Relationship Between Mean Median and Mode – Formula, FAQs

Relationship Between Mean Median and Mode

We should know about mean, median and mode before we learn about their relationship. 

Mean is the average of the given values or data set. We can calculate it by adding all the values and dividing the sum by the total number of data sets.

Median is the middle value among the given values. First, we arrange the values in ascending or descending order and then choose the middle value to calculate the median.

Mode is the highest frequency of a number from the given values. By counting the number of times each value occurs, we can calculate it.

The relation between mean, median and mode in a moderately skewed distribution can be represented by the formula given below –                                      

                                      3(Median) = (Mode) + 2(Mean) 

Proof of the Mean, Median, Mode Formula

It can be understood by the Karl Pearson’s formula, which states:

     (Mean – Median) = 1/3 (Mean – Mode)

⇒  3 (Mean – Median) = (Mean – Mode)

⇒  3 Mean – 3 Median = Mean – Mode

⇒  3 Median = 3 Mean – Mean + Mode

⇒  3 Median = 2 Mean + Mode

Empirical Relation Between Mean Median and Mode

Now, we will understand the mean, median, and mode empirical relation employing a frequency distribution graph. We can divide it into four different cases:

• Moderately Skewed Distribution

In this case, the mode is equal to the difference between three times the median and two times the mean. Therefore, in this case, we represent the empirical relationship as

      Mode = 3 (Median) – 2 (Mean) 

Or,  (Mean) – (Mode) = 3 (Mean – Median)

• Symmetrical Frequency Curve

In this condition, the empirical relation is expressed as mean = median = mode.

• Positively Skewed Frequency Distribution Curve

In this condition, mean > median > mode.

• Negatively Skewed Frequency Distribution Curve

In this condition, mean < median < mode.

Examples

1. In a moderately skewed distribution, median = 11 and mean = 13. Find the value of the mode.

In a moderately skewed distribution, that the relationship between mean, median, and mode is 3 (median) = mode + 2 (mean)

Let mode be x 

Substituting the values,

      3 × 11 = x + (2 × 13)

⇒  33 = x + 26

⇒  x = 33 – 26

⇒  x = 7

Therefore, the value of mode is 7.

2. In a positively skewed distribution, calculate the median range if the mean and mode values are 40 and 30, respectively?

We know that for positively skewed frequency distribution, the empirical relation is mean > median > mode. 

Based on this, the range of the median if the mean is 40 and mode is 30 is 40 > median > 30. 

It means that the median will be greater than 30 and less than 40.