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Class 11 Mathematics · Chapter 13
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Summary
Measures of central tendency (mean, median, mode) tell where data are centred but not how spread out they are. This chapter studies measures of dispersion: range, mean deviation, variance and standard deviation. The range is simply maximum minus minimum. The mean deviation about a central value \(a\) (mean or median) is the average of the absolute deviations: for ungrouped data \(\text{M.D.}(\bar x)=\dfrac{1}{n}\sum|x_i-\bar x|\), with frequency-weighted versions for grouped data. Because absolute values are awkward to manipulate, variance squares the deviations: \(\sigma^2=\dfrac{1}{n}\sum (x_i-\bar x)^2\), and the standard deviation \(\sigma=\sqrt{\sigma^2}\) restores the original units. For a frequency distribution, \(\sigma=\dfrac{1}{N}\sqrt{N\sum f_i x_i^2-\left(\sum f_i x_i\right)^2}\), and a step-deviation (shortcut) method simplifies the arithmetic. Adding a constant to every observation leaves the variance unchanged, while multiplying by \(k\) multiplies the variance by \(k^2\). These measures let us compare the consistency of two data sets with the same mean.
Measures of dispersionRangeMean deviation for ungrouped and grouped dataVariance and standard deviationShortcut (step-deviation) methodEffect of changing observations on variance
Key terms
Dispersion
The scatter or spread of data about a central value.
Range
Maximum value minus minimum value of the data.
Mean deviation
Average of the absolute deviations from a central value \(a\): \(\dfrac{1}{n}\sum|x_i-a|\).
Variance
Mean of the squared deviations from the mean: \(\sigma^2=\dfrac{1}{n}\sum(x_i-\bar x)^2\).
Standard deviation
The positive square root of the variance, \(\sigma=\sqrt{\sigma^2}\).
Effect of change
Adding a constant leaves variance unchanged; multiplying by \(k\) multiplies variance by \(k^2\).
Extra questions & answers
Try one from this chapter
The range of a data set is: