TITLE: Standard Deviation vs Population Standard Deviation
QUESTION [5 upvotes]: In GRE study guide, it gives difference between standard deviation and sample or population standard deviation.
I understand the mechanics of this, i.e. in standard deviation you divide all differences of the mean by $n$, but in population standard deviation, you divide by $n - 1$
Honestly, why not just divide by $n$? I am really not understanding the logic of this.
When I google for an explanation, I find the same mechanical difference.
Kindly explain.

REPLY [4 votes]: You have this backwards: It is in the sample variance that you divide by $n-1$.  If $n$ is the size of the whole population then the population variance is the average of the squares of the deviations, and that is the sum of those squares divided by $n$.
Division by $n-1$, if done at all, should be done ONLY when using the sample to ESTIMATE the variance of the whole population.
This section of a Wikipedia article explains why it is done.
Whether it ought to be done, i.e. whether estimates ought to be unbiased, is debatable.
You may have read that if $X_1,\ldots,X_n$ are independent random variables then $$\operatorname{var}(X_1+\cdots+X_n) = \operatorname{var}(X_1) + \cdots + \operatorname{var}(X_n). \tag 1$$  That works with the one where you divide by $n$, but not with the one where you divide by $n-1$.  The identity $(1)$ is the reason why standard deviation rather than mean absolute deviation or some other measure of dispersion is used.