Relation between the Mean, Median, and Mode is the difference between the thrice of the median and twice of the mean gives Mode. Mode Mean, median, and mode are fundamental statistics measurements that give valuable insights into a given dataset. They represent the central tendency of the dataset. Although their values differ from each other for a given dataset they are closely related to each other.

Relation between Mean, Median and Mode

In this article, we’ll learn the concepts of relation between mean, median, and mode and explore the connections between them.

Mean Definition

Mean, often called the average, is a measure of central tendency. It calculates the centre or middle data value of a given dataset. It can be calculated by summing up all the values in the dataset and dividing the sum by the total number of data points. For given data X, the formula for calculating the mean (μ) is:

where:

• μ is Mean
• x_i is the i^th data value
• n is Number of data points

Median Definition

Median is the middle value of a dataset when it is arranged in ascending or descending order. For odd number of data points, the median is directly taken as the middle value but in case of even number of data points, median is the average of the two middle values.

If n is the total number of observations,

• If n is odd, median is the (n+1)/2^th value.

• If n is even, median is the average of n/2^th and (n/2)+1^th value.

Mode Definition

Mode is the most frequently occurring data point in the dataset. It does not take any consideration of the magnitude of the data points. It just tells us the data point with maximum frequency. There can be multiple mode if there are more then one data points with the maximum frequency.

Relation between Mean, Median and Mode

The relation between these three statistical measures depends upon the skewness of the data. For a moderately skewed frequency distribution, the empirical relation between the Mean, Median and Mode can be written as:

Mode = 3 Median – 2 Mean

It can be derived from Karl Pearson’s formula, which states:

(Mean – Median) = 1/3 (Mean – Mode)

It can be written as:

3 (Mean – Median) = (Mean – Mode)

3 Mean – 3 Median = Mean – Mode

3 Median = 2 Mean + Mode

Relation between Mean, Median and Mode with Frequency Distribution

We can also compare the mean, median and mode by looking at the frequency distribution curve of the data. There are commonly three types of distribution:

Symmetrical Frequency Distribution

In a symmetrical frequency distribution, values are equally distributed on both sides of the central point, creating a balanced and mirror-like pattern in the histogram or frequency polygon.

Positively Skewed Frequency Distribution

A positively skewed frequency distribution is characterized by a longer right tail. The majority of values cluster on the left side, while a few higher values extend the distribution to the right.

Negatively Skewed Frequency Distribution

In a negatively skewed frequency distribution, or left-skewed, the bulk of values cluster on the right side, and a longer left tail is observed. This indicates that there are fewer lower values, and the distribution is pulled toward the left.

Here is the comparison between mean, median and mode for these frequency distribution types: