# Standard Deviation Formula

This formula calculates the standard deviation of a normal distribution from **population** data. If you are working with **sample** data, see the sample standard deviation formula. The difference between population and sample data is that sample data is a subset of the whole population.

Expression | Description |
---|---|

The population standard deviation | |

The size of the population | |

Element of the population | |

Average value (mean) of the population |

This formula calculates the standard deviation of a normal distribution from **population** data. The standard deviation is a numeric measure of the distribution of data around the mean. A smaller standard deviation means that most of the data is close to the mean. A bigger standard deviation means that the data is more spread out over the sample space. This is illustrated in the figure below:

The distribution of data around the mean for any normal distribution is the same. One standard deviation away from the mean on either side contains approximately of the data, two standard deviations contain approximately of the samples, and so on. This is what makes the standard deviation a useful measure for normal distributions, it can be used to look up and calculate probabilities of a normal distribution.

The symbols (Latin small letter s) and (sigma) are used to differentiate between the sample and population data when calculating the standard deviation of a distribution. The difference between the two formulas, shown below, is **Bessel’s Correction** which corrects for bias in the sample data.

The normal distribution is a continuous probability distribution that appears naturally in applications of statistics and probability. The shape of the normal distribution forms a "bell curve".