Mathematics and Statistics · Ch 11 — Measures of Dispersion

Mean Deviation

The range and quartile deviation both ignore most of the data. The Mean Deviation (M.D.) uses every value: it is the average of how far each value lies from a central value (the mean or the median), counting distances as positive.

Why distances must be made positive: the ordinary deviations from the mean always add up to zero (the positives and negatives cancel), so their plain average would be $0$ and tell us nothing. We therefore take the absolute value $|\;\cdot\;|$ of each deviation, which turns every distance positive before averaging.

Note

Mean Deviation (from the mean $\bar{x}$ )

Raw data: $\text{M.D.} = \frac{\sum |x_i - \bar{x}|}{n}$
Frequency distribution: $\text{M.D.} = \frac{\sum f_i\,|x_i - \bar{x}|}{N}, \qquad N = \sum f_i$ The same formulas, with the median $M$ in place of $\bar{x}$ , give the mean deviation from the median.

Note

Coefficient of Mean Deviation

$\text{Coefficient of M.D.} = \frac{\text{M.D.}}{\text{the average used}}$

i.e. divide by $\bar{x}$ if the deviations were taken from the mean, or by $M$ if from the median.

Note

A useful fact

The mean deviation is smallest when taken about the median — of all central values, the median minimises the total absolute distance. So if a problem does not specify, taking deviations about the median gives the tightest (smallest) mean deviation. …

Definition 8Mean deviation (M.D.)

The average of the absolute deviations of the values from a central value: $\dfrac{\sum |x_i - \bar{x}|}{n}$ (raw) or $\dfrac{\sum f_i|x_i - \bar{x}|}{N}$ (frequency); may be tak …

Definition 9Coefficient of mean deviation

$\dfrac{\text{M.D.}}{\text{mean or median used}}$ , the unit-free relative measure of dispersion based on …

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