Standard Deviation

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

$$\sigma = \sqrt{E{(X - \mu ) }^{2}}.$$

(1)

That is, the standard deviation σ is the square root of the average value of (X − μ)². The standard deviation of a continuous...