Standard Deviation Calculator

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 “s²“/“σ²” 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
s² = sample variance
x₁, ..., 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
σ² = population variance
x₁, ..., 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).