To compute the mean, you simply have to sum all the elments in the data set and the divide the sum by the number of elements:
[tex] M = \frac{1+4+9+12+13+13}{6} = \frac{52}{6} = 8.6 [/tex]
To compute the variance, we first need to compute the distance of each element from the mean. To do so, we build a "parallel" dataset, given by the difference of every value and the mean:
[tex] D' = 1-8.6,9-8.6,4-8.6,12-8.6,13-8.6,13-8.6 [/tex]
[tex] D' = -7.6, 0.4, -4.6, 3.4, 4.4, 4.4 [/tex]
Now we need those difference squared:
[tex] (D')^2 = 57.76, 0.16, 21.16, 11.56, 19.36, 19.36 [/tex]
The variance is the mean of this new vector, so
[tex] \sigma^2 = \frac{57.76+ 0.16+ 21.16+ 11.56+ 19.36+ 19.36}{6} = \frac{129.36}{6} = 21.6 [/tex]
Finally, the standard deviation is simply the square root of the variance, so you have
[tex] \sigma = \sqrt{21.6} = 4.6 [/tex]
