Unadjusted sample variance

The unadjusted sample variance measures the average dispersion of a sample of observations around their mean.

It is computed by averaging the squared deviations from the mean.

Table of contents

Synonyms
Definition
Example
The bias
The cause of the bias
How to correct the bias
Maximum likelihood estimator
More details
Keep reading the glossary

Synonyms

It is also often called biased sample variance, because, under standard assumptions, it is a biased estimator of the population variance.

Definition

It is defined as follows:

Definition Given observations , their unadjusted sample variance is [eq2] where is their sample mean: [eq3]

Example

Suppose that the sample consists of the following observations: [eq4]

We arrange the observations into a table and use the table to do all the necessary calculations:

[eq5]

Thus, the sample variance is [eq6]

The bias

Suppose that are independent realizations of random variables having the same mean and the same variance .

It can be proved (see Variance estimation) that the unadjusted sample variance $S_{n}^{2}$ is a biased estimator of , that is,where is the expected value of $S_{n}^{2}$ .

The cause of the bias

The cause of the bias is that we use the sample mean instead of the true mean in the calculation.

If were known and used in the formula in place of , then the resulting estimator would be unbiased.

How to correct the bias

In order to obtain an unbiased estimator of the true variance , we need to perform a so-called degrees of freedom adjustment:

The estimator $s_{n}^{2}$ is unbiased (see Adjusted sample variance).

Maximum likelihood estimator

If the observations are not only independently and identically distributed, but they are also normal, then the biased sample variance coincides with the maximum likelihood estimator of the variance (see Maximum likelihood estimation of the parameters of a normal distribution).

More details

The lecture entitled Variance estimation provides a thorough introduction to the concept of unadjusted sample variance, including a detailed analysis of its statistical properties (e.g., its bias as an estimator of the population variance).

Keep reading the glossary

Previous entry: Type II error

Next entry: Unbiased estimator

How to cite

Please cite as:

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