# The standard deviation

The standard deviation is the sum of squared deviations (or errors) of a data set with respect to the mean divided by the number of data.

By far the most common formula for computing variance in a sample is:

Where:

- s² = standard deviation
- X = value
- M = mean
- N = sample size

Since samples are usually used to estimate parameters, s² is the most commonly used measure of variance. Calculating the variance is an important part of many statistical applications and analyses. It is the first step in calculating the standard deviation.

## Example:

if your set is 1, 2, 3, 4, 6, 8 we can compute the standard deviation as follow:

X |
Freq. |
M |
(X-M) |
(X-M)^2 |

1 | 1 | 4 | 3 | 9 |

2 | 1 | 4 | 2 | 4 |

3 | 1 | 4 | 1 | 1 |

4 | 1 | 4 | 0 | 0 |

6 | 1 | 4 | 2 | 4 |

8 | 1 | 4 | 4 | 16 |

24 |
6 |
34 |

- Sample size N = 6 (frequency)
- The mean : 24/6 = 4
- The sum of (X-M)^2 = 34
- 34/6 = 5.67