Standard deviation

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

Definition
Interpretation
Sample standard deviation
Corrected standard deviation
Standard error
More details
Keep reading the glossary

Definition

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 .

Interpretation

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.

Sample standard deviation

Let 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 are independent and have the same mean and variance, $s_{n}^{2}$ is an unbiased estimator of .

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

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

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How to cite

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.