As far as everyday usage goes, both mean and average mean the same thing. Usually, in normal speech, we use the term 'average' much more than the term 'mean', which usually sounds more technical.
In statistics, average is used as a name for a general statistic that usually falls around the center of the data. Mean, on the other hand, is usually the sum of all values divided by the total number of values. But this itself applies only for arithmetic means. You can also have differently defined geometric and harmonic means.
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