# Standard Deviation Formula ## Formula

### Summary

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

## Usage

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.

### Population vs Sample

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.