# Standard Deviation Formula

The standard deviation formula represents the distribution of the sample data around the mean of the sample. A smaller standard deviation means that most of the data is close to the mean, sometimes called the expected value. A bigger standard deviation means that the data is more spread out over the sample space.

Notation | Description |
---|---|

sample standard deviation. | |

size of the sample. | |

i-th element in the data set. | |

average value of the data set. |

This formula represents an approximation of the population standard deviation. Of course, as the size of the sample increases the closer the sample and population standard deviation become. The symbols μ (mu) and σ (sigma) are used to differentiate between the sample which a subset of the whole population.

## Significance of the Standard Deviation

The value for the standard deviation can be used to understand how closely the data set is to the mean. One standard deviation away from the mean on either side contains approximately 68.3% of the samples, two standard deviations contains approximately 95.4% of the samples, and so on.