Relation between Mean, Median and Mode

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 but they are closely related to each other. In this article, we’ll learn the concepts of relation between mean, median, and mode and explore the connections between them.

Mean

The mean, often called the average, is a measure of central tendency. It calculates the center 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 the mean
• x_i is the i^th data value
• n is the number of data points

Median

The 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

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:

Frequency Distribution	Relation
Symmetrical	Mean = Median = Mode
Positively Skewed	Mean > Median > Mode
Negatively Skewed	Mean < Median < Mode

Read More

• Mean in Statistics
• Median
• Mode

Solved Examples on Relation between Mean, Median and Mode

Example 1: The median and mode for a given set of data points is 20 and 30 respectively. Find out the mean. (Assume a moderately skewed distribution)

Solution:

Given,

Mode = 30, Median = 20

To find the Mean:

Mode = 3×Median − 2×Mean

30=3×20−2×Mean

30=60−2×Mean

2×Mean=30

Mean=15

Example 2: The mean and mode for a given set of data points is 20 and 30 respectively. Find out the mean. (Assume a moderately skewed distribution)

Solution:

Given:

Mode = 25, Mean = 12

To find the Median:

Mode=3×Median−2×Mean

25=3×Median−2×12

25=3×Median−24

3×Median=49

Median≈16.33

Example 3: The median and mean for a given set of data points is 15 and 10 respectively. Find out the mean. (Assume a moderately skewed distribution)

Solution:

Given:

Median = 15, Mean = 10

To find the Mode:

Mode=3×Median−2×Mean

Mode=3×15−2×10

Mode=45−20

Mode=25

Example 4: For a symmetrical distribution, the value of mean is 42. What can we say about the value of median and mode?

Solution:

For a symmetrical distribution, the value of mean, median and mode are approximately equal.

Hence,

Median = Mode = Mean = 42

Example 5: For a positively skewed distribution, the value of mean is 42 and mode is 20. What can we say about the value of median?

Solution:

For a positively skewed frequency distribution, the value of mean, median and mode has the following relation:

Mean > Median > Mode

For Mean = 42, and Mode = 20, we can say that the value of median lies in the range of 20 to 42.

Practice Questions on Relation between Mean, Median and Mode

1. Given: Mean = 24, Mode = 28. Calculate the Median and verify if the distribution is positively or negatively skewed.

2. Given: Mean = 40, Median = 35. Calculate the Mode and comment on the data distribution’s symmetry.

3. Given: Mode = 48, Mean = 45. Calculate the Median and determine the implications for the data’s central tendency.

Relation between Mean, Median and Mode – FAQs

1. What is Mean?

The mean is center value of a given dataset calculated by taking the average of all the values of dataset.

2. What is Median?

It is the middle value of a ordered dataset.

3. What is Mode?

It is the highest occurring value in the dataset.

4. What is the Empirical Relation Between the Mean, Median and Mode?

For a moderately skewed frequency distribution, 3 Median = 2 Mean + Mode

5. In What Condition, the Mean, Median and Mode will be Equal?

For completely symmetrical frequency distribution, Median = Mean = Mode

6. In What Scenarios would the Mean, Median, and Mode Differ Significantly?

Differences among mean, median, and mode can occur in skewed distributions or when the dataset contains outliers.

7. What is 3 Median Equal to?

3 Median is equal to to the sum of Mode and twice the Mean i.e. 3 Median = Mode + 2 Mean

8. How do you Find Mode when Mean and Median are Given?

For a moderately skewed frequency distribution, Mode = 3 Median – 2 Mean