Bessel's Correction - Alternate Proof of Correctness

Alternate Proof of Correctness

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Recycling an identity for variance,

\begin{align} \sum_{i=1}^n \left(x_i - \overline{x} \right)^2 &= \sum_{i=1}^n \left(x_i - \frac 1 n \sum_{j=1}^n x_j \right)^2 \\ &= \sum_{i=1}^n x_i^2 - n \left(\frac 1 n \sum_{j=1}^n x_j \right)^2 \\ &= \sum_{i=1}^n x_i^2 - n \overline{x}^2 \end{align}

so

\begin{align} \operatorname{E}\left(\sum_{i=1}^n \left^2 \right) &= \operatorname{E}\left(\sum_{i=1}^n (x_i-\mu)^2 - n (\overline{x}-\mu)^2 \right) \\ &= \sum_{i=1}^n \operatorname{E}\left((x_i-\mu)^2 \right) - n \operatorname{E}\left((\overline{x}-\mu)^2\right) \\ &= \sum_{i=1}^n \operatorname{Var}\left(x_i \right) - n \operatorname{Var}\left(\overline{x} \right) \end{align}

and by definition,

\begin{align} \operatorname{E}(s^2) & = \operatorname{E}\left(\sum_{i=1}^n \frac{(x_i-\overline{x})^2}{n-1} \right)\\ & = \frac{1}{n-1} \operatorname{E}\left(\sum_{i=1}^n \left^2 \right)\\ &= \frac{1}{n-1} \left \end{align}

Note that, since x1, x2, · · ·, xn are a random sample from a distribution with variance σ2, it follows that for each i = 1, 2, . . ., n:

and also

This is a property of the variance of uncorrelated variables, arising from the Bienaymé formula. The required result is then obtained by substituting these two formulae:

\operatorname{E}(s^2) = \frac{1}{n-1}\left = \frac{1}{n-1}(n\sigma^2-\sigma^2) = \sigma^2. \,


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