Wikipedia:Reference desk/Archives/Mathematics/2022 November 26

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November 26

Concerning specific properties of the arithmetic mean

So to calculate the mean squared error of a data set we divide the sum of (Sn - Sa)^2 for every Sn in S by the size of the set, N, where Sa is the arithmetic mean of S. But oddly enough, it also works if we replace (Sn - Sa)^2 with (Sn^2 - Sa^2). Why is that? What special property of the mean allows for that sort of rearrangement? Earl of Arundel (talk) 03:51, 26 November 2022 (UTC)

If you expand (Sn - Sa)^2 you get (Sn^2 - 2Sa Sn + Sa^2). Summing the middle term becomes Sum (2 Sa Sn) which is 2 Sa Sum (Sn). Dividing by n gives 2 Sa Sum (Sn) / n. But [Sum (Sn) / n] is just the mean, Sa, so it becomes 2Sa^2. Hence the correspondence. 5.81.136.7 (talk) 04:07, 26 November 2022 (UTC)

OK, that's simple enough. Thanks! Earl of Arundel (talk) 05:02, 26 November 2022 (UTC)