Most pure statisticians prefer to use mean instead of average. The two words describe the same statistical quantity and as such are synonymous. However, the median is a related but different concept.
The mean and median are each centers of measure for values in a data set. Whereas the mean describes the average value in a data set, the median describes the value in the middle value of a sorted data set.
For example, suppose we have a data set containing five values: 1, 3, 4, 7 and 11. The average of these values is
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