Empirical Relation between Mean, Median and Mode

The empirical relation between mean, median, and mode is one of statistics' most important shortcuts. It is a single formula that connects all three. The word "empirical" means "based on observation". In statistics, an empirical formula isn't derived from a rigid mathematical proof but from patterns that statisticians noticed in real-world data. The empirical relation between mean, median, and mode is a formula introduced by the statistician Karl Pearson, which helps you estimate one measure of central tendency when you know the other two, making data analysis easier and more practical. In this guide, you will learn the empirical relation between mean, median, and mode, along with the formula, explanation, and examples for easy understanding.

Table of Contents

• What is the Empirical Relation between Mean, Median, and Mode
• Relation between Mean, Median, and Mode in Frequency Distribution
• Solved Examples on the Empirical Relation between Mean, Median, and Mode

What is the Empirical Relation between Mean, Median, and Mode

Karl Pearson, a pioneer of modern statistics, studied the pattern across hundreds of real-world frequency distributions. He noticed that for any moderately skewed dataset, the gap between the mean and the mode is almost exactly three times the gap between the mean and the median. This insight led to what we now call the empirical relation. Karl Pearson’s Empirical relation:

Mean − Mode = 3 (Mean − Median)

Equivalent forms:

• Mode = 3 Median − 2 Mean

• 3 Median = 2 Mean + Mode

Relation between Mean, Median, and Mode in Frequency Distribution

The formula applies specifically to moderately skewed distributions. Here's how the three measures sit relative to each other in different distribution types:

Symmetrical Frequency Curve

If the frequency distribution has a perfectly symmetrical curve, the mean, median, and mode all coincide and have the same value. Data is balanced perfectly on both sides of the centre.

Mean = Median = Mode

Positively Skewed Frequency Distribution

In a positively skewed distribution, the mean is greater than the median and the median is greater than the mode. A very few high values pull the mean towards the right, creating a long tail on the right side of the distribution.

Mean > Median > Mode

Negatively Skewed Frequency Distribution

In a negatively skewed distribution, the mean is less than the median, and the median is less than the mode. A very few low values pull the mean towards the left, creating a long tail on the left side of the distribution

 

Mean < Median < Mode

Solved Examples on the Empirical Relation between Mean, Median, and Mode

Example 1: A distribution has mean = 50 and median = 45. Find the mode using the empirical formula.

Solution: Using Karl Pearson’s Empirical relation Mode = 3 Median − 2 Mean

Mode = 3(45) − 2(50) = 135 - 100 = 35
Mode = 35

Example 2: A distribution has a mean of 40.5 and a mode of 27. Find the median using the empirical formula.

Solution: Given, Mean = 40.5 and Mode = 27

Mode = 3 Median − 2 Mean 27 = 3 Median − 2(40.5)

Median = (27 + 81)/3 = 36.
Median = 36.

Example 3: A company HR report states that the median monthly salary of employees is ₹45,000 and the mean monthly salary is ₹52,000. Use the empirical relation to estimate the most common (modal) salary.

Solution: Given, median salary = ₹45,000 and mean salary = ₹52,000
Mode = 3 Median − 2 Mean
Modal Salary = 3(₹45,000) - 2(₹52,000) = ₹31,000
The most common (modal) salary is ₹31,000.