Variance and standard deviation

In a nutshell

The variance and the standard deviation are two measures that can be used to evaluate the spread of the data in a given set.

Variance

This measure considers how much each point $x$ deviates from the mean $\overline x$ with the calculation $(x-\overline x)^2$ .

It is given in square units.

$\sigma^2=\dfrac{\Sigma(x-\overline x)^2}{n}=\dfrac{\Sigma(x^2)}{n}-\left(\dfrac{\Sigma x}{n}\right)^2$

$\sigma^2$	Variance of a data set
${\Sigma (x-\overline x)^2}$	Sum of the squares of each point's deviation from the mean
$n$	Number of elements in the data set

Note: For raw data, it is easier to use $\dfrac{\Sigma x^2}{n}-\left(\dfrac{\Sigma x}{n}\right)^2$ : "the mean of the squares minus the square of the mean".

Finding the variance in grouped data

To find the variance in a grouped data set, such as in a frequency table, you can use the following formulae:

$\sigma^2=\dfrac{\Sigma f(x-\overline x)^2}{\Sigma f}=\dfrac{\Sigma fx^2}{\Sigma f}-\left(\dfrac{\Sigma fx}{\Sigma f}\right)^2$

Standard deviation

This is the square root of the variance.

$\sigma=\sqrt\dfrac{\Sigma(x-\overline x)^2}{n}=\sqrt{\dfrac{\Sigma(x^2)}{n}-\left(\dfrac{\Sigma x}{n}\right)^2}$

$\sigma$	Standard deviation of the data set
${\Sigma (x-\overline x)^2}$	Sum of the squares of each point's deviation from the mean
$n$	Number of elements in the data set

Finding the standard deviation in grouped data

To find the standard deviation in a frequency table, use the following formulae:

$\sigma=\sqrt\dfrac{\Sigma f(x-\overline x)^2}{\Sigma f}=\sqrt{\dfrac{\Sigma fx^2}{\Sigma f}-\left(\dfrac{\Sigma fx}{\Sigma f}\right)^2}$

Example 1

A company keeps track of the number of vacation days taken by their employees. The results are shown below.

Number of vacation days	$18$	$19$	$20$	$21$	$22$	$23$	$24$
Number of employees	$1$	$4$	$5$	$8$	$10$	$3$	$1$

Find the variance and the standard deviation.

Start by finding $\Sigma fx$ and $\Sigma fx^2$ :

$\Sigma fx=(18\times1)+(19\times4)+(20\times5)+(21\times8)+(22\times10)+(23\times3)+(24\times1)=675$

${\Sigma fx^2=(18^2\times1)+(19^2\times4)+(20^2\times5)+(21^2\times8)+(22^2\times10)+(23^2\times3)+(24^2\times1)=14299}$ 

$\sigma^2={\dfrac{\Sigma fx^2}{n}}-\left({\dfrac{\Sigma fx}{n}}\right)^2=\dfrac{14299}{32}-\left(\dfrac{675}{32}\right)^2=\underline{1.89746...}$

$\sigma=\sqrt{1.89746...}=\underline{1.37748...}$

Note: If the data was given in a grouped data set, you could have calculated the variance and standard deviation with the midpoint of each class interval.