# Variance and Standard deviation

## Variance

The variance of some data is the arithmetical mean of the square of the absolute deviations. It is symbolized as $$\sigma ^2$$ and it is calculated by applying the formula $$\sigma^2=\displaystyle \frac{\displaystyle\sum_{i=1}^N (x_i-\overline{x})^2}{N}=\frac{(x_1-\overline{x})^2+(x_2-\overline{x})^2+\ldots+(x_N-\overline{x})^2}{N}$$$which it is possible to simplify as: $$\sigma^2=\displaystyle \frac{\displaystyle \sum_{i=1}^N x_i^2}{N}-\overline{x}^2=\frac{x_1^2+x_2^2+\ldots+x_N^2}{N}-\overline{x}^2$$$

Same as with the average, it is not always possible to find the variance, and it is a parameter that is very sensitive to the extreme scorings. We can see that, with the deviation being squared, the variance cannot have the same units as the data.

Comparing with the same type of information, a high variance means that the data is more dispersed. And a low value of the variance indicates that the values are in general closer to the average.

A value of the variance equal to zero means that all the values are equal, and therefore they are also equal to the arithmetical average.

In a basketball match, we have the following points for the players of a team: $$0, 2, 4, 5, 8, 10, 10, 15, 38$$. Calculate the variance of the scorings of the players of the team.

Applying the formula $$\overline{x}=\displaystyle \frac{0+2+4+5+8+10+10+15+38}{9}=\frac{92}{9}=10.22$$ the average is obtained.