Spread - Variation

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There are various ways of displaying or calculating the spread, or variation, of a collection of data. From the simpliest to calculate, range, to those summarising all the data, standard deviation. Displays of the distribution of the data can be completed quickly ising stem-and-leaf diagrams or in more detail using frequency density plots, or histograms.

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Range
(a measure of spread)

The difference between the highest and lowest elements.

$$\text{range}=\text{maximum Value}-\text{minimum value}$$

Example:

The set

$$\{33,31,42,45,32,13,36,40\}$$

has a highest value of 45 and lowest value of 13 therefore the range is $45-13=32$.

For group data, an estimate of the range can be worked out using $\mathrm{maximum\; possible\; value}-\mathrm{minimum\; possible\; value}$

Example:

Fish Length / cm	Frequency
1 - 10	1
11 - 20	7
21 - 30	5
31 - 40	4
41 - 50	3

Here the $\mathrm{minimum\; possible\; value}$ is $1\mathrm{cm}$ and the $\mathrm{maximum\; possible\; value}$ is $50\mathrm{cm}$ so the range is 49.

Range Video

Range (frequency tables) Video

Practice Questions

Textbook Exercise

(Care of corbettmaths.com)

Questions
(Care of thirdspacelearning.com)

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IQR - interquartile range

$$\mathrm{IQR}=\mathrm{UQ}-\mathrm{LQ}$$

Averages: Quartiles Video

Practice Questions

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Stem & Leaf Diagrams

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Histograms

$$\mathrm{frequency\; density}=\frac{\mathrm{frequency}}{\mathrm{class\; width}}$$

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Videos and Questions:

Graphs: histograms (draw) Video:

Graphs: histograms (interpret) Video:

Graphs: histograms harder Video:

Practice Questions;

Textbook Exercise.

Textbook Exercise.

(Care of corbettmaths.com)

Histograms:

Questions [ MS ].

(Care of www.mathsgenie.co.uk)

Histograms - GCSE Maths - Steps, Examples & Worksheet
(Care of thirdspacelearning.com)

Histograms - Math is Fun
(Care of www.mathsisfun.com)

Histograms | Edexcel A Level Maths: Statistics Revision Notes 2018
(Care of www.savemyexams.com)

Search for GCSE Maths Histograms

Search for A Level Maths Histograms

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Cumulative Frequency Graphs

Cumulative Frequency Graphs Worksheet [ MS ];

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Box Plots

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Normal Distibution

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Variance and Standard Deviation - measures of spread:

$$\mathrm{var}={\sigma }^2$$ $${\sigma }^2=\frac{\Sigma {(x_i-\stackrel{\_}{x})}^2}{n}$$ $${\sigma }^2=\frac{\Sigma ({x_i}^2-2x_i\stackrel{\_}{x}+{\stackrel{\_}{x}}^2)}{n}$$ $${\sigma }^2=\frac{\Sigma {x_i}^2-2\Sigma x_i\stackrel{\_}{x}+\Sigma {\stackrel{\_}{x}}^2}{n}$$ $${\sigma }^2=\frac{\Sigma {x_i}^2}{n}-\frac{2\stackrel{\_}{x}\Sigma x_i}{n}+\frac{n{\stackrel{\_}{x}}^2}{n}$$ $${\sigma }^2=\stackrel{\_}{x^2}-2{\stackrel{\_}{x}}^2+{\stackrel{\_}{x}}^2$$ $${\sigma }^2=\stackrel{\_}{x^2}-{\stackrel{\_}{x}}^2$$

or "the standard deviation is the mean of the square minus the square of the mean".

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