#### Abstract

In this post, we will quickly go through the math behind Bessel’s correction.

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### Bessel’s correction

First, let’s assume we have n independent observations from a population with mean $\mu$ and variance $\sigma^2$. The definition of population variance $\sigma^2$ is:

Given the observation, we can estimate $\sigma^2$ with the sample variance $\sigma_2^2$ from textbook:

Bessel’s correction is the usage of $n-1$ instead of $n$ in the denominator for the sample variance. It’s unintuitive to think that $\sigma_s^2$ is actually an unbiased estimation of $\sigma^2$:

### Some useful identities

To prove (3), we need to prove a few more useful definitions, namely $\mathrm{E}(x_i)$, $\mathrm{Var}(x_i)$, $\mathrm{E}(x_i^2)$, $\mathrm{E}(\bar{x})$, $\mathrm{Var}(\bar{x})$ and $\mathrm{E}(\bar{x}^2)$. By the population definition, we have:

For the sample mean $\bar{x}$, we have expected value:

Similarly, for variance of sample mean:

Given (7) and (8), we have:

### Proof

Given the above identities, proving (3) is straight forward. Let’s ignore the denominator $n-1$ for now:

Given (10), it’s not hard to see that: