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:
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:
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:
