# Show the two forms of the sample variance are equivalent

## Homework Statement

Showthe two forms of the sample variance are equivalent:
$\frac{1}{n-1}$$\sum_{i=1}^\n (Yi-Ybar)2$ = $\frac{1}{n(n-1)}$$\sum_{i=1}^\n \sum_{j>i}\n (Yi-Yj)2$

The first summation is from i=1 to n, the second is i=1 to n and the third is j>i to n. Sorry, I don't know how to format those.

## The Attempt at a Solution

I don't really know where to begin, so I tried just expanding and cancelling where I could. I know the (n-1) in the denominator on both sides cancel, and then after expanding I get

$\sum (Yi2-2YiYbar + Ybar2$ = $\frac{1}{n}$$\sum \sum (Yi2-2YiYj+Yj2)$

Then, I can distribute the 1/n and the summations on the right side. If I do that, and I have a term that does not have j (such as Yi), I can essentially drop the j summation from that term, correct? After I do that, I have the following:

$\sum Yi2$ -2$\sum YiYbar$ + $\sum Ybar2$ = $\frac{1}{n}$$\sum Yi2 - \frac{2}{n}$$\sum \sum YiYj$+$\sum \sum Yj2$

And here's where I'm stuck (assuming I even did everything right to get here, which I doubt). I don't know how to deal with the Yj, among other things. Any help would be greatly appreciated.

edit: I'm sorry about the terrible formatting. I tried using the LaTex tag buttons and I've looked at the FAQ; not sure what I'm doing wrong.

Last edited:

Mark44
Mentor

## Homework Statement

Showthe two forms of the sample variance are equivalent:
$\frac{1}{n-1}$$\sum_{i=1}^\n (Yi-Ybar)2$ = $\frac{1}{n(n-1)}$$\sum_{i=1}^\n \sum_{j>i}\n (Yi-Yj)2$

The first summation is from i=1 to n, the second is i=1 to n and the third is j>i to n. Sorry, I don't know how to format those.

## The Attempt at a Solution

I don't really know where to begin, so I tried just expanding and cancelling where I could. I know the (n-1) in the denominator on both sides cancel, and then after expanding I get

$\sum (Yi2-2YiYbar + Ybar2$ = $\frac{1}{n}$$\sum \sum (Yi2-2YiYj+Yj2)$

Then, I can distribute the 1/n and the summations on the right side. If I do that, and I have a term that does not have j (such as Yi), I can essentially drop the j summation from that term, correct? After I do that, I have the following:

$\sum Yi2$ -2$\sum YiYbar$ + $\sum Ybar2$ = $\frac{1}{n}$$\sum Yi2 - \frac{2}{n}$$\sum \sum YiYj$+$\sum \sum Yj2$

And here's where I'm stuck (assuming I even did everything right to get here, which I doubt). I don't know how to deal with the Yj, among other things. Any help would be greatly appreciated.

edit: I'm sorry about the terrible formatting. I tried using the LaTex tag buttons and I've looked at the FAQ; not sure what I'm doing wrong.

What you wrote is very hard to read. The problem you're having with formatting comes from mixing HTML tags (e.g., and ) with LaTeX script. Use one or the other, but not both.

Here's your first equation, cleaned up:
$\frac{1}{n-1} \sum_{i=1}^n (Y_i - Ybar)^2 = \frac{1}{n(n-1)}\sum_{i=1}^n \sum_{j>i}^n (Y_i-Y_j)^2$

1 person
Ray Vickson
Homework Helper
Dearly Missed
What you wrote is very hard to read. The problem you're having with formatting comes from mixing HTML tags (e.g., and ) with LaTeX script. Use one or the other, but not both.

Here's your first equation, cleaned up:
$\frac{1}{n-1} \sum_{i=1}^n (Y_i - Ybar)^2 = \frac{1}{n(n-1)}\sum_{i=1}^n \sum_{j>i}^n (Y_i-Y_j)^2$

Even better:
$$\frac{1}{n-1} \sum_{i=1}^n (Y_i - \bar{Y})^2 = \frac{1}{n(n-1)}\sum_{i=1}^{n-1} \sum_{j=i+1}^n (Y_i-Y_j)^2$$
Note that in the second form, if we have j>i with j going up to n, then i can only go up to (n-1).

1 person
Ah, I see. I was just using the buttons at the top and side. Hopefully I'll get this right this time.

The original equation.

$$\frac{1}{n-1} \sum_{i=1}^n (Y_i - \bar{Y})^2 = \frac{1}{n(n-1)}\sum_{i=1}^{n-1} \sum_{j=i+1}^n (Y_i-Y_j)^2$$

Like I said, the (n-1) from both sides cancel, and then I expand both sides

$$\sum_{i=1}^n(Y_i^2 -2Y_i \bar{Y} + \bar{Y}^2 = \frac{1}{n}\sum_{i=1}^{n-1} \sum_{j=i+1}^n(Y_i^2 - 2Y_iY_j + 2Y_j^2)$$

I then distribute the summations to all the terms

$\sum_{i=1}^nY_i^2-2 \sum_{i=1}^nY_i\bar{Y} + \sum_{i=1}^n\bar{Y}^2 = \frac{1}{n}\sum_{i=1}^{n-1} \sum_{j=i+1}^nY_i^2 - \frac{2}{n}\sum_{i=1}^{n-1} \sum_{j=i+1}^n Y_iY_j + \frac{1}{n}\sum_{i=1}^{n-1} \sum_{j=i+1}^nY_j^2$

I don't really know where to go from here. I know that $\frac{1}{n}\sum_{i=1}^nY_i=\bar{Y}$, but I'm not sure what that does from me. I'll still have the j summations to worry about on the right hand side, as well as the Yj terms.