Question

How do you find the range, variance, and standard deviation for 1, 2, 3, 4, 5, 6, 7?

Answer 1

The range is `6` the variance is `14/3`, and the standard deviation is `sqrt(14/3)`.

Explanation:

For a sample set `S={1,2,3,4,5,6,7}`

The range is `max(S)-min(S)=7-1=6`

The average `barx=1/nsum_(k=1)^nk` where `n=7`

Then `barx=1/7sum_(k=1)^7k=(1+2+3+4+5+6+7)/7=28/7=4`

This is important because the variance

`s^2=1/(n-1)sum_(k=1)^n(x_k-barx)_k^2`, and since `n=7` and `barx=4`

`s^2=1/6sum_(k=1)^7(x-4)_k^2=1/6((-3)^2+(-2)^2+(-1)^2+0^2+1^2+2^2+3^2)=((-3)^2+(-2)^2+(-1)^2+0^2+1^2+2^2+3^2)/6=(9+4+1+0+1+4+9)/6=(2+8+18)/6=28/6=14/3`

And since the standard deviation is the square root of the variance `s=sqrt(s^2)`

`s=sqrt(14/3)`