# Alternative formula for variance problem

1. Jan 31, 2013

### Biosyn

1. The problem statement, all variables and given/known data
Which of the following formulas represents the variance of the data set {x1,x2,x3,...,x10}?
(μ denotes the mean of the data set)

Here is a photo that I took of the problem for better understanding.

http://i.imgur.com/PqEKajx.jpg?1?7734

I understand why the answer I chose is wrong.
What I don't understand is how e) is the answer. I did the calculations by hand with that formula and it is correct.

Would someone please show me how that formula is derived from the variance formula we usually see:

2. Relevant equations

Formula for variance

3. The attempt at a solution

2. Jan 31, 2013

### SammyS

Staff Emeritus

$\displaystyle \sigma^2=\frac{\sum_{i=1}^{10}(x_i-\mu)^2}{10}$​
Expand the square.

$\displaystyle (x_i-\mu)^2=x_i^2-2x_i\mu+\mu^2$

Now, take the sum of each term. Then recall how you calculate μ .

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3. Jan 31, 2013

### Biosyn

I get $\frac{{x_1 + x_2 + x_3....+x_10} + { 2x_1μ + 2x_2μ...+2x_10μ} + 10μ^2}{10}$

I can see where $\displaystyle \sigma^2=\frac{\sum_{i=1}^{10}(x_i)^2}{10}$ comes from and that 10μ^2 cancels out.

What about the ${2x_1\mu + 2x_2μ...+2x__10\mu}$ ?

p.s. Sorry, having a hard time with Latex. I hope you understand what I mean! :O

Last edited: Jan 31, 2013
4. Jan 31, 2013

### SammyS

Staff Emeritus
Divide that by N, i.e. 10 .

That's $\displaystyle \ \ 2\mu\frac{\sum_{i=1}^{10}x_i}{10}$

5. Jan 31, 2013

### Biosyn

Oh, stupid me. >.>

I understand now:
http://i.imgur.com/NpdcN86.jpg

6. Jan 31, 2013

### Ray Vickson

You should not get $\frac{{x_1 + x_2 + x_3....+x_10} + { 2x_1μ + 2x_2μ...+2x_10μ} + 10μ^2}{10}$, which is wrong. You should not get $\displaystyle \sigma^2=\frac{\sum_{i=1}^{10}(x_i)^2}{10}$ because that is also wrong unless μ = 0. Start over, and proceed carefully!

7. Jan 31, 2013

### Biosyn

Sorry, I was a bit lazy typing out my work. But in the post before this I solved it!
http://i.imgur.com/NpdcN86.jpg
Thanks anyways. :P