Search for probability and statistics terms on Statlect

Standard deviation

by , PhD

Standard deviation is a measure of how much the realizations of a random variable are dispersed around its mean. It is equal to the square root of variance.

Table of Contents


A precise definition follows.

Definition Let X be a random variable and let [eq1]be its variance. The standard deviation of X is[eq2]

The standard deviation is usually denoted by [eq3] or by [eq4].


Standard deviation is often deemed easier to interpret than variance because it is expressed in the same units as the random variable X.

For example, if X is the height of an individual extracted at random from a population, measured in inches, then the deviation[eq5]is also expressed in inches.

However, the squared deviation[eq6]and the variance [eq7]are expressed in squared inches, which makes variance hard to interpret.

Instead, the standard deviation[eq8]is again expressed in inches. Therefore, it is easier to interpret.

Sample standard deviation

Let [eq9] be a sample of observations having sample mean[eq10]and sample variance[eq11]

The square root of the sample variance $sqrt{S_{n}^{2}}$ is usually called sample standard deviation.

However, it is sometimes also simply called standard deviation, which might create confusion (sample or population?).

Corrected standard deviation

Note that $S_{n}^{2}$ is the biased sample variance.

An alternative is to use the unbiased sample variance[eq12]

The square root $sqrt{s_{n}^{2}}$ is often called corrected sample standard deviation.

When the observations [eq9] are independent and have the same mean and variance, $s_{n}^{2}$ is an unbiased estimator of [eq14].

However, by Jensen's inequality, $sqrt{s_{n}^{2}}$ is not an unbiased estimator of [eq15].

This is the reason why $sqrt{s_{n}^{2}}$ cannot be called unbiased sample standard deviation and it is instead called corrected sample standard deviation.

Standard error

The standard deviation of an estimator (e.g., the OLS estimator) is often called standard error.

More details

More details about the standard deviation can be found in the lecture entitled Variance.

Keep reading the glossary

Previous entry: Size of a test

Next entry: Stationary sequence

How to cite

Please cite as:

Taboga, Marco (2021). "Standard deviation", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix.

The books

Most of the learning materials found on this website are now available in a traditional textbook format.