So glad I looked here today.
And OMG … I was, just last night, trying to get my head wrapped around Breit–Wigner in another matter altogether.
The arithmetic mean of minimum and maximum temperatures isn’t a reliable indicator of the day’s radiative capacity to space. Either extreme may be for only a few minutes or for many hours of the day. The resulting total radiated energy is substantially different to the arithmetic mean for any substantial range of daily temperature. And of course; it gets much worse if one averages the averages over a month.
This isn’t as ‘cheeky’ a question as it might seem.
All the statistical manipulation I’ve seen done on temperatures tends to presume they have a Standard Normal Distribution and that it is a valid statistical operation to compute a mean. While this might seem reasonable for a single thermometer in a single place ( but even there can ‘have issues’ ) as soon as you start doing an arithmetic mean over a geographic field of thermometers spread over 1200 km (as is done in codes like GIStemp) you are making the implicit assumption that the mean is defined.
For those of us who had some, but not extensive, statistics, that is a natural assumption as we spent months (years?) doing various kinds of problems all universally based on a Standard Normal Distribution. But there are other kinds of distribution…
Why this matters is that it is a property…
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