Unbiased sample variance formula equvivalent

[Aedrha2]

Hi there I just started a course in mathematical statistics. And have already run into some questions.

I'm not sure about the terms here since the course is not in english but I think the formula below is for unbiased sample variance:

$$s^2=\frac{1}{n-1}\sum_{i=1}^n(x_i-\overline x)$$
Our text book says "The variance is often easier to calculate from the mathematicly equivalent expression:"

$$s^2=\frac{1}{n-1}\left(\left(\sum_{i=1}^nx_i^2 \right)-n\overline x^2\right)$$
And I simply can't see how they are equivalent. Seems to me they used the difference of two squares when they should have used the binomial theorem.

 

[Dr.Peterson]

This isn't a difference of squares! And they do use the binomial theorem (though it's easier if you don't omit the exponent as you did ...).

Just expand the square of the binomial, and do some work with the summation.

Here's an explanation (though it's for population variance):

Alternate Variance Formulas | Courses.com

Explore alternative formulas for calculating population variance and their implications in statistics.

www.courses.com

 

[Aedrha2]

This isn't a difference of squares! And they do use the binomial theorem (though it's easier if you don't omit the exponent as you did ...).

Just expand the square of the binomial, and do some work with the summation.

Here's an explanation (though it's for population variance):

Alternate Variance Formulas | Courses.com

Explore alternative formulas for calculating population variance and their implications in statistics.

www.courses.com

Opps didn't mean (pun intended) to leave out the square
Thanks a bunch! The part where the middle term is the mean is what i missed!