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[Community Question] Statistics: Possible simplified variance

One of our user asked:

I noticed during calculating variance:

$$S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\overline{X})^2$$

that the sum $\sum_{i=1}^n(X_i-\overline{X})^2$ is equal to $\sum_{i=1}^n(X_i-\overline{X})*X_i$.

However I wasn't able to prove why this is true, or if there are cases in which it won't be equal.


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