# Alternative formula for variance problem

• Biosyn
In summary, the formula for variance of a data set {x1,x2,x3,...,x10} is given by σ^2 = ∑(x_i – μ)^2 / 10, which can be derived by expanding the square and taking the sum of each term. This is equivalent to σ^2 = ∑(x_i)^2 / 10, with the difference being the inclusion of 2μ(∑x_i / 10) in the first formula.
Biosyn

## Homework Statement

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:

## Homework Equations

Formula for variance

Biosyn said:

## Homework Statement

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:

## Homework Equations

Formula for variance

## The Attempt at a Solution

$\displaystyle \sigma^2=\frac{\sum_{i=1}^{10}(x_i-\mu)^2}{10}$​
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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SammyS said:

$\displaystyle \sigma^2=\frac{\sum_{i=1}^{10}(x_i-\mu)^2}{10}$​
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 .
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:
Biosyn said:
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 ${2x_1\mu + 2x_2\mu...+2x_10\mu}$ ?

Divide that by N, i.e. 10 .

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

SammyS said:
Divide that by N, i.e. 10 .

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

Oh, stupid me. >.>

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

Biosyn said:
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

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!

Ray Vickson said:
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!

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

## 1. What is the purpose of an alternative formula for variance?

The alternative formula for variance is used to calculate the spread or variability of a set of data. It is an alternative to the traditional formula for variance and is often used when the data set is large or when the traditional formula may be prone to errors.

## 2. How is the alternative formula for variance different from the traditional formula?

The traditional formula for variance uses the mean of the data set in its calculation, while the alternative formula uses the median. This means that the alternative formula is less influenced by extreme values in the data set and can be more accurate in certain situations.

## 3. When should the alternative formula for variance be used?

The alternative formula for variance should be used when the data set is large or when there are extreme values present. It can also be used when the data set is not normally distributed, as the traditional formula assumes a normal distribution.

## 4. How do you calculate the alternative formula for variance?

The alternative formula for variance is calculated by finding the difference between each data point and the median, squaring these differences, and then finding the average of these squared differences. This value is the alternative variance.

## 5. Can the alternative formula for variance be used for any type of data?

The alternative formula for variance can be used for any type of data, as long as the data set is numerical. It is not limited to only normally distributed data, unlike the traditional formula for variance.

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