# Standard Deviation

Reference work entry

First Online:

**DOI:**https://doi.org/10.1007/978-3-642-04898-2_535

## Introduction

Standard deviation is a measure of variability or dispersion. The term *Standard deviation* was first used in writing by Karl Pearson in 1894. This was a replacement for earlier alternative names for the same idea: for example, “mean error” (Gauss), “mean square error,” and “error of mean square” (Airy) have all been used to denote standard deviation. Standard deviation is the most useful and most frequently used measure of dispersion. It is expressed in the same units as the data. Standard deviation is a number between 0 and *∞*. A large standard deviation indicates that observations/data points are far from the mean and a small standard deviation indicates that they are clustered closely around the mean.

## Definition

If

*X*is a random variable with mean value*μ*=*E*(*x*), the standard deviation of*X*is defined by
$$\sigma = \sqrt{E{(X - \mu ) }^{2}}.$$

This is a preview of subscription content, log in to check access.

## References and Further Reading

- Pearson Karl (1894) On the dissection of asymmetrical curves. Philos Tr R Soc S-A 185:719–810Google Scholar
- Miller J. Earliest known uses of some of the words of mathematics. http://jeff560.tripod.com/mathword.html
- Das Gupta A, Haff L (2006) Asymptotic expansions for correlations between measures of spread. J Stat Plan Infer 136: 2197–2213MathSciNetGoogle Scholar
- Yule GU, Kendall MG (1958) An introduction to the theory of statistics, 14th edn. 3rd Impression. Charles Griffin & Company, LondonGoogle Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2011