Variance Explanation

Question

I'm trying to learn statistics and I've been introduced to Variance, say I have the following...

values = 5, 6, 22, 89, 54, 7

mean = 30.5

variance = 974.916666666666

Now, how does 974 relate to the values that were used to get these numbers?

I've read and read and I can't seem to get a clear answer on what the variance is used for that someone like me, just starting out can understand.

Answer 1

Variance is the mean (average) of the squared differences from the mean.

That is, you first compute the difference between each of your values and the mean. In your example:

 5 - 30.5 = -25.5
 6 - 30.5 = -24.5
22 - 30.5 =  -8.5
etc...

You notice that we have some negative numbers here, so to overcome this problem we square each distance, i.e.

-25.5 ^2 = 650.25
-24.5 ^2 = 600.25
-8.5  ^2 =  72.25
etc...

The mean of all these values is the variance. But as you as will notice, this does not have any straightforward relationship with your original numbers. That is what standard deviation is for!

Standard deviation is defined as the square root of variance, which in your case would be:

31.22

This amount is on the same scale as your original values/
But what does this number mean? It is a measure of how dispersed your numbers are compared with the mean.

So if all your numbers were similar to the mean, the value of standard deviation would be very small. If your numbers vary wildly, its value would be larger.

You can further read here: http://www.mathsisfun.com/data/standard-deviation.html That is a great website with simple explanations.