2B. Measures of Spread

Range

The range of a dataset can describe the difference between the largest and smallest values in the dataset ( $m a x - m i n$ ).

Interquartile Range (IQR) describes the difference between the upper quartile $(Q_{3})$ and the lower quartile $(Q_{1})$

Interpercentile range is the range between any two given percentiles.

Variance

Variance is the average of squared deviations from the mean. it shows how spread out the data is. It is represented by $σ^{2}$

There are three versions of the formulae to find variance $σ^{2}$
1)

σ^{2} = \frac{\sum (x - \overset{―}{x})^{2}}{n}

This is easier to use when given raw data. You can remember this as "mean of squares minus square of means"

σ^{2} = \frac{\sum x^{2}}{n} - (\frac{\sum x}{n})^{2}

$σ^{2} = \frac{S_{x x}}{n}$

where $S_{x x} = \sum (x - \bar{x})^{2} = \sum x^{2} - \frac{(\sum x)^{2}}{n}$

$S_{x x}$ is a summary statistic used to simplify formulae.

Standard Deviation

The standard deviation $(σ)$ is the square root of variance; larger value = more spread

$\sqrt{σ^{2} = \frac{\sum (x - \overset{―}{x})^{2}}{n}} =$ $\sqrt{σ^{2} = \frac{\sum x^{2}}{n} - (\frac{\sum x}{n})^{2}} =$ $\sqrt{σ^{2} = \frac{S_{x x}}{n}}$ $= σ$

Frequency Tables

Variation

For data displayed in a frequency table

σ^{2} = \frac{\sum f (x - \overset{―}{x})^{2}}{\sum f}

σ^{2} = \frac{\sum f x^{2}}{\sum f} - (\frac{\sum f x}{\sum f})^{2}

Standard Deviation

For data displayed in a frequency table

σ = \sqrt{σ^{2} = \frac{\sum f (x - \overset{―}{x})^{2}}{\sum f}}

σ = \sqrt{σ^{2} = \frac{\sum f x^{2}}{\sum f} - (\frac{\sum f x}{\sum f})^{2}}

Five-number summary

minimum, lower quartile, median, upper quartile, maximum

Source

Mogull, R. G. (1990). Popular Measures of Central Tendency. The Mathematics Teacher, 83(9), 744–746. Retrieved from https://pmt.physicsandmathstutor.com/download/Maths/A-level/Statistics/Data-Presentation-and-Interpretation-1/Cheat-Sheets/Measures of Location and Spread.pdf

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