In probability and statistics, the **standard deviation**, generally denoted σ ('sigma'), is the most commonly used measure of statistical dispersion which is measured with the same units as the data. It is calculated as the positive square root of the variance and is therefore always a non-negative number.

The standard deviation of a sample, as opposed to a population, is denoted *s*.

See also: mean, skewness, kurtosis, raw score, standard score.

## Geometric Interpretation of Mean and Standard Deviation

Given a set of numbers , it is desired to define the mean and standard deviation of these numbers. We will imagine an*n*-dimensional hypercube in

*R*. Let the hypercube be large enough to contain all the numbers, so let it have sides of length . Let the point be a point inside this hypercube. For convenience, we will visualize this by means of a three dimensional diagram, in which point

^{n}**A**is inside a cube.

Now draw the main diagonal of the cube, which goes through points **O**=(0,0,0) and point **M**=(*L*,*L*,*L*) and call it *OM*.

Now find a point on line *OM* such that line *OB* and line *BA* are perpendicular:

- .

*B*

_{0},

*OB*is

*n*. For any set of numbers , their mean is

**B**is a point on line

*OM*such that (recapitulating):

*OM*, and

*OB*and

*BA*be equal to 0 means that lines

*OB*and

*BA*are perpendicular.

The standard deviation can then be defined as

**A**) and the vector mean of the event.

*The mean is the distance from the origin to the projection of the event onto the main diagonal. The standard deviation is the distance between the event and the main diagonal.*The mean is the projected distance away from the origin, the standard deviation is the distance away from the mean.