What Is Variance and Standard Deviation?

The standard deviation is a measure of dispersion. It tells you how much the data varies in relation to the mean. When you’re going to find the standard deviation from a data set, you can use the standard deviation formula.

Rule

Instructions to Finding the Standard Deviation

1.: Calculate the mean of the $N$ observations in the data set.
2.: Calculate how much each of the observations deviates from the mean value.
3.: Square the deviations and calculate the sum of the squared deviations.
4.: Divide the sum by the number $N - 1$
5.: Take the square root of the answer.

σ = \frac{\sqrt{{(x_{1} - \overset{}{x})}^{2} + \dots + {(x_{N} - \overset{}{x})}^{2}}}{\sqrt{N - 1}}

σ = \frac{\sqrt{{(x_{1} - \overset{}{x})}^{2} + {(x_{2} - \overset{}{x})}^{2} + \dots + {(x_{N} - \overset{}{x})}^{2}}}{\sqrt{N - 1}}

Example 1

The Russian national gymnastics team is comprised of Anoushka, Alexandra, Triana, Mirella, Jana and Fiona. They are all close friends. Anoushka is $167 cm$ tall, Alexandra is $159 cm$ tall, Triana is $162 cm$ tall, Mirella is $155 cm$ tall, Jana is $160 cm$ tall and Fiona is $157 cm$ tall. Find the standard deviation of the heights of the Russian national gymnastics team.

First, you find the mean height $\overset{}{x}$ by adding all of the girls’ heights and dividing by the number of observations, that is the number of girls on the team:

\frac{167 + 159 + 162 + 155 + 160 + 157}{6}

This gives you a mean height of $160 cm$ . Then, you determine how much the height of each of the girls deviates from the mean height. You do this by taking the height of each girl minus the mean height.

Furthermore, you must find the square of each of the deviations. This is done by multiplying the deviation by itself. Remember that minus multiplied by minus becomes positive. The square deviation is always a positive number! You can see the results in the table below.


Name	Height (cm)	Deviation	Squared Deviation

Anoushka	167	7	49

Alexandra	159	-1	1

Triana	162	2	4

Jana	160	0	0

Fiona	157	-3	9

Mirella	155	-5	25

You must add all the squared deviations. Then, divide the sum by $N - 1$ observations. Finally, take the square root of this value:

\begin{matrix} σ = \sqrt{\frac{49 + 1 + 4 + 0 + 9 + 25}{6 - 1}} \\ σ = \sqrt{\frac{88}{5}} \\ σ = \sqrt{17, 6} \\ σ \approx 4.2 cm \end{matrix}

The standard deviation of the heights of the Russian gymnasts is $4.2$ cm.

The variance and the standard deviation are both measures of how much the data deviates from the expected value. This is called statistical dispersion, or simply dispersion. Dispersion is a measurement of the spread within your data. Variance and standard deviation are examples of measures of dispersion.

Theory

Variance

The variance, $Var (X)$ , is calculated like this:

Var (X) = \sum_{i = 1}^{m} {(x_{i} - μ)}^{2} \cdot P (X = x_{i})

Theory

Standard Deviation

The standard deviation $σ$ is

σ = SD (X) = \sqrt{Var (X)}

Example 2

A die is rolled many times, and you are going to find the variance and standard deviation of the die rolls. You already know that the expected value of a die roll is $3.5$ . You put this into the formula and get:

\begin{array}{l} Var (X) \\ = \sum_{i = 1}^{6} {(x_{i} - 3.5)}^{2} \cdot \frac{1}{6} \\ = {(1 - 3.5)}^{2} \cdot \frac{1}{6} + {(2 - 3.5)}^{2} \cdot \frac{1}{6} \\ + {(3 - 3.5)}^{2} \cdot \frac{1}{6} + {(4 - 3.5)}^{2} \cdot \frac{1}{6} \\ + {(5 - 3.5)}^{2} \cdot \frac{1}{6} + {(6 - 3.5)}^{2} \cdot \frac{1}{6} \\ \approx 2.92 \end{array}

\begin{array}{l} Var (X) & = \sum_{i = 1}^{6} {(x_{i} - 3.5)}^{2} \cdot \frac{1}{6} \\ = {(1 - 3.5)}^{2} \cdot \frac{1}{6} + {(2 - 3.5)}^{2} \cdot \frac{1}{6} + {(3 - 3.5)}^{2} \cdot \frac{1}{6} \\ + {(4 - 3.5)}^{2} \cdot \frac{1}{6} + {(5 - 3.5)}^{2} \cdot \frac{1}{6} + {(6 - 3.5)}^{2} \cdot \frac{1}{6} \\ \approx 2.92 \end{array}

The variance of repeated die roll is about

2.92

. You find the standard deviation by taking the square root of the variance:

SD (X) \approx \sqrt{2.92} \approx 1.71

The standard deviation of repeated die roll is about $1.71$ .

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Interquartile Range and Semi-interquartile Range

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