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Measures of location and spread

Variance and standard deviation

Variance and standard deviation

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Explainer Video

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 xxx​ deviates from the mean x‾\overline xx with the calculation (x−x‾)2(x-\overline x)^2(x−x)2.​

It is given in square units.


​σ2=Σ(x−x‾)2n=Σ(x2)n−(Σxn)2\sigma^2=\dfrac{\Sigma(x-\overline x)^2}{n}=\dfrac{\Sigma(x^2)}{n}-\left(\dfrac{\Sigma x}{n}\right)^2σ2=nΣ(x−x)2​=nΣ(x2)​−(nΣx​)2​​

​σ2\sigma^2σ2​​

Variance of a data set

​Σ(x−x‾)2{\Sigma (x-\overline x)^2}Σ(x−x)2​​

Sum of the squares of each point's deviation from the mean

nnn​​

Number of elements in the data set


Note: For raw data, it is easier to use Σx2n−(Σxn)2\dfrac{\Sigma x^2}{n}-\left(\dfrac{\Sigma x}{n}\right)^2nΣx2​−(nΣx​)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:


​σ2=Σf(x−x‾)2Σf=Σfx2Σf−(ΣfxΣf)2\sigma^2=\dfrac{\Sigma f(x-\overline x)^2}{\Sigma f}=\dfrac{\Sigma fx^2}{\Sigma f}-\left(\dfrac{\Sigma fx}{\Sigma f}\right)^2σ2=ΣfΣf(x−x)2​=ΣfΣfx2​−(ΣfΣfx​)2​​


Standard deviation

This is the square root of the variance.


​σ=Σ(x−x‾)2n=Σ(x2)n−(Σxn)2\sigma=\sqrt\dfrac{\Sigma(x-\overline x)^2}{n}=\sqrt{\dfrac{\Sigma(x^2)}{n}-\left(\dfrac{\Sigma x}{n}\right)^2}σ=nΣ(x−x)2​​=nΣ(x2)​−(nΣx​)2​​​

​σ\sigmaσ​​

Standard deviation of the data set

​Σ(x−x‾)2{\Sigma (x-\overline x)^2}Σ(x−x)2​​

Sum of the squares of each point's deviation from the mean

nnn​​

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:


​σ=Σf(x−x‾)2Σf=Σfx2Σf−(ΣfxΣf)2\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}σ=ΣfΣf(x−x)2​​=ΣfΣfx2​−(ΣfΣfx​)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

​181818​​

​191919​​

​202020​​

​212121​​

​222222​​

​232323​​

​242424​​

Number of employees

​111​​

​444​​

​555​​

​888​​

​101010​​

​333​​

​111​​


Find the variance and the standard deviation.


Start by finding Σfx\Sigma fxΣfx and Σfx2\Sigma fx^2Σfx2​:


Σfx=(18×1)+(19×4)+(20×5)+(21×8)+(22×10)+(23×3)+(24×1)=675\Sigma fx=(18\times1)+(19\times4)+(20\times5)+(21\times8)+(22\times10)+(23\times3)+(24\times1)=675Σfx=(18×1)+(19×4)+(20×5)+(21×8)+(22×10)+(23×3)+(24×1)=675

Σfx2=(182×1)+(192×4)+(202×5)+(212×8)+(222×10)+(232×3)+(242×1)=14299{\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}Σfx2=(182×1)+(192×4)+(202×5)+(212×8)+(222×10)+(232×3)+(242×1)=14299​​​


σ2=Σfx2n−(Σfxn)2=1429932−(67532)2=1.89746...‾\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...}σ2=nΣfx2​−(nΣfx​)2=3214299​−(32675​)2=1.89746...​


​σ=1.89746...=1.37748...‾\sigma=\sqrt{1.89746...}=\underline{1.37748...}σ=1.89746...​=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.


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Measures of location: Quartiles, percentiles, deciles

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Measures of spread: Range

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Variance and standard deviation

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Variance and standard deviation

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FAQs - Frequently Asked Questions

What is the standard deviation?

It is the square root of the variance.

What are the units of the variance?

Square units.

What type of measures are the variance and standard deviation?

They are measures of spread.

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