Relation Between Mean Median and Mode

In statistics, for a moderately skewed distribution, there exists a relation between mean, median, and mode. This mean median and mode relationship is known as the “empirical relationship” which have been discussed in detail below.

To recall,

  • Mean is the average of the data set which is calculated by adding all the data values together and dividing it by the total number of data sets.
  • Median is the middle value among the observed set of values and is calculated by arranging the values in ascending order or in descending order and then choosing the middle value.
  • Mode is the number from a data set which has the highest frequency and is calculated by counting the number of times each data value occurs.

Empirical Relationship between Mean, Median and Mode

In case of a moderately skewed distribution, the difference between mean and mode is almost equal to three times the difference between the mean and median. Thus, the empirical mean median mode relation is given as:

Mean – Mode = 3 (Mean – Median)

Mean Median Mode Relation With Frequency Distribution

  • Frequency Distribution with Symmetrical Frequency Curve

If a frequency distribution graph has a symmetrical frequency curve, then mean, median, and mode will be equal.

Relation between Mean Median Mode

Mean = Median = Mode
  • For Positively Skewed Frequency Distribution

In case of a positively skewed frequency distribution, the mean is always greater than median and the median is always greater than the mode.

Positively skewed mean median mode

Mean > Median > Mode
  • For Negatively Skewed Frequency Distribution

In case of a negatively skewed frequency distribution, the mean is always lesser than median and the median is always lesser than the mode.

Negatively Skewed mean median mode

Mean < Median < Mode


Also Check:
Mean Median Mode Formula

Example Question Using the Mean, Median and Mode Relationship

Question: In a moderately skewed distribution, the median is 20 and the mean is 22.5. Using these values, find the approximate value of the mode.

Solution:

Given,

Mean = 22.5

Median = 20

Mode = x

Now, using the relationship between mean mode and median we get,

(Mean – Mode) = 3 (Mean – Median)

So,

22.5 – x = 3 (22.5 – 20)

22.5 – x = 7.5

∴ x = 15

So, Mode = 15.

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2 Comments

  1. Very good, this is going to be useful for some central tendency estimator I need to implement. Thanks!

  2. Srinivas Godisela

    Simple and good explanation.

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