(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that the sample variance of a sample is given by

S^{2}= [tex]\frac{\sum^{n}_{j=1}x_j^2 - \frac{1}{n} (\sum^{n}_{j=1}x_j)^2}{n-1}[/tex]

2. Relevant equations

N/A

3. The attempt at a solution

For my purposes it is sufficient to show that:

[tex]\sum^{n}_{j=1}(x_j -x_j^2) = \sum^{n}_{j=1}x_j^2 - \frac{1}{n}\left(\sum^{n}_{j=1}x_j\right)^2[/tex]

I got as far as this:

[tex]= \sum^{n}_{j=1}x_j^2 - \sum^{n}_{j=1}2\bar{x}x_j + \sum^{n}_{j=1}\bar{x}^2[/tex]

I need help getting from there to here:

[tex] = \sum^{n}_{j=1}x_j^2 - \frac{1}{n}\left(\sum^{n}_{j=1}x_j\right)^2[/tex]

Thanks in advance and I apologize for any coding error.

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# Homework Help: Prove the sample variance formula.

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