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

##### 1 Answer

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 **expected value**) of *average* of the set:

#mu = E[X] = 1/n sum_(i=1)^n x_i#

The standard deviation *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

For probability distributions (as well as data sets), the standard deviation is a measure of the *spread* of the distribution (data). The larger

In the name "standard deviation":

standard

#-># typical

deviation#-># distance

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