Question

What are the variance and standard deviation of {2,9,3,2,7,7,12}?

Answer 1

Variance (population): `sigma_"pop"^2=12.57`
Standard Deviation (population): `sigma_"pop" = 3.55`

Explanation:

The Sum of the data values is `42`

The Mean (`mu`) of the data values is `42/7=6`

For each of the data values we can calculate the difference between the data value and the mean and then square that difference.

The sum of the squared differences divided by the number of data values gives the population variance (`sigma_"pop"^2`).

The square root of of the population variance gives the population standard deviation (`sigma_"pop"`)

Note: I've assumed the data values represent the entire population.
If the data values are only a sample from a larger population then you should calculate the sample variance, `s^2`, and sample standard deviation, `s`, using the method above with the only difference being that the division to find the variance needs to be by (1 less than the number of data values).

Note 2: Normal statistical analysis is done with the aid of computers (e.g. using Excel) with built-in functions to provide these values.