The sample mean as a formula makes intuitive sense, but seems to come from nowhere. In this page we derive the sample mean from first principles as the value for the data that makes the deviation from the estimate as small as possible (in a special way).
The idea is that we have data
We can think of the distance of each observation to our chosen number, it will be
Consider the sum of the squared differences, i.e. sum up
So this is where the equation comes from. If we chose a different function besides
squaring the differences, we would get a different estimator for the center. For example
if we took instead the absolute value of the difference,
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