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It is abundantly clear that February will be the coldest month this winter, which doesn't happen very often. As a mater of fact, over the past 55 years it has only occurred 5 times in my area. This is under 10%. For my area, I picked Kamloops (~51N, 121W) because of the long data set.
Using 115 years of data (mean temperature):
November is the coldest month of winter 2% of the time
December is the coldest month of winter 24% of the time
January is the coldest month of winter 57% of the time
February is the coldest month of winter 17% of the time
I used November and December of the previous year. Starting the year in January gives a slightly different result, namely tripling the number of times that November was the coldest month of the year. Note: 100 years ago, February was twice as likely to be the coldest month of the year as it is today, but it appears to me based upon the last 5 to 10 years that February is starting once again to rise in popularity.
For the hottest month of the year, I get:
June is the hottest month of the year 5% of the time
July is the hottest month of the year 71% of the time
August is the hottest month of the year 23% of the time
March has never been the coldest month over there? I'm not aware of the same information being easily available here, and I don't have the time to work it all out manually, but I had a look for March in our CET data (bear in mind this is inland - March being colder is even more likely on the coast), and March has been the coldest month of the season in 15 of the past 354 years, though last year was the only time since 1950.
I did a Google search for November being the coldest month, and if you give half points for when it was joint coldest with another month, it has been the coldest 8.5 times in 354 years, though this data is for it being the coldest in a calendar year, so not quite the same thing.
With our low seasonal range it isn't surprising that the data set throws up some odd results, but even if you only use data from the era of modern equipment we have had the mild December of 1974 come in only 1C below the cold May of 1996. Perhaps the most surprising thing of all is that there have been three occasions when the two coldest months of a winter season were November and March, with all three "winter" months coming in milder than either of them.
March has never been the coldest here. Not even close. March and February are the driest months of the year, and typically quite sunny, which pushes the average temperatures up. The absolute coldest Marches were was still 2.2 and 2.7 degrees C warmer than the coldest month those years (February). The average March is 10.3 C (18.5F) warmer than the coldest month each winter, and more than 2 degrees C warmer than November. In 1916 and 1950 March was more than 24C warmer than January.
I calculated the probability of the coldest and warmest month for my hometown as well.
Available data is from 1948 on, so there are 66 years of records. Used also the monthly means here.
My gut instinct would say Februar would have the highest chance being the coldest and August being the warmest before looking at that data.
The span for the coldest month goes here from November (happend only once in 1993) to March (which happened three times in 1949, 1958 and just recently in 2013).
Month Occurences Probability being the coldest month
November 1x (1993) 2%
December 18x 27%
January 22x 33%
Februar 22x 33%
March 3x (1949, 1958, 2013) 5%
So January and February have an equal chance to become the coldest month in a year, followed closely by December. Having March as the coldest month here is not as rare as i thought so the cold March 2013 despite rare was not really a once in a lifetime event. The very cold November in 1993 probably is.
And now the same with the warmest months- here the span goes from June to September in the past 66 years.
Month Occurrences Probability being the warmest month
June 5x (1960, 1961, 1966, 1970, 1992) 8%
July 29x 44%
August 31x 47%
September 1x (1949) 2%
Here the distribution is different from those in the winter.
91% of all warmest summer months either occur in July and August, its less likely to have to warmest
month in June with only 8%, and even less likely in September with only one occurrence in 1949.
Here's another interesting thing. September is slowly gaining ground on May. At one time May was warmer than September, but not any more. Trend line is the 5 year running average.
The Coldest month here in Central Indiana is almost always January and the Hottest Month is Usually July, although this past year the hottest month was in August.
I've updated Kamloops with graphs. Note that for the coldest month of the year I mean winter (November and December of the previous year vs the current year). Here we go:
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