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I hope nobody is actually redefining the word "average" to mean median, because that would confuse all I thought I knew about statistics.

Is there a less allegedly ambiguous term for the "traditional" definition of average, that is, the sum of many measurements of the same characteristic, divided by the number of elements measured?

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Average can refer to any of the common measures of central tendency, as Wikipedia calls them: https://en.wikipedia.org/wiki/Average#Harmonic_mean

Average intelligence, for example, refers to median intelligence (equivalent to an IQ of 100). Mean intelligence, however, is likely the same as median intelligence due to its normal distribution.

Mean, surely?

Very true. Thank you.

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