Expected and actual weather

This is a somewhat odd post and is pretty much me trying to understand part of what I have observed with the weather.  If fiddling with numbers is not your thing, I suggest you close this post now!

Plain explanation of what has been seen.

The post covers some playing with current maximum and minimum temperatures and the averages of those attributes for the previous 27 years for each day.  The detail of that follows.  However what I realised was that there was a fair degree (r = 0.71) of correlation between the 2 values.   In essence when the minimum is high so is the maximum.

In this area frontal activity is usually a cold front so it should not be very unusual to get an above average minimum followed by a below average maximum.  However a warm front is most unusual  so getting a below average minimum followed by an above average maximum would be unexpected.

That leads to the following situation

% of observations
	min above	min below
max above	27	18
max below	24	31

In future months I shall use those proportions, applied to the total number of days in the month, to assess the pattern for the month.

Detail and Methods

It arose as a result of my calculating the temperature anomaly for each month.  That is the difference between the current value of a variable and the long term average of that variable.   Thus for March 2020 the average temperature for the month was was 18.34^oC while the long term average temperature for April is 18.44 so the month was very slightly below average temperature.

In compiling those averages I have listed the actual minimum and maximum temperatures for the day and calculated the average temperatures for that day (all data being that collected by the Bureau of Meteorology - BoM).  In August 2019 I started coding these data according to whether the minimum and maximum were above or below average.  That gave a table such as this (for March 2020):

	min above	min below	Total
max above	10	1	11
max below	8	12	20
Total	18	13

I started to wonder if this distribution was different to "normal" which immediately suggested a Chi-square test.  To do that I needed a set of expected values and set them as 7.75 for each cell (adding to 31) which I felt would be an approximation to normal.  The resultant values of Chi square were significantly different to 0 in each month.  The March 2020 result gives a probability of the value of Chi square calculated by EXCEL being due to chance of 0.002 .  That is a high likelihood the data is different to assumed equal distribution.

However it seemed that every month the low achiever was the minimum below, maximum above cell suggesting that the assumption of equality was not sound.  So I calculated the actual table for the 7 months for which I have data (January was not available as the BoM site was not working for several days due to power problems after the fire).  That gave a table 

	min above	min below
max above	57	38
max below	52	66

The low value for the cell of interest is very obvious.  It is significantly different to both of the cells showing the extremes moving in the same direction.  Overall, when compared to a table with equal cells, this table is very close having a value of Chi square significant at the 5% probability.

My next step was to create an expected table with the same proportions in each cell (but with a total for the table of 31) and then calculating the values of Chi square for each month.  The results of that exercise follow: I compared what the analysis seemed to show against the monthly reports and they were (fortunately) quite compatible.

1. August 2019:  Chi square significant at 2% level.   This was mainly due to a high proportion of maxima below normal.
2. September 2019: Chi square significant at 10% level.   Not a strong signal and difference was mainly due to a high proportion of minima below normal.
3. October 2019 Chi square significant at 10% level.   Not a strong signal and difference was mainly due to a high proportion of minima below normal.
4. November 2019: Chi square not significant at 10% level.   A little surprising as most minima below average.
5. December 2019 Chi square significant at 0.01% level.   This was mainly due to a very high proportion (17/31) of results with both maximum and minimum above normal.
6. January 2020:  no analysis - see above.
7. February 2020: Chi square significant at 1% level.   The pattern was closer to average but with cooler days and warmer nights, possibly reflecting a lot of cloud.
8. March 2020: Chi square significant at 5% level.   The pattern was closer to average but with cooler days and warmer nights