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Recently, there has been much talk in the media of it being the hottest day of the year so far. It has always seemed to me that there are likely many more of these in the northern hemisphere than the southern.

In the northern hemisphere "the year" starts close to the local minimum and so for a good half of the year there is a fairly reasonable chance that a given day will be the hottest so far.

However, in the southern hemisphere after an initial cluster of hot days it is very unlikely there will be another so hot until the end of the year, limiting the number "hottest so far". (And vice versa for cold).

I'm interested in quantitative models of this kind of phenomenon (not just for temperature, but similar patterns). But I don't know how to start developing a model which is tractable. Maybe the year could be modeled as a sinusoid, with a superimposed zero-mean normally distributed random element. It seems that the likely number of "highest"/"lowest" so-fars should be expressible in terms as a function of starting phase and variance.

Is this something that has been studied, or is maybe trivial to those with the relevant skills?

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    $\begingroup$ It also depends a lot on the geographic location within the hemisphere. Even between places at the same latitude there are great differences. $\endgroup$ Jul 2, 2015 at 21:34
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    $\begingroup$ Instead of being attentive to an arbitrary date such as that on which the calendar year begins (there was a time in the past when the day after March 24 in the year $X$ was March 25, in the year $X+1$, the first day of the year) one should consider the distribution of the number of days since last time there was a hotter day than today. $\endgroup$ Jul 2, 2015 at 21:36
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    $\begingroup$ paris agreement who $\endgroup$ Jul 26, 2017 at 22:22
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    $\begingroup$ Even with something completely random (i.e., without seasonal trend underlying), in a random permutation of $\{1,2,\ldots, n\}$, you will have about $\ln n$ "hottest so far" points (namely, the $k$th entry is one with probability $\frac1k$) $\endgroup$ Aug 20, 2019 at 6:35

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Meteorological winter begins in December and ends in February; summer begins in June and ends in August. The highest temperatures are in June in Tucson (summer monsoon has increased humidity and decreases the high temperature average in July) and early August in Eugene, where such does not occur. The normal highs and lows are typically 30 year averages, which are recomputed every decade. I once tried to get all the temperatures for each day for 120 years to see if the "normal" high really was. Those temperatures were not available. To compare both hemispheres, one might consider comparing comparable meteorological seasons.

It's complicated further by the fact that the low temperatures, while increasing in the northern hemisphere as the year progress, don't peak when the high temperatures do; in Tucson, the highest low temperature occurs about a month after the highest high, due to cloudiness and less heat radiating outward at night.

The human factor is that meteorologists don't start referring to the hottest day of the year so far until at least the "shoulder season" or spring in the Northern hemisphere. In December, were the coldest day of the year recorded, it would make the annual summary, but it wouldn't likely be noted as the coldest at the time.

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