Question

What is the standard deviation of `1, 2, 3, 4` and `5`?

Answer 1

The standard deviation of `{1, 2, 3, 4, 5}`
`= [(5^2-1)/(12)]^(1/2) = sqrt2`

Explanation:

Let's develop a general formula then as a particular you get standard deviation of `1, 2, 3, 4` and `5`. If we have `{1, 2,3, .... , n}` and we need to find the standard deviation of this numbers.

Note that

`"Var" (X) = 1/n sum_{i=1}^n x_i^2 - (1/n sum _(i=1)^n x_i)^2`

`implies "Var" (X) = 1/n sum_{i=1}^n i^2 - (1/n sum _(i=1)^n i)^2`

`implies "Var" (X) = 1/n * (n(n+1)(2n+1))/(6) - (1/n *(n(n+1))/2)^2`
`implies "Var" (X) = ((n+1)(2n+1))/(6)- ((n+1)/2)^2`
`implies "Var" (X) = (n+1)/(2)[(2n+1)/3- (n+1)/2]`

`implies "Var" (X) = (n+1)/(2) * (n-1)/6`
`implies "Var" (X) = (n^2-1)/(12)`

So, Standard deviation of `{1, 2,3, .... , n}` is `["Var" (X)]^(1/2) = [ (n^2-1)/(12)]^(1/2)`

In particular, your case the standard deviation of `{1, 2, 3, 4, 5}`
`= [(5^2-1)/(12)]^(1/2) = sqrt 2`.