The weighted standard deviation is a useful way to measure of values in a dataset when some values in the dataset have higher weights than others.

The formula to calculate a weighted standard deviation is:

where:

• N: The total number of
• M: The number of non-zero weights
• w_i: A vector of weights
• x_i: A vector of data values
• x: The weighted mean

The easiest way to calculate a weighted standard deviation in Python is to use the function from the statsmodels package:

DescrStatsW(values, weights=weights, ddof=1).std

The following example shows how to use this function in practice.

Example: Weighted Standard Deviation in Python

Suppose we have the following array of data values and corresponding weights:

#define data values 
values = [14, 19, 22, 25, 29, 31, 31, 38, 40, 41]

#define weights
weights = [1, 1, 1.5, 2, 2, 1.5, 1, 2, 3, 2]

The following code shows how to calculate the weighted standard deviation for this array of data values:

from statsmodels.stats.weightstatsimport DescrStatsW

#calculate weighted standard deviation
DescrStatsW(values, weights=weights, ddof=1).std

8.570050878426773

The weighted standard deviation turns out to be 8.57.

Note that we can also use var to quickly calculate the weighted variance as well:

from statsmodels.stats.weightstatsimport DescrStatsW

#calculate weighted variance
DescrStatsW(values, weights=weights, ddof=1).var

73.44577205882352

The weighted variance turns out to be 73.446.

Additional Resources

The following tutorials explain how to calculate weighted standard deviation in other statistical software: