if we just keep averaging averages, sooner or later the variation — whether error, uncertainty, or even just plain bad data — becomes so small as not to matter anymore”.  That way leads to scientific madness.

There would be different arguments if we actually had an accurate and precise average of temperatures from weather stations.  Many would still not agree that the temperature record alone indicates a change in retention of solar energy in the climate system.  Energy entering the system is not auto-magically turned into sensible heat in the air at 2-meters above the ground, or in the skin temperature of the oceans.  Changes in sensible heat in the air measured at 2-meters and as ocean skin temperature do not necessarily equate to increase or decrease of retained energy in the Earth’s climate system.

There will be objections to the conclusions of this essay — but the facts are what they are.   Some will interpret the facts differently,  place different importance values on different facts and draw different conclusions.  That’s science.

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via Watts Up With That?


October 2, 2018 at 01:06PM

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