# Is it possible to mathematically approximate the most extreme value experienced in the average year from the most extreme values experienced in each average month, if a statistical distribution is assumed? If so, how would you do that? : meteorology

3 min read(Not exactly sure if this would belong here or r/MathHelp, but I’m posting here because I think there’s too much a conceptual gap for them to be of much help given their ruleset—I don’t think I’ve showed enough of an initial attempt.)

For example… Let’s say I have data that looks like this:

Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | Year |
---|---|---|---|---|---|---|---|---|---|---|---|---|

52.1 | 57.2 | 72.4 | 82.1 | 88.9 | 94.2 | 96.3 | 94.2 | 90.9 | 82.6 | 68.4 | 57.1 | 97.4 |

31.6 | 35.3 | 46.0 | 59.0 | 70.5 | 80.6 | 84.7 | 82.7 | 75.6 | 64.1 | 48.4 | 35.7 | 59.5 |

The top data row contains the highest temperatures (in °F) recorded in the average month and year, and the bottom row contains the average highs over those periods, all for the weather station 4 miles southwest of Midway International Airport in Chicago (111577), over its entire period of record. As you can see, while July has the highest average monthly record high, the fact that other months occasionally experience the hottest temperature in the year (17 out of 41 years on record, in fact, with May, June, August, and September all making successful shots) leads to the average yearly record high being somewhat higher.

Whoops! *crashing sounds* *boom*

Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | Year |
---|---|---|---|---|---|---|---|---|---|---|---|---|

52.1 | 57.2 | 72.4 | 82.1 | 88.9 | 94.2 | 96.3 | 94.2 | 90.9 | 82.6 | 68.4 | 57.1 | |

31.6 | 35.3 | 46.0 | 59.0 | 70.5 | 80.6 | 84.7 | 82.7 | 75.6 | 64.1 | 48.4 | 35.7 | 59.5 |

I lost the average annual record high! But I still want to use the data! **What can I do?**

Now, at least if we know or assume a statistical distribution (say a normal distribution, which meteorological temperature generally very roughly follows), we can calculate standard deviations for the daily highs in each month, as statistically, average monthly record highs are just the high temperatures for which 27.25/28.25 to 30/31 of the high temperature curve lies below:

Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | Year |
---|---|---|---|---|---|---|---|---|---|---|---|---|

20.5 | 21.9 | 26.4 | 23.1 | 18.4 | 13.6 | 11.6 | 11.5 | 15.3 | 18.5 | 20.0 | 21.4 | (37.9 |

11.1 | 12.1 | 14.3 | 12.6 | 10.0 | 7.4 | 6.3 | 6.2 | 8.3 | 10.0 | 10.9 | 11.6 | (13.6 |

(Top data row is the difference between the average record highs and average highs, while the bottom data row is the calculated standard deviation.)

This can in turn be used to roughly estimate the *probability* that a given month will exceed the average record high in July (not necessarily the yearly July record high in the years that happens):

Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec |
---|---|---|---|---|---|---|---|---|---|---|---|

64.7 | 61.0 | 50.3 | 37.3 | 25.8 | 15.7 | N/A | 13.6 | 20.7 | 32.2 | 47.9 | 60.6 |

5.84 | 5.03 | 3.52 | 2.96 | 2.58 | 2.12 | N/A | 2.19 | 2.48 | 3.22 | 4.39 | 5.24 |

1 in ~12.1 Myr | 1 in ~147 Kyr | 1 in ~151 years | 1 in ~22 years | 1 in ~6–7 years | ~4–5 in 10 years | N/A | ~4–5 in 10 years | 1 in ~5 years | 1 in ~50 years | 1 in 5.95 Kyr | 1 in ~393 Kyr |

(Top data row is the maximum temperature anomaly required to exceed the average July record high, the second row is the Z-score of that anomaly, and the third row is the estimated period of return of that anomaly assuming a normal distribution with no skewness [again, not exactly realistic, but ehh.])

**But I’m unsure on how to go any further, if doing so is even possible. Anyone have any insights?**

(And yes, I know that is *very* rare for average monthly extremes to be provided and not either average yearly extremes or their immediate precursors [extremes for individual years/months], *but I am dealing with just such an instance…*)

/u/GrantExploit

2022-04-09 02:00:01

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