Formula

Empirical relation between mean, median and mode

Statistics • Mode

Formula

Mode = 3 Median - 2 Mean

For a moderately asymmetrical (skewed) distribution,

Mode \approx 3 Median - 2 Mean

; equivalently

Mean - Mode = 3 (Mean - Median)

How to use it

1Given any two of the three averages, rearrange for the third.
2Check the arithmetic with the second form: $Mean - Mode$ must come to three times $Mean - Median$ .
3The three averages coincide in a symmetrical distribution, so a large gap between mean and median signals strong skew.

Worked example

A skewed distribution has mean

32

and median

30

. Estimate its mode.

Step-by-step

1
$Mode = 3 (30) - 2 (32)$
Why: Substitute the median and the mean into the relation.
2
$= 90 - 64 = 26$

Answer

26

Watch out

It is empirical, not exact: it holds only for moderately skewed unimodal data and can fail badly on a strongly skewed or multimodal set. Interchanging the median and the mean gives

3 Mean - 2 Median

, which is the standard error.

Builds on

Median of a data set Mean of a frequency distribution

Also known as

mode = 3 median - 2 mean, empirical relation, Karl Pearson relation, mean median mode relation, skewness relation

Appears in past papers

2017 (I)

Q55