## About This Tool

The online Standard Deviation Calculator is used to calculate the mean, variance, and standard deviation of a set of numbers.

## Standard Deviation

Standard deviation is a widely used measurement of variability or diversity used in statistics and probability theory. It shows how precise your data is.

The standard deviation is the square root of its variance. A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

Variance and standard deviation depend on the mean of a set of numbers. Calculating them depends on whether the set is a population or sample.

## Variance and Standard Deviation of a Population

The population standard deviation is measures the variability of data in a population. It is usually an unknown constant. The variance σ^{2} and standard deviation σ of the population are given by:

Where:

σ = population standard deviation

σ^{2} = population variance

x_{1}, ..., x_{N} = the population data set

μ = mean of the population data set

N = size of the population data set

## Variance and Standard Deviation of a Sample

The sample standard deviation is an estimate, based on a sample, of a population standard deviation. The variance s^{2} and standard deviation s of the sample are given by:

Where:

s = sample standard deviation

s^{2} = sample variance

x_{1}, ..., x_{N} = the sample data set

x̄ = mean value of the sample data set

N = size of the sample data set