Question

What are the variance and standard deviation of {1, -1, -0.5, 0.25, 2, 0.75, -1, 2, 0.5, 3}?

Answer 1

If the given data is the entire population then:
`color(white)("XXX")sigma_"pop"^2 = 1.62; sigma_"pop"=1.27`
If the given data is a sample of the population then
`color(white)("XXX")sigma_"sample"^2 = 1.80; sigma_"sample"=1.34`

Explanation:

To find the variance (`sigma_"pop"^2`) and standard deviation (`sigma_"pop"`) of a population

1. Find the sum of the population values
2. Divide by the number of values in the population to obtain the mean
3. For each population value calculate the difference between that value and the mean then square that difference
4. Calculate the sum of the squared differences
5. Calculate the population variance (`sigma_"pop"^2`) by dividing the sum of the squared differences by the number of population data values.
6. Take the (primary) square root of the population variance to obtain the population standard deviation (`sigma_"pop"`)

If the data represents only a sample extracted from a larger population then you need to find the sample variance (`sigma_"sample"^2`) and sample standard deviation (`sigma_"sample"`).
The process for this is identical except in step 5 you need to divide by `1` less than the sample size (instead of the number of sample values) to get the variance.

It would be unusual to to all of this by hand. Here's what it would look like in a spreadsheet: