Thread: Prove the sample variance formula.

1. Prove the sample variance formula.

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

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

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

Help is appreciated. Thanks in advance.

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

2. Originally Posted by Iceflash234
Basically I have to prove that the sample variance of a sample is given by:

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

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

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?

3. 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.

4. Originally Posted by Iceflash234
Basically I have to prove that the sample variance of a sample is given by:

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

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

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?