During extreme weather events there are at least 3 different ways of assessing the temperature against other historical extreme events. We could immediately turn that into 9 different ways if you isolate the max, min, and mean temperatures. Further, we could isolate the time period as daily, weekly, biweekly, monthly, seasonally, etc.
But let's simplify things, and only refer to the question of which months were the hottest and coldest on record. A common refrain that we've all heard is that on the Prairies they experienced their coldest month on record in February 1936, and then their warmest month on record 5 months later in July of the same year. This is based upon the absolute coldest months.
Another way of looking at and comparing months would be to compare how much the temperature deviated from the normal. Let's say that Pheonix had some freak July where the temperature averaged 30 degrees below average, say something more like a typical February. Now, in my mind this would be the coldest month on record because it is the most out of the ordinary anomaly.
Better yet, we should really be calculating the standard deviations of each month, and then figuring out which months were the most number of deviations from average.
Let's start with Calgary since it has records back to 1885, and it's on the prairies, which was apparently at its coldest and hottest in 1936. Because Calgary is further west, the absolute average temperature of February 1936 was beaten by the great January 1950 extreme cold snap that those west of the Rockies would know about, but February 1936 was 3.5 deviations from the average. this compares to January 1950 which was only 2.9 deviations from average.
Another month that anyone in Seattle or Vancouver would know about would be November 1896, which was so cold that no December has ever been that cold. In Calgary this same month, though 8 degrees warmer than January 1950, was 3.7 deviations from average.
Similarly, July 1936 is the 2nd hottest month on record in Calgary in terms of the absolute average since August 1971 edged it out by 0.2 degrees. These two months deviated by 2.7 and 3.0, respectively.
One month beats the August deviation, and by quite a bit, and that is June of 1961, which was 3.4 deviations from the average.
Therefore, the coldest and hottest months on record in Calgary were November 1896 and June 1961.
ASIDE: Further west where I live in British Columbia, the extreme cold snap of 1896 was beat in November of 1985. Interestingly, the coldest temperature recorded in my lifetime in southern BC occurred then. We are talking -52F in southern British Columbia in November.