# Standard Deviation

**Standard Deviation** is a concept in statistics that is one way of showing how much *variation* there is in a
set,
multiset
or
random variable.

*Standard Deviation* is sometimes just shortened to SD
and is often denoted by the lowercase greek letter
** σ**.

## Prerequisites

A deep understanding of *standard deviation* requires an understanding of one (other) statistical topic.
Specifically, it requires knowledge of one type of average known as the mean.

## Definition

The most basic definition of *standard deviation* is for a discrete set whose elements are considered equal in terms of probability.

Thus, for a discrete set `X` the *standard deviation*,
which we will denote as ** σ**,
is defined as in

*figure 1*.

Note that there is nothing special about using the indeterminate `X`.
We could have used `Y`, `Z`, `C`, etc.
It is just that it is a convention that for some things, the variable we start with is the letter "X".
But really, we could have wrote the same definition in *figure 1* with another indeterminate;
for example, in *figure 2* we use the indeterminate `Y`.

## Tutorial on Standard Deviation

Let us consider 3 different discrete random variables -- `X`, `Y` and `Z` --
as shown in *figure 3*, *figure 4* and *figure 5*.

I have choosen these 3 discrete random variables
(shown in *figure 3*, *figure 4* and *figure 5*)
carefully to make for a good example.

Let us calculate the means of these 3 random variables.
This is done in
*figure 6*,
*figure 7*,
and
*figure 8*.

We can see that the mean for each of these 3 discrete random variable are all 20, even though there are important differences between these 3 discrete random variables.

For example, with the discrete random variable `Y`, there is no **variation**;
all of its elements are 20.
Alternatively, With the discrete random variables `X` and `Z` there is **variation**.

Using the definition for **standard deviation** in *figure 1*
we calculate the *standard deviation* of the discrete random variables
`X`, `Y`, and `Z`
in *figure 9*, *figure 10*, and *figure 11* respectively.