Adjusted sample variance

by Marco Taboga, PhD

The adjusted sample variance is a measure of the dispersion of a sample around its mean.

It is obtained by:

1. summing the squared deviations from the mean;

2. dividing the result thus obtained by the number of observations minus one.

Below we provide a precise definition, we illustrate its calculation with an example, and we introduce some of its properties.

Synonyms

It is also often called unbiased sample variance because, under certain conditions, it is an unbiased estimator of the population variance.

Definition

It is defined as follows.

The adjective "adjusted" refers to the fact that the sum of squared deviations is divided by rather than by .

Example

Suppose that we have observed the following sample of six observations:

The sample mean is

The adjusted sample variance is

Adjusted vs unadjusted

Another common way to compute the sample variance is which is called unadjusted or biased sample variance.

The main difference is that the sum of squared deviations:

• is divided by in the unadjusted variance;

• is divided by in the adjusted variance.

Degrees of freedom adjustment

We can write the adjusted variance in terms of the unadjusted one:

The ratio is called a degrees of freedom adjustment. It is also sometimes called Bessel's correction.

Unbiased estimator

Suppose that the observations are all drawn from probability distributions having the same mean and the same variance .

It is possible to prove (see Variance estimation) that, if are independent, then the adjusted sample variance is an unbiased estimator of .

Bias-variance trade-off

The degrees of freedom adjustment is not a free lunch: it eliminates the bias, but it usually increases the variance of the sample variance.

This can be proved rigorously when are drawn from a normal distribution (see Variance estimation).

More details

The lecture entitled Variance estimation presents more details about the adjusted sample variance and its properties (e.g., its consistency and lack of bias).

For more details about the biased version, you can instead see the glossary entry on the Unadjusted sample variance.

Keep reading the glossary

Previous entry: Absolutely continuous random vector

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How to cite

Please cite as:

Taboga, Marco (2021). "Adjusted sample variance", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/adjusted-sample-variance.