GNDU B.Com (Bachelor of Commerce)

Chapter 3: Measures of Dispersion and Coefficient of Variation

Summary

Dispersion measures how far the values in a data set are spread out from a central value; two series may have the same average yet very different scatter, so a measure of dispersion is needed to judge the reliability of the average and to compare consistency. The range is the simplest measure, the difference between the largest and smallest values, \(Range = L - S\); it is easy to compute but uses only the extreme values. The quartile deviation, or semi-interquartile range, is \(Q.D.=\dfrac{Q_3 - Q_1}{2}\) and covers the middle fifty per cent of the data, avoiding the extremes. The mean deviation is the average of the absolute deviations of the items from a central value such as the mean or median. The most important measure is the standard deviation, the positive square root of the mean of the squared deviations from the arithmetic mean, written \(\sigma=\sqrt{\dfrac{\sum (x-\bar{x})^2}{n}}\); it is based on all observations and is suitable for further mathematical treatment. To compare the variability of two series with different units or averages, the coefficient of variation is used, \(C.V.=\dfrac{\sigma}{\bar{x}}\times 100\); the series with the lower coefficient of variation is more consistent or uniform, while the higher value indicates greater variability. Dispersion thus complements the average by showing how representative that average really is.

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Practice quiz