Post from Philip Eden explaining the methods
People can work out their monthly mean whatever way they like. But if you do not conform to a standard then those figures will only be of interest to yourself ... they'll be largely useless and meaningless to anyone else.
The standard, for historical reasons, is the average of the mean monthly maximum and the mean monthly minimum (as recorded at 09z) as John and others have explained.
The main reason is simply that, before AWSs arrived, most climatological stations were read only once per day, at 09z, and that was the case for over a century. It's only in the last 10 years or so that relatively cheap and reasonably reliable AWSs have provided means of logging data throughout the day.
The mean monthly temperature, as described above, is of course only an approximation to the true mean, but the main use of these figures is for comparing one month with another, and one site with another. Consider it as a monthly temperature index if you like, rather than a true mean temperature.
If you think it's also important to have the closest approximation to a true mean temperature by integrating your hourly or 10-minute or 5-minute loggings throughout the month, then by all means do so. It's not forbidden to have the standard monthly temperature index AND an integrated mean temperature. Do both.
But for purposes of geographical and/or temporal comparison, use the standard.
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