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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.

σ = i = 1 | X | ( X i - X _ ) 2 | X | - 1
Figure 1. The definition of standard deviation for a disrete set X.

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.

σ = i = 1 | Y | ( Y i - Y _ ) 2 | Y | - 1
Figure 2. The same definition of standard deviation as in figure 1, but instead of using the indeterminate X to denote the discrete set, the indeterminate Y is used.

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.

X = 0 18 22 40
Figure 3. Example discrete set X.
Y = 20 20 20 20
Figure 4. Example discrete set Y.
Z = 10 18 19 21 22 23 27
Figure 5. Example discrete set Z.

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.

X _ = 0 + 18 + 22 + 40 4 = 80 4 = 20
Figure 6. The mean of the random variable X, from figure 3.
Y _ = 20 + 20 + 20 + 20 4 = 80 4 = 20
Figure 7. The mean of the random variable Y, from figure 4.
Z _ = 10 + 18 + 19 + 21 + 22 + 23 + 27 4 = 140 7 = 20
Figure 8. The mean of the random variable Z, from figure 5.

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.

σ X = i = 1 | X | ( X i - X _ ) 2 | X | - 1 = i = 1 4 ( X i - 20 ) 2 4 - 1 = ( X 1 - 20 ) 2 + ( X 2 - 20 ) 2 + ( X 3 - 20 ) 2 + ( X 4 - 20 ) 2 4 - 1 = ( 0 - 20 ) 2 + ( 18 - 20 ) 2 + ( 22 - 20 ) 2 + ( 40 - 20 ) 2 4 - 1 = 400 + 4 + 4 + 400 4 - 1 = 808 3
Figure 9. The standard deviation for a disrete set X from figure 3.
-- Mirza Charles Iliya Krempeaux
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