Mean Median Mode

Introduction to Mean, Median, and Mode

The concepts of mean, median, and mode are some of the most important topics in statistics. They are called measures of central tendency because they help us to find a central or specific value in a set of numbers. These measures are widely used to understand and compare data in mathematics, economics, science, and even our daily lives.

The mean is the average value, which is found by adding all the numbers in the dataset and dividing by the number of values. The median is the middle value when the numbers are arranged in order, and the mode is the value that most often appears. Together, they give us a complete picture of how the data spreads and where the most values lie.

For example, if we have the numbers 5, 6, 9, 9, and 10, the average is 7.8, the median is 9, and the mode is also 9. These measures are useful in real-life situations, such as calculating average markings in the exam, detecting the most common shoe size in a store, or identifying the middle income in a city. In this topic, we will learn what the mean, median, and mode are, how to calculate them, and how they are related to each other with simple rules and examples.

 

Table of Contents

• What are the mean, median, and mode in statistics?
• Measures of central tendency
• Mean
• Median
• Mode
• Example of Mean, Median, and Mode
• Relation
• Solved Problem
• Practice Questions
• FAQs on Mean, Median, and Mode

 

What are the mean, median, and mode in statistics?

We often see numbers & data in our daily lives, such as cricket scores, exam marks, or even the number of goals scored in a football match. But instead of looking at all numbers one by one, it is easy to find just one number that represents the whole set. This is where mean, median, and mode come from. These are called measures of central tendency because they show the middle, average, or most common value of data.

For example:

• In cricket, the running frequency is the average number of runs scored per over.

• At school, average marks show how you did in general.

• The most common score on a test can also tell us something important about the performance of the entire class.

Statistics is the subject related to collecting, organising, and displaying data. We can represent data in many ways, such as tables, graphs, bar charts, or pie charts. But before that, we often need to find a single value of the mean, median, or mode to better understand the data.

 

Measures of central tendency

Measurements of central tendency are special numbers that tell us about the centre or middle of a group of numbers.

The three most common measures are:

1. Mean

2. Median

3. Mode

We use them to summarise a lot of data with one value, so it's easy to understand.

1. Mean

The mean is the most common type we use in mathematics. This tells us what value we get if all numbers are shared equally.

How to find the mean:

• Add all the numbers to the dataset.

• Divide the amount by the total number of values.

• If our dataset has an n value:

Mean  = (Sum of all observations) ÷ (Number of observations)

Example:

• Numbers: 5, 6, 7, 10

• Step 1: Add → 5 + 6 + 7 + 10 = 28

• Step 2: Divide → 28 ÷ 4 = 7  

• So, the mean is 7.

2. Median

The middle value is the median when the numbers are arranged in order (either the smallest to largest or the smallest).

How to Find the Median:

• For odd numbers of observations:

Median = Middle term

• • Example: 5, 8, 10 

  • The middle number is 8, so median = 8.

• For an even number of observations:

Median = [ ( Middle term 1 + Middle term 2 ) / 2 ]

• • Example: 4, 7, 9, 12

  • Middle two numbers → 7 and 9

  • Median = [ (7 + 9) / 2 ] = 8

3. Mode

The mode is the number that is displayed most often in datasets.

How to Find the Mode:

Formula:

Mode = Value that appears most often

• Example: 4, 3, 4, 6, 7, 4, 9. The number 4 is often visible → Mode is 4.

• If no number is repeated, there is no mode in the data.

• If two or more numbers appear equally in maximum numbers, the data may have two or more modes.

 

Example of Mean, Median, and Mode

Let's understand the mean, median, and mode through an example.

Example:

The table below shows the race scores by different players in the cricket match. Find the mean, median, and mode of the given data.

 

Mean 

The mean is an average of all values.

Formula:

Mean  = (Sum of all observations) ÷ (Number of observations)

• Step 1: Add all the runs. 
  85 + 52 + 30 + 55 + 60 + 1 + 6 = 289

• Step 2: Divide by the number of players.
  Mean =  289 / 7 = 41

• Mean = 41

Median

The median is the middle value when the data is arranged in order (from smallest to largest).

• Step 1: Arrange for races in ascending order: 1, 6, 40, 52, 52, 70, 80

• Step 2: Since the number of players is 7 (an odd number), the middle value is the 4th value.

• Median = 52

Mode 

The mode is the number that appears most often in data.

• Looking at the runs:
  1, 6, 40, 52, 52, 70, 80 → 52 appears twice, more than any other number.

• Mode = 52

 

Relation

• The mean, median, and mode are three important ways to find the central value of a set of numbers. They help us understand what is "average" or "general" in a group of data.

• For a moderately skewed distribution (where the data is not perfectly symmetrical, but not extremely uneven), there is a special formula that connects the mean, medium, and mode. This formula is:

Mode = 3 × Median − 2 × Mean

• This is called an empirical relationship between mean, median, and mode.

Why is this formula helpful?

• If we know any of these two values, we can easily find the third one.

• It is mostly used when working with a large set of data and wants to save time.

Example: If the dataset mean is 20 and the median is 25, then:

• Mode = 3 × 25 – 2 × 20

• Mode = 75 – 40 = 35

• So the mode is 35.

Range in Statistics 

The range tells us how data is spread. This is the difference between the highest value and the lowest value in a set of numbers.

The formula for the area is:

Range = highest value - best value

Why is the range important?

• This shows how diverse the data is.

• A small range means that the numbers are close to each other.

• A large area means that the number extends far away.

Example:

In a test, the students will get marks: 12, 18, 25, 30, 35. Then:

• Highest Value = 35

• Lowest value = 12

• Range = 35 - 12 = 23

• So the range is 23.

 

Solved problem 

By solving mean, median, and mode questions, you can easily build the conceptual fluency required for exams and competitive tests.

Question:

Find the mean, median, mode, and range for the following data:
50, 40, 65, 90, 75, 60, 55, 45, 70, 95, 55, 30, 55, 60, 45

Number of Observations (n) = 15

1. Mean

Formula: Mean = Sum of all observations / Number of observations

Sum of observations = 50 + 40 + 65 + 90 + 75 + 60 + 55 + 45 + 70 + 95 + 55 + 30 + 55 + 60 + 45

Let's add step-by-step: 

• 50 + 40 = 90

• 90 + 65 = 155

• 155 + 90 = 245

• 245 + 75 = 320

• 320 + 60 = 380

• 380 + 55 = 435

• 435 + 45 = 480

• 480 + 70 = 550

• 550 + 95 = 645

• 645 + 55 = 700

• 700 + 30 = 730

• 730 + 55 = 785

• 785 + 60 = 845

• 845 + 45 = 890

Mean = 890 / 15 = 59.333

2. Median

The median is the middle value when the numbers are arranged in ascending order.

• Ascending order: 30, 40, 45, 45, 50, 55, 55, 55, 60, 60, 65, 70, 75, 90, 95

• Here, n = 15 (odd number)

• Middle position = [ (n + 1) / 2 ] = [ (15 + 1) / 2 ] = 8

• The 8th number in the list is  55 

• So, Median = 55

3. Mode

• The mode is the most repeated value in the data.

• Here, 55 appears 3 times 

• So, Mode = 55

4. Range 

• Formula: Range = Highest value − Lowest value

• Highest value = 95

• Lowest value  = 30 

• Range = 95 - 30 = 65

 

Practice Questions

These practice problems will help you test your understanding of mean, median, and mode. Try to resolve them step by step.

Question 1: Cricket Scores

The runs scored by a cricket team in 12 matches are as follows:
12, 26, 40, 18, 22, 15, 35, 40, 18, 35, 28, 26

Find the mean, median, and mode of the run scores of the team.

Question 2: Missing value in Median

The following marks are organized in ascending order:

15, 20, 25, x, x + 4, 40, 42, 48, 50

If the median of the data is 32, find the value of x.

Question 3: Mode from a frequency table

The table below shows the number of books read by a group of students within a month:

Find the mode of the given data.

Question 4: Range Calculation 

The following data shows the ages of the players in a football team:

17, 20, 18, 21, 19, 20, 22, 18, 21, 20

Find the age range limit.

Question 5: Combined Question

In the test, marks were scored by 15 students:

28, 35, 40, 35, 50, 45, 40, 30, 35, 50, 45, 28, 40, 35, 50, 50

Find the mean, median, mode, and range of marks.

 

FAQs on Mean, Median, and Mode

1. What are the median, mode, and mean?

• Mean (Average):

• Mean  = (Sum of all values) ÷ (Number of values)

• Median (Middle Value):

• Arrange numbers in ascending order and find the middle value.

• If the count is odd → The middle number is the median.

• If the count is even → Median = Average of 2 middle numbers.

• Mode (Most Frequent Value):

• The number that occurs most often in the data.

2. Median of 4, 5, 9, 2, 6, 8, 7

• Step 1: Arrange in ascending order: 2, 4, 5, 6, 7, 8, 9

• Step 2: Count = 7 (odd) → Middle value is 4th number = 6

• Median = 6

3. Mean, Median, and Mode of 13, 16, 12, 14, 19, 12, 14, 13, 14

• Step 1: Mean

  • Sum = 13 + 16 + 12 + 14 + 19 + 12 + 14 + 13 + 14 = 127

  • Count = 9

  • Mean = 127 / 9 ≈ 14.11

• Step 2: Median

  • Arrange: 12, 12, 13, 13, 14, 14, 14, 16, 19

  • Count = 9 (odd) → middle value = 5th number = 14

  • Median = 14

• Step 3: Mode

• 14 occurs 3 times → Mode = 14

4. Median of 8, 5, 7, 9, 11, 6, 10

• Arrange: 5, 6, 7, 8, 9, 10, 11

• Count = 7 → middle number = 8

• Median = 8

5. Median of 12, 8, 5, 6, 1, 10, 13, 11

• Arrange: 1, 5, 6, 8, 10, 11, 12, 13

• Count = 8 (even) → middle two numbers = 8 and 10

• Median = (8 + 10) / 2 = 9