This **standard deviation calculator** can help you calculate the population & sample standard deviations, standard variance, sum and the mean of a data set. Below the form you can find the equations used.

## Understanding the concept of standard deviation

In statistics, standard deviation refers to an indicator that shows by how much the individual members of a data set/group vary from the mean value for the data set. It can be calculated both for a population case in which it is referred to as population standard deviation and for a sample case in which is called sample standard deviation. Its formula takes account of the:

■ Data set members (x_{i});

■ Mean of the data being analyzed x̄ in case of sample; respectively the expected mean of the population data set denoted with μ;

■ Total number of members in the group which is the count of values in the data set (N).

Standard deviation is denoted by “s” in case of a sample, respectively “σ” in case of a population. In statistics this measure is related to the concept of variance symbolized with “s^{2}“/“σ^{2}” since it is the square root of its variance.

## Variance and standard deviation of a sample

### Sample standard deviation formula

### Sample variance formula

Where:

s = sample standard deviation

s^{2} = sample variance

x_{1}, ..., x_{i} = individual members of the sample data set

x̄ = data set mean value

N = size of the sample data set

## Variance and standard deviation of a population

### Population standard deviation formula

### Population variance formula

Where:

σ = population standard deviation

σ^{2} = population variance

x_{1}, ..., x_{i} = members of the data set

μ = expected mean of the population data set

N = size of the population

## Interpretation of the levels of the standard deviation

In regard of the interpretation of the standard deviation levels, typically a low standard deviation indicates that the members/values within the group/data set tend to be close to the mean, while a high standard deviation is an evidence of the contrary (values tend to differ significantly in comparison to the average).

15 Apr, 2015