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.

A precise definition follows.

Definition
Let
be a random variable and let
be
its variance. The **standard deviation** of
is

The standard deviation is usually denoted by or by .

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

For example, if is the height of an individual extracted at random from a population, measured in inches, then the deviationis also expressed in inches.

However, the squared deviationand the variance are expressed in squared inches, which makes variance hard to interpret.

Instead, the standard deviationis again expressed in inches. Therefore, it is easier to interpret.

Let be a sample of observations having sample meanand sample variance

The square root of the sample variance
is usually called **sample standard deviation**.

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

Note that is the biased sample variance.

An alternative is to use the unbiased sample variance

The square root
is often called **corrected sample standard deviation**.

When the observations are independent and have the same mean and variance, is an unbiased estimator of .

However, by Jensen's inequality, is not an unbiased estimator of .

This is the reason why cannot be called unbiased sample standard deviation and it is instead called corrected sample standard deviation.

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

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

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Please cite as:

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

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