# Calculation of mode by using Relationship of mean and median that is empirical formula

20/10/2022

The relation between mean, median and mode is very important to understand in statistics and is useful while dealing with similar problems. In the case of a moderately skewed distribution, i.e. in general, the difference between mean and mode is equal to three times the difference between the mean and median.

The arithmetic mean refers to the average of a data set of numbers. It can either be a simple average or a weighted average. To calculate a simple average, we add up all the numbers given in the data set and then divide it by the total frequency. The median is the middle number of a given data set when it is arranged in either a descending order or ascending order. If there is an odd amount of numbers, the median value is the number that is in the middle whereas if there is an even amount of numbers, the median is the simple average of the middle pair in the dataset. Median is much more effective than a mean because it eliminates the outliers. The mode refers to the number that appears the most in a dataset. A set of numbers may have one mode, or more than one mode, or no mode at all.

The formula to define the relation between mean, median, and mode in a moderately skewed distribution is 3 (median) = mode + 2 mean. The proof of the mean, median, mode formula can be understood using 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 Relationship

In statistics, there is a relationship between the mean, median and mode that is empirically based. Observations of countless data sets have shown that most of the time the difference between the mean and the mode is three times the difference between the mean and the median. This relationship in equation form is:

Mean – Mode = 3(Mean – Median)