If you model x_i as a random sequence generated by a random variable:

    X ~ s + eps

Where eps is normal distributed with mu=0,sigma>0. The probability density of the observation x_1, \dots, x_n is:

   p(x_1,\dots,x_n) = C * exp(-(s-x_1)^2) * exp(-(s-x_2)^2) ... * exp(-(s-x_n)^2)

Taking the log you get

   Log(p(x_1,\dots,x_n)) = - C' * sum_i (s-x_i)^2

So the probability density of your observation is maximized if the sum-of-squares is minimized.

This means if you set s to mean(x_i), you will maximize the "log-likelyhood" of your observation.