Business Mathematics and Statistics · Ch 9 — Measures of Dispersion

Mean Deviation and its Coefficient

Range and Quartile Deviation each use only a couple of specific data points (the extremes, or the quartiles) and ignore the rest. Mean Deviation fixes this by using every observation: it is the average of the absolute distances of every value from a chosen central value — usually the mean, sometimes the median. Absolute values are used deliberately, because the plain (signed) deviations from the mean always add up to exactly zero, which would make an unmodified average deviation useless.

Note

Mean Deviation about the Mean

$\text{MD}(\bar{x}) = \dfrac{\sum |x_i - \bar{x}|}{n} \quad \text{(individual data)}, \qquad \text{MD}(\bar{x}) = \dfrac{\sum f\,|x - \bar{x}|}{N} \quad \text{(discrete/grouped data)}$

where $N=\sum f$ , and for grouped/continuous data $x$ is the class mark of each class.

Note

Mean Deviation about the Median

$\text{MD}(M) = \dfrac{\sum |x_i - M|}{n}, \qquad \text{MD}(M) = \dfrac{\sum f\,|x - M|}{N}$

For continuous data, the median $M$ itself must first be found by the interpolation formula ( $M = L + \left(\frac{N/2-cf}{f}\right)h$ ) before the deviations can be measured from it.

A well-known result (used without proof at this level) is that mean deviation about the median is always the smallest mean deviation obtainable about any single point — smaller than mean deviation about the mean — which is one reason it is a genuinely useful alternative, not just a variant computed for its own sake.

As with every other absolute measure, Mean Deviation is expressed in the data's own unit, so its Coefficient — Mean Deviation divided by the value it was measured about — gives the unit-free relative version needed to compare two series fairly:

Note

Coefficient of Mean Deviation …

Definition 1Mean Deviation

The average of the absolute deviations of every observation from a chosen central value (mean or median); uses ALL observations, unlike Ran …

Definition 2Coefficient of Mean Deviation

Mean Deviation divided by the value (mean or median) it was measured about — a unit-free r …

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