Prove the sample variance formula.

**Iceflash234** · September 7th 2010, 11:51 AM

Basically I have to prove that the sample variance of a sample is given by:

$s^2 =$ ${\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2\over n-1}$

For my purposes it's sufficient to show that ${\sum_{j=1}^n(x_j-x_j^2) = {\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2$

Help is appreciated. Thanks in advance.

If I made any mistakes inputting the sums please let me know.

**mr fantastic** · September 7th 2010, 12:51 PM

Originally Posted by Iceflash234

Basically I have to prove that the sample variance of a sample is given by:

$s^2 =$ ${\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2\over n-1}$

For my purposes it's sufficient to show that ${\sum_{j=1}^n(x_j-x_j^2) = {\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2$

Help is appreciated. Thanks in advance.

If I made any mistakes inputting the sums please let me know.

This is a standard proof found in most textbooks on mathematical statistics. Have you consulted any?

**Iceflash234** · September 7th 2010, 01:52 PM

Originally Posted by mr fantastic

This is a standard proof found in most textbooks on mathematical statistics. Have you consulted any?

I checked all the ones I have access to. I even tried googling it.

**mr fantastic** · September 9th 2010, 03:06 AM

Originally Posted by Iceflash234

Basically I have to prove that the sample variance of a sample is given by:

$s^2 =$ ${\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2\over n-1}$

For my purposes it's sufficient to show that ${\sum_{j=1}^n(x_j-x_j^2) = {\sum_{j=1}^nx_j^2-{1\over n}}(\sum_{j=1}^nx_j)^2$

Help is appreciated. Thanks in advance.

If I made any mistakes inputting the sums please let me know.

Your starting point should be the definition of sample variance: Sample Variance -- from Wolfram MathWorld

What progress have you made from this?

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