Empirical Relationship among mean, median and mode

Empirical Relationship among mean, median and mode

A frequency distribution in which the values of arithmetic mean, median and mode coincide is known of symmetrical distribution, when the values of mean, median and mode are not equal the distribution is known as asymmetrical or skewed. In moderately skewed asymmetrical distributions a very important relationship exists among arithmetic mean, median and mode.

Karl Pearson has expressed this relationship as follows

Mode = 3 Median – 2 Arithmetic Mean

Example 5.28

In a moderately asymmetrical frequency distribution, the values of median and arithmetic mean are 72 and 78 respectively; estimate the value of the mode.

Solution:

The value of the mode is estimated by applying the following formula:

Mode = 3 Median – 2 Mean = 3 (72) – 2 (78)

216 - 156 = 60 Mode = 60

Example 5.29

In a moderately asymmetrical frequency distribution, the values of mean and mode are 52.3 and 60.3 respectively, Find the median value.

Solution:

The value of the median is estimated by applying the formula:

Mode = 3 Median – 2 Mean

 60.3 = 3 Median – 2 × 52.3

 3 Median = 60.3 + 2 × 52.3

 60.3 + 104.6 = 164.9

 Median = 164.9/3 = 54.966 = 54.97