Question

What is the variance of {-4, 5, 8 ,-1, 0 ,4 ,-12, 4}?

Answer 1

The population variance of the data set is

`sigma^2=35`

Explanation:

First, let's assume that this is the entire population of values. Therefore we are looking for the population variance . If these numbers were a set of samples from a larger population, we would be looking for the sample variance which differs from the population variance by a factor of `n//(n-1)`

The formula for the population variance is

`sigma^2=1/N sum_(i=1)^N(x_i-mu)^2`

where `mu` is the population mean, which can be calculated from

`mu = 1/N sum_(i=1)^N x_i`

In our population the mean is

`mu = (-4+ 5+ 8 -1+ 0 +4 -12+ 4)/8=4/8=1/2`

Now we can proceed with the variance calculation:

`sigma^2=((-4-1/2)^2+ (5-1/2)^2+ (8-1/2)^2+( -1-1/2)^2+ (0-1/2)^2 +(4-1/2)^2+( -12-1/2)^2+ (4-1/2)^2)/8`

`sigma^2 = 35`