# Thread: Need help with variance proof

1. ## Need help with variance proof

(a) Show that the sample variance is unchanged if a constant c is added to or subtracted from each value in the sample.

(b)Show that the sample variance becomes $\displaystyle c^2$ times its original value if each observation in the sample is multiplied by c.

This was something our professor gave us to do. I showed both by using problem examples from our textbook, but he said that it would be wise to be able to give a general proof for each, should it come up on our exam. I've never really done any proofs so I'm at a bit of a loss as where to start and whatnot. I'll be trying to figure this out but any help would definitely be appreciated.

2. I don't know if this is a correct approach though...

Let x represent the values in a set of sample.
Let y be the new set of values, where y = x+c.

y = x + c

So, Sd(y) = Sd(x + c)

Which is:

Sd(y) = Sd(x) + Sd(c)

Does a constant has any standard deviation? No, so, Sd(c) = 0

And

Sd(y) = Sd(x)

which implies that Var(y) = Var(x)

As $\displaystyle Sd(x) = \sqrt{Var(x)}$

~~~~~~~~~~~

y = xc

Sd(y) = Sd(xc)

Sd(y) = c(Sd(x))

which implies that:

$\displaystyle Var(y) = c^2Var(x)$

3. Thanks for the reply. I haven't really looked at your answer yet but the professor posted hints that make me believe you're on the right track. He wrote:

Part A. Hint: Define y_i=x_i+c. Express the sample variance of the y’s and simplify the expression.
Part B. Hint: Define y_i=cx_i . Express the sample variance of the y’s and simplify the expression.
Just glancing at what you typed, it seems that you did the same thing. I'm going to look over what you typed to make sure I understand what's going on. If I have further questions I'll be sure to post them.

Thanks

4. I think I just got part a. If someone could look it over it'd be greatly appreciated. Also, I not too sure on the proper format of a proof, so if someone could point out any errors regarding that it'd be much appreciated.

Solution is attached as a .pdf. I'm going to work on getting typed out on here because my handwriting isn't the best.

Solution for part a:

$\displaystyle Var[x+c]=E[(x+c^2)]-(E[x+c])^2)$
$\displaystyle =E[x^2 + 2cx+c^2]-[(E[x])^2 +2cE[x]+c^2]$
$\displaystyle =E[x^2]+2cE[x]+c^2 - E[x]^2 -2cE[x]-c^2$
$\displaystyle =E[x^2]-(E[x])^2$
$\displaystyle =Var[x]$

I think I got the solution for part b as well. Just need someone to look over it. Thanks

I'm starting to think what I did was wrong. I'm too tired to look it over right now though.

Okay, I went ahead and redid part a. I believe my mistake was that I was using population variance earlier instead of proving the case for sample various. Please take a look at the attached "part a revised.pdf"

If it's correct, I'll move forward with part b. Thanks