Absolute and Relative Measures of Dispersion

All the measures described so far — range, quartile deviation, mean deviation and standard deviation — are absolute measures of dispersion. They give a value that can be difficult to interpret, for two reasons.

First, an absolute measure can mislead when the averages differ. Consider the daily sales of an ice-cream vendor, Set A = ₹500, ₹700, ₹1000, and of a big departmental store, Set B = ₹1,00,000, ₹1,20,000, ₹1,30,000. The range of Set A is ₹500 while that of Set B is ₹30,000 — far higher. Yet the variation in Set A is really greater: its largest value is double its smallest, whereas Set B's largest is only about 30% higher than its smallest. Second, an absolute measure gives the answer in the units of the original data — express the values in metres instead of kilometres and the measure of dispersion becomes 1000 times as large, though nothing real has changed.

To overcome these problems we use relative measures of dispersion. Each absolute measure has a relative counterpart, obtained by dividing it by a suitable average, so that the result is a pure number free of units and can be compared even across groups measured in different units.

• Coefficient of Range $= \dfrac{L - S}{L + S}$, where $L$ is the largest value and $S$ the smallest.
• Coefficient of Quartile Deviation $= \dfrac{Q_3 - Q_1}{Q_3 + Q_1}$, where $Q_3$ is the third quartile and $Q_1$ the first quartile.
• Coefficient of Mean Deviation $= \dfrac{M.D.}{\bar{x}}$ if the mean deviation is taken from the mean, or $\dfrac{M.D.}{Median}$ if it is taken from the median.
• Coefficient of Variation (C.V.), the relative measure corresponding to standard deviation: $C.V. = \dfrac{\sigma}{\bar{x}} \times 100$ …