Economics · Ch 16 — Measures of Dispersion

Measures of Dispersion from Average

Dispersion was defined as the extent to which values differ from their average. Range and quartile deviation, though useful, measure only the spread of values and do not directly measure how far the values lie from their average. Two measures that are based on the deviation of the values from their average are the Mean Deviation and the Standard Deviation.

Since the average is a central value, some deviations from it are positive and some negative. If they are added as they are, they cancel out — in fact the sum of the deviations from the arithmetic mean is always zero. Mean deviation overcomes this by ignoring the signs (treating all deviations as positive); standard deviation overcomes it by squaring the deviations, averaging them, and then taking the square root.

Mean Deviation

Mean deviation is the arithmetic mean of the absolute differences of the values from their average (either the arithmetic mean or the median; the mode, not being a stable average, is not used). The absolute deviation of a value is written $|d|$ .

For ungrouped data, the steps are: (i) calculate the average (mean or median); (ii) find the absolute deviation $|d|$ of each value from that average; (iii) take the arithmetic mean of these deviations. That is,

M.D. = \dfrac{\sum |d|}{n}

For a continuous (grouped) distribution, use the class mid-points and weight each deviation by the class frequency:

M.D. = \dfrac{\sum f|d|}{\sum f}

Mean deviation — a note. Mean deviation is based on all the values, so a change in even one value affects it. It is least when calculated from the median (it is larger when taken from the mean). Its weaknesses are that it ignores the signs of deviations — which appears mathematically unsound — and that it cannot be calculated for open-ended distributions.

Standard Deviation

Standard deviation is the positive square root of the mean of the squared deviations from the mean. Given a set of values, first the mean is found; then the deviations from the mean are calculated; these deviations are squared; the mean of the squared deviations is the variance; and the positive square root of the variance is the standard deviation, denoted $\sigma$ . (Note that standard deviation is always calculated on the basis of the mean only.)

Standard deviation for ungrouped data. Four alternative methods all give the same value:

Actual Mean Method — with $d = X - \bar{X}$ :

\sigma = \sqrt{\dfrac{\sum d^2}{n}}

Assumed Mean Method — taking deviations $d = X - A$ from any arbitrary value $A$ :

\sigma = \sqrt{\dfrac{\sum d^2}{n} - \left(\dfrac{\sum d}{n}\right)^2}

The sum of deviations from a value other than the actual mean is not zero, but standard deviation is unaffected by the constant chosen — it is independent of origin.

Direct Method — using the values directly, without taking deviations (i.e. taking deviations from zero):

\sigma = \sqrt{\dfrac{\sum X^2}{n} - (\bar{X})^2}

Step-Deviation Method — when the values (or deviations) share a common factor $c$ , divide by it to simplify. Dividing the values by $c$ (so $x' = \dfrac{X}{c}$ ):

\sigma = \sqrt{\dfrac{\sum d'^2}{n}} \times c

or, dividing the deviations (from an assumed mean) by $c$ :

\sigma = \sqrt{\dfrac{\sum d'^2}{n} - \left(\dfrac{\sum d'}{n}\right)^2} \times c

Standard deviation for a continuous distribution. Using class mid-points $m$ and frequencies $f$ :

Actual Mean Method (with $d = m - \bar{X}$ ):

\sigma = \sqrt{\dfrac{\sum fd^2}{n}}

Assumed Mean Method (with $d = m - A$ ): …

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