What is the Relation Between Mean, Median and Mode: CBSE Class 11 Maths

Mean, median, and mode are the three fundamental measures of central tendency in statistics. Mean is the average value of a given data set, the median is the middle value of the given set of data, and mode is the value repeated the highest number of times. Let us look at these measures in detail and establish their relationship.

What are Mean, Median, and Mode?

What is Mean?

In mathematics, the term mean generally refers to the “average” of a set of numbers. It is calculated by dividing the sum of all the values in a given set of data by the total number of values. The mean is a measure of central tendency, which indicates the central or typical value in a dataset. There are different types of means, but the most common one is the arithmetic mean.

Arithmetic Mean

The arithmetic mean is calculated by adding up all the numbers in a dataset and then dividing the sum by the total number of values. Here’s the formula:

Arithmetic Mean=Sum of all values/Number of values​

Example:

Suppose you have the following numbers: 5, 7, 10, 15, and 3.

1. Add the numbers: $5+7+10+15+3=40$
2. Count the numbers: There are 5 numbers in the set.
3. Calculate the mean: 405=8\frac{40}{5} = 8540​=8

So, the arithmetic mean of the set is 8.

Other Types of Means

• Geometric Mean: The geometric mean is used when you want to find the average rate of growth or the central tendency of a set of numbers that are multiplicatively related. It is calculated by multiplying all the numbers together and then taking the $n$th root, where $n$ is the number of values.
• Harmonic Mean: The harmonic mean is used in situations where you want to calculate the average of rates, such as speeds. It is calculated by dividing the number of values by the sum of the reciprocals of the values.

Use of Mean in Data Analysis

The mean is widely used in statistics, economics, finance, and many other fields to summarize data, compare different sets of data, and make predictions based on historical data. However, the mean can be influenced by extreme values (outliers), which can skew the result, so it’s often used in conjunction with other measures of central tendency, such as the median and mode, to provide a more complete picture of the data.

What is Median?

In mathematics, the median is a measure of central tendency that represents the middle value in a dataset when the numbers are arranged in ascending or descending order. The median is particularly useful because it is not affected by extreme values (outliers), making it a robust measure of central tendency.

How to Calculate the Median?

1. Arrange the Data:

• • First, arrange the data in order from smallest to largest (or largest to smallest).

2. Identify the Middle Value:

• • If the dataset has an odd number of values, the median is the middle number.
  • If the dataset has an even number of values, the median is the average of the two middle numbers.

Examples:

Example 1: Odd Number of Values

Consider the dataset: 3, 7, 9, 15, 20.

• Arrange in order (already sorted): 3, 7, 9, 15, 20.
• The middle value is 9, so the median is 9.

Example 2: Even Number of Values

Consider the dataset: 4, 8, 10, 12, 18, 22.

• Arrange in order (already sorted): 4, 8, 10, 12, 18, 22.
• The two middle values are 10 and 12.
• The median is the average of these two values: 10+122=11\frac{10 + 12}{2} = 11210+12​=11.

So, the median is 11.

Properties of the Median

• The median is less sensitive to outliers compared to the mean.
• In a symmetric distribution, the mean and median are usually the same.
• In a skewed distribution, the median is often a better representation of central tendency than the mean.

Use of Median in Data Analysis

The median is commonly used in various fields such as economics, social sciences, and real estate, especially when dealing with income data, home prices, or any data that may have significant outliers.

What is Mode?

In mathematics, the mode is a measure of central tendency that refers to the value or values that appear most frequently in a data set. Unlike the mean or median, which are calculated, the mode is simply the most common value in the set.

Key Points About Mode:

• Single Mode (Unimodal): A data set may have one mode, known as unimodal. For example, in the data set $3,5,7,7,9$, the mode is 7 because it appears most frequently.
• Multiple Modes (Multimodal): A data set can have more than one mode if two or more values occur with the same highest frequency. For example, in $2,4,4,6,6$, both 4 and 6 are modes, making it bimodal.
• No Mode: If all values in a data set occur with the same frequency, the data set may be considered to have no mode.

Use of Mode:

• The mode is particularly useful in categorical data where we want to know which is the most common category.
• It is also helpful when analyzing the frequency of numbers or events in a data set.

Relation between Mean, Median and Mode

Formula 1: Mode = 3 Median – 2 Mean

For example, If median = 12, mean = 8, mode = ?

We know that mode = 3 median – 2 mean

So, let us put the values we know in the formula

Mode = 3 (12) – 2 (8)

Mode = 36 – 16 = 20

Thus, the mode is equal to 20. Very simple!

Formula 2: Mode = Mean – 3 (mean – median)

For example, If mode = 6, mean – median = 12, mean = ?

We know that, mode = mean – 3 (mean – median)

So, let us put the values we know in the formula

6 = mean – 3 (12)

6 = mean – 36

6 + 36 = mean

Thus, the mean is equal to 42.

Formula 3: Mode = Median – ⅔ (mean – mode)

For example, mode = 13, mean – mode = 18, median = ?

We know that, mode = median – ⅔ (mean – mode)

So, let us put the values we know in the formula

13 = median – ⅔ (18)

13 = median – 12

13 + 12 = median

Median = 25

Thus, the median is equal to 25.

Simple solved question: Calculate the Mean, Mode and Median for the following set of values:

3, 5, 7, 9, 8, 4, 3, 8, 7, 9, 7

Ans: 

1. We know that mode is the value repeated the maximum number of times in the given values. So, here mode is 7.
2. Mean = Sum of all the values/ total number of values

                      = 70/11

                      = 6.36

    So, here mean is equal to 6.36

1. To calculate the median for an odd set of values, we first need to arrange all the given numbers in ascending order.

           3, 3, 4, 5, 7, 7, 7, 8, 8, 9, 9, 9

• Now let’s add the total number of values (n) to 1 and divide the result by 2. 

            11+½ = 12/6 = 6

• The 6th number in the given set of numbers is the median. So, here the median is 7.

If you go through the whole topic on mean, median and mode carefully from NCERT Solutions from Class 11 Maths and practise as many questions as possible, no one can stop you from securing full marks in the questions that come from this topic in the exam. Also, don’t forget to solve the previous years’ question papers for better understanding.

Also See: CBSE Class 12 Term 2 Syllabus Maths: 10 High Scoring Topics for Class 12 Board Exam Maths

Difference Between Mean, Median & Mode FAQs