What is Empirical Relationship Between Mean, Median And Mode?

In a symmetrical distribution, mean, median and mode are identical and have the same value. However, in actual life most distributions are not symmetrical—they are skew. (These concepts of symmetry and skewness would be discussed in the chapter on Skewness). In cases of moderately skew (or moderately asymmetrical) distributions the value of the mean, median and mode have the following empirical relationship which is given by Karl Pearson :

(i) Mode = Mean—3 (Mean—Median)

= Mean—3 Mean + 3 Median

= 3 Median — 2 Mean

Thus the value of Mode — 3 Median—2 Mean. From this relationship,

Mode = Mean—3 (Mean—Median) we can derive

(ii) Mean—Mode = 3 (Mean—Median) or

(iii) Mean—Median = 1/3 (Mean—Mode)

Thus we see that the difference between Mean and Mode is three times, the difference between mean and median. In other words median is closer to the mean than mode. This relationship between Mean (X), Median (M) and Mode (Z) is shown in the following diagram.


Relationship between Mean (X) Median (M) and Mode (Z)

Relationship between Mean

Mode (Z) touches the peak of the curve indicating maximum frequency ; Median (M) divides the area of the curve in two equal halves and Mean (X) is the centre of gravity.

These relationships in the values of mean, median and mode are of great value. In cases where mode is ill-defined, or the series is bi-modal and mode cannot be calculated by the formulae discussed earlier, this relationship can give an empirical value of the mode.

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