Proof that mean squared deviations are minimised by the sample mean.

Define a function to equal the mean of the squared deviations of a sample from a variable x.
mean square deviation from x
Take a binomial expansion of the square
binomial expansion

Now use the definition of the sample mean
definition of the mean
to rewrite this equation in the form

where N is the number of measurements in the sample and Xi is the value of the ith measurement.

Finally minimise this function with respect to the variable x.
minimisation
which has the solution x = m.

Thus the function f(x), measuring the mean squared deviations from x has its minimum value at the sample mean m and hence the mean squared deviations about any other point, including the true mean, must be larger than s2 = f(m). QED.

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