Business Mathematics and Basic Statistics · Ch 7 — Measures of Dispersion

Mean Deviation about the Mean — Discrete Data

The measures of central tendency covered earlier — mean, mode, median — each summarise a data set with a single representative value, but they say nothing about how spread out the data is around that value. Two classes could have the identical average mark of 60 yet look completely different: one where every student scored within a couple of marks of 60, and another where scores range wildly from 20 to 95. Measures of dispersion answer exactly this question — how much do the individual observations typically deviate from a central value?

The mean deviation about the mean is the most direct such measure: it is the average of the absolute distances of every observation from the arithmetic mean. Absolute values are used deliberately — if plain (signed) deviations were averaged, the positive and negative deviations would always cancel out to exactly zero (a fact already noted when the mean was introduced), which would make the measure useless.

Note

Mean Deviation about the Mean — Individual Observations

For $n$ individual observations $x_1, x_2, \ldots, x_n$ with arithmetic mean $\bar{x}$ ,

$\text{MD}(\bar{x}) = \dfrac{\sum |x_i - \bar{x}|}{n}$

When the data is organised as a discrete frequency distribution — distinct values $x_i$ against frequencies $f_i$ — every deviation $|x_i - \bar{x}|$ is simply weighted by how many times that value occurs:

Note

Mean Deviation about the Mean — Discrete Frequency Distribution

$\text{MD}(\bar{x}) = \dfrac{\sum f_i\,|x_i - \bar{x}|}{\sum f_i} = \dfrac{\sum f\,|x-\bar{x}|}{N}$

where $N = \sum f$ is the total number of observations, and $\bar{x}$ is computed first, exactly as in the measures-of-central-tendency chapter.

A larger mean deviation signals a more widely spread-out data set; a mean deviation close to zero signals that most observations cluster tightly around the mean. Computing the mean deviation about the mean — for both a plain list of observations and a frequency table — is a standard skill assessed in the WBCHSE Class 12 Business Mathematics and Basic Statistics semester examination, and the same idea of averaging absolute deviations from the mean is taught under measures of dispersion in statistics courses across every Indian curriculum.

Definition 1Mean Deviation about the Mean

The average of the absolute deviations of every observation from the arithmetic mean; $\text{MD}(\bar{x}) = \sum|x-\bar{x}|/n$ for individual data, $\sum f|x-\bar{x}|/N$ for a frequency distribution.

Definition 2Absolute Deviation

The quantity $|x_i - \bar{x}|$ — the (always non-negative) distance of an observation from the mean, ignoring the sign of the difference.

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