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