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> The concept of an arithmetic mean is quite artificial - there's other ways you could define "average" such as the median. I'd give some pushback on that. It
by rocqua 8y ago
> The concept of an arithmetic mean is quite artificial - there's other ways you could define "average" such as the median.
I'd give some pushback on that. It is the only linear mean, and linearity matters.
More importantly though, it corresponds to the Expected Value. That is, the average of n samples of a distribution f converges to E[f]. In fact, I believe the arithmetic mean is the best unbiased estimator for the expected value of a distribution.
Now, expected values matter because they are the basis of robust decision-making.
- FabHK 8y agoYou can consider mean, mode, and the other centrality measures as points that minimise some sort of average deviation. And depending on which norm you use to measure your deviation, you get different centrality measures: * L_0 -> mode * L_1 -> median * L_2 -> mean * L_infinity -> midrange, I think where roughly L_p(x) = [ sum_i |x-x_i|^p]^(1/p) ]
- rocqua 8y agoAmong these norms, again the L_2 norm stands out. It being the standard measure of distance. Here, it is important because it is invariant under rotation. We can define rotation without any appeal to length as the "special linear group". This consists of all matrices that have determinant 1. I wonder where exactly the connection between 'rotation invariance' and 'estimate of the expected value' comes from.