# What is standard deviation? Why is it called that?

Jan 7, 2017

It is a measure of the expected distance between an element of a set and the set's mean.

#### Explanation:

The standard deviation of a set of numbers can be thought of as how far, on average, each number in the set is from the mean of the set. In other words, if we pick a number from a set at random, the mean tells us what we should expect that number to be, and the standard deviation tells us how far we should expect that number to be from the mean.

For a random variable $X$ which is equally likely to be any element of $\left\{{x}_{1} , {x}_{2} , \ldots , {x}_{n}\right\}$, the mean $\mu$ (aka the expected value) of $X$ is the average of the set:

$\mu = E \left[X\right] = \frac{1}{n} {\sum}_{i = 1}^{n} {x}_{i}$

The standard deviation $\sigma$ of $X$ is the expected value of the distance between an element of the set and the set's average:

sigma = sqrt(E[(X-mu)^2])=sqrt(1/n sum_(i=1)^n(x_i- mu)^2

(The square/square root is necessary because we want to calculate a positive average distance, and some of the elements in our set are below $\mu$, and so would make $X - \mu$ negative. But no matter the value of $X$, ${\left(X - \mu\right)}^{2}$ is always positive.)

For probability distributions (as well as data sets), the standard deviation is a measure of the spread of the distribution (data). The larger $\sigma$ is, the more flat (spread out) the distribution (data) will be.

In the name "standard deviation":

standard $\to$ typical
deviation $\to$ distance

Hence, the standard deviation of a set measures the typical distance between elements of the set and the set's average.