# Population Variance.

• Mar 16th 2013, 11:28 AM
Joyeux
Population Variance.
Ok, so I messed up displaying the questions in a previous thread now. Here is the question straight from my worksheet.

For the data set s={x_1,x_2,...,x_n}

Show /sigma x^2= (\sum \frac{x_i^2}{N})-\mu x^2

Where /mu x= /frac{\sum x_i}{N}

All sums between i=1 (Worried)
• Mar 16th 2013, 11:29 AM
Joyeux
Re: Population Variance.
Sorry, I have no clue as to how to make the website display the actual formula visually.
• Mar 16th 2013, 11:57 AM
ILikeSerena
Re: Population Variance.
Quote:

Originally Posted by Joyeux
Ok, so I messed up displaying the questions in a previous thread now. Here is the question straight from my worksheet.

For the data set $\displaystyle s={x_1,x_2,...,x_n}$

Show $\displaystyle \sigma_x^2= (\sum \frac{x_i^2}{N})-\mu_x^2$

Where $\displaystyle \mu_x= \frac{\sum x_i}{N}$

All sums between i=1 (Worried)

Hi Joyeux! :)

I took the liberty to make your formulas work.
If you click Reply with quote, you'll be able to see what I did.

I suspect you are supposed to start with the definition of variance.

$\displaystyle \sigma_x^2= \frac{\sum (x_i - \mu_x)^2}{N}$

Can you simplify this?
• Mar 16th 2013, 12:09 PM
Joyeux
Re: Population Variance.
I don't think that is what the question is asking, the question right after that one in the sheet asks for that. This one is just to substitute the /mu x into the first equation and simply to get the equation displayed.
• Mar 16th 2013, 12:18 PM
ILikeSerena
Re: Population Variance.
• Mar 16th 2013, 12:51 PM
Shakarri
Re: Population Variance.
From the definition of variance which Serena gave

$\displaystyle \sigma_x^2= \frac{\sum (x_i - \mu_x)^2}{N}$

Expand the squared values
$\displaystyle \sigma_x^2= \sum \frac{x_i^2- 2x_i\mu_x+\mu_x^2}{N}$

$\displaystyle \sigma_x^2= \sum \frac{x_i^2}{N}- \sum \frac{2x_i\mu_x}{N}+\sum \frac{\mu_x^2}{N}$

Use what you are given to simplify this.

Note: Since $\displaystyle \mu_x$ does not change with i

$\displaystyle \sum \mu_x= N\mu_x$
• Mar 16th 2013, 01:15 PM
Joyeux
Re: Population Variance.
How do I get from

This: /sigma x^2= (\sum \frac{x_i^2}{N})-(\frac {\sum x_i}{N})^2

To this: /sigma x^2= (\sum \frac{x_i^2}{N})-\mu x^2
• Mar 16th 2013, 07:47 PM
Joyeux
Re: Population Variance.