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Measures of location and spread

Variance and standard deviation

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Tutor: Bilal

Summary

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.

Learn with Basics

Length:

Unit 1

Measures of location: Quartiles, percentiles, deciles

Unit 2

Measures of spread: Range

Jump Ahead

Optional

Variance and standard deviation

Videos

Summary

Exercises

Select Lesson

Exam Board

Further kinematics

Application of forces

Projectiles

Forces and friction

Moments

Forces and motion

Variable acceleration

Constant acceleration

Modelling in mechanics

Normal distribution

Conditional probability

Correlation and hypothesis testing

Hypothesis testing I

Binomial distribution

Probability

Representation of data

Measures of location and spread

Data collection

Vectors II

Numerical methods

Integration II

Differentiation II

Trigonometry and modelling

Trigonometric functions

Radians

Sequences and series

Binomial expansion II

Parametric equations

Functions

Algebraic fractions

Proof II

Vectors I

Integration I

Differentiation I

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Trigonometric equations

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Binomial expansion I

Circles

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Explainer Video

Summary

Variance and standard deviation

In a nutshell

Variance

Finding the variance in grouped data

Standard deviation

Finding the standard deviation in grouped data

Example 1

Number of vacation days

Number of employees

Learn with Basics

Unit 1

Measures of location: Quartiles, percentiles, deciles

Unit 2

Measures of spread: Range

Jump Ahead

Unit 3