What is Variance all about?

If you want to summarise a set of numbers the most obvious start point is roughly how big are the numbers - and the mean solves this nicely. It is sometimes described as a measure of location because it tells you where the numbers are on the scale.

The next most obvious thing that you might want to know is how spread out the numbers are. The range gives you some idea but it only actually tells you about two numbers, the highest and the lowest. You really want a measure that tells you something about every number.

So we could try measuring how far each number is from the mean: (x_i - μ)

We could average this for each data value:

Σ (x_i - μ) /n

but a moment's rearranging will show you that you end up with 0. You can think of this being because the values are negative for the numbers below the mean and positive for the numbers above.

What if we square each number - this makes all the numbers positive and also is more sensitive to numbers which are further away from the mean.

Σ (x_i - μ)² /n

This is the variance! It is pretty much the simplest thing you could come up with that measures the spread of the numbers and takes account of all of the numbers in the data.

It can easily be rearranged to form this expression,

Σ x_i² /n - μ²

which is easier to calculate, but the first expression makes it clearer where it comes from.

The only remaining problem is that it is a squared quantity. If your original data were weights in kg, for example, then the variance is measured in kg², which doesn't mean a lot. So we take the square root to get the standard deviation which has the same units as the original data and can be compared with it. Indeed, if the data comes from a normal distribution, then 68% of the data values are within 1 standard deviation of the mean.