Question

What are the variance and standard deviation of {1, 1, 1, 1, 1, 80, 1, 1, 1, 1, 1, 1}?

Answer 1

The population variance is:

`sigma^2 ~= 476.7`

and the populations standard deviation is the square root of this value:

`sigma ~= 21.83`

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 = (1+1+1+1+1+80+1+1+1+1+1+1)/12=91/12=7.58bar3`

Now we can proceed with the variance calculation:

`sigma^2=(11*(1-7.58bar3)^2+ (80-7.58bar3)^2)/12`

`sigma^2 ~= 476.7`

and the standard deviation is the square root of this value:

`sigma ~= 21.83`