Whenever we want to compare the variability of two series which differ widely in their averages or which are measured in different units, we do not merely calculate the measures of dispersion but we calculate the coefficients of dispersion which are pure numbers independent of the units of measurement. The coefficient of dispersion based on quartiles of the given set of data is known as the coefficient of quartile deviation.
Formula:
The coefficient of quartile deviation can be calculated using the formula, \text{Coefficient of Quartile Deviation }= \frac{\frac{Q_3-Q_1}{2}}{\frac{Q_3+Q_1}{2}} = \frac{Q_3-Q_1}{Q_3+Q_1}. Here, Q_3 denotes the upper quartile and Q_1 denotes the lower quartile.
Interpretation:
Notice that in the above formula the numerator is always less than the denominator. In particular, this implies that the value of the coefficient is always less than 1. In fact, the coefficient always lies between 0 and 1. The closer the value of the coefficient is to 1 the greater is the degree of spread/variation in the data.
Example Calculation:
Consider the following series of ungrouped data given as: 21, 25, 28, 32, 36, 57, 59, 63. Since we have 8 data values we have that N=8. We first calculate the upper and lower quartiles as follows, \text{Lower Quartile }Q_1 = \left(\frac{N}{4}\right)^{th}\text{ term } = \left(\frac{8}{4}\right)^{th}\text{ term } = 2^{nd}\text{ term } = 25 . \text{Upper Quartile }Q_3 = \left(\frac{3N}{4}\right)^{th}\text{ term } = \left(\frac{3\times 8}{4}\right)^{th}\text{ term } = 6^{th}\text{ term } = 57 . We now calculate the coefficient of quartile deviation using the formula, \text{Coefficient of Quartile Deviation } = \frac{Q_3-Q_1}{Q_3+Q_1} = \frac{57-25}{57+25} = \frac{32}{82} = 0.3902. Since the value of the coefficient for this data set is closer to 0 compared to 1, we conclude that the above data set has a lesser degree of dispersion/variation.
