Question

The standard deviation of 1,2,3,....25 is?

Answer 1

The standard deviation is `sigma = 7.2111`. This is the population standard deviation, not the sample standard deviation. (The difference is explained here).

Explanation:

I did this by writing a formula into a spreadsheet to find the standard deviation of the integers from 1 to 25.

Alternatively, we could calculate it by hand - click here for details.

`"Standard deviation " sigma = sqrt(frac{sum_(i=1)^(N)(x_i - mu)^2}{N})`

We know the mean of the dataset is:
`mu = frac{sum_(i=1)^25 (i)}{25} = frac{1 + 2 + 3 + ... +25}{25}`

`mu = frac{1+25}{2} = 13`

Substitute values:
`sigma = sqrt(frac{sum_(i=1)^(25)(x_i - 13)^2}{25})`

`sigma= sqrt(frac{(1-13)^2+(2-13)^2+(3-13)^2 + ... + (25-13)^2}{25})`

`sigma = sqrt(frac{12^2+11^2 + ... +2^2+1^2+0^2+1^2 + ... + 11^2+12^2}{25})`

`sigma = sqrt(1300/25) = sqrt52`

`sigma = 7.2111`