Introduction
Quartiles
Percentiles
Quantiles
Standard Deviation and Variance
8. Standard Deviation

Instruction

The standard deviation is probably the most frequently used measure of the dispersion of a set of data values. The interpretation is quite easy to understand. The standard deviation measures how concentrated the data are around the mean. A low value of the standard deviation indicates that the data are close to the mean, while a high values indicates the opposite – data points are spread out are spread out farther from the mean.

The formula for the standard deviation (SD) looks difficult:

SD=\sqrt{\frac{1}{n}((x_1-\overline{x})^2+(x_2-\overline{x})^2+\ldots+(x_n-\overline{x})^2)}

In fact, it is quite simple.

We will walk through the formula in the example below. Let us consider the set of the following numbers:

{-1,0,1,2}

To calculate the standard deviation, we first need to calculate the arithmetic mean: (-1+0+1+2)/4 = 0.5.

x_i x_i-\overline{x} (x_1-\overline{x})^2
-1 -1.5 2.25
0 -0.5 0.25
1 0.5 0.25
2 1.5 2.25
SUM 5

Therefore, the standard deviation can be computed as follows:

SD = \sqrt{5/4} ~= 1.1180