Mean, Median and Mode
Imagine your class just finished a maths test, and the teacher announces the results. Some students scored very high, some scored very low, and most scored somewhere in between. How do you describe the "typical" score of the class with just one number? This is where mean, median, and mode come in. They are called measures of central tendency because they try to find the "centre" or the most representative value of a set of data.
In Class 7 NCERT Maths, under the chapter Data Handling, you will learn these three measures. The mean (or average) is what most people think of when they hear "average". The median is the middle value when the data is arranged in order. The mode is the value that appears most frequently. Each measure is useful in different situations, and knowing when to use which one is an important skill.
Think about cricket! If a batsman scores 45, 78, 12, 56, and 89 in five innings, what is his "average" score? That is the mean. If your school announces the "most popular" lunch choice, that is the mode. If you want to know the "middle" value of house prices in a colony, that is the median. These concepts are used everywhere: in sports, weather reports, business, science, and everyday decision-making.
In this chapter, we will learn the definition, formula, and calculation method for each measure. We will work through many examples using relatable scenarios like pocket money, test scores, and cricket runs. By the end, you will know how to find the mean, median, and mode of any data set, and understand when each measure is most appropriate.
What is Mean, Median and Mode?
Mean (Arithmetic Mean or Average):
The mean is the sum of all the values divided by the number of values. It gives a balanced "centre" of the data by considering every value.
Mean = Sum of all observations / Number of observations
For example, if 5 friends have pocket money of Rs. 50, Rs. 60, Rs. 40, Rs. 70, and Rs. 80, the mean = (50 + 60 + 40 + 70 + 80) / 5 = 300 / 5 = Rs. 60.
Median:
The median is the middle value when all the data is arranged in ascending (or descending) order. If there is an odd number of values, the median is the single middle value. If there is an even number of values, the median is the average (mean) of the two middle values.
For example, for the data 12, 15, 18, 22, 30 (5 values, odd), the median is the 3rd value = 18. For the data 10, 15, 20, 25 (4 values, even), the median is the average of the 2nd and 3rd values = (15 + 20) / 2 = 17.5.
Mode:
The mode is the value that occurs most frequently in a data set. A data set can have one mode (unimodal), two modes (bimodal), or more. If no value repeats, the data has no mode.
For example, in the data 5, 7, 3, 7, 9, 2, 7, the mode is 7 because it appears 3 times, more than any other value.
Mean, Median and Mode Formula
Mean (Average):
Mean = (x1 + x2 + x3 + ... + xn) / n
where x1, x2, ..., xn are the observations and n is the total number of observations.
Median:
If n is odd: Median = ((n + 1) / 2)th observation
If n is even: Median = Average of (n/2)th and (n/2 + 1)th observations
(Data must be arranged in ascending or descending order first.)
Mode:
Mode = The observation with the highest frequency
Range (Bonus):
Range = Maximum value - Minimum value
The range tells you how spread out the data is. For example, if test scores range from 35 to 95, the range is 95 - 35 = 60. A small range means the data values are close together. A large range means the values are widely spread.
Quick Summary Table:
| Measure | What It Tells You | How to Calculate |
|---|---|---|
| Mean | The "average" or balanced centre | Sum of all values / Count of values |
| Median | The middle value | Arrange in order, find the middle |
| Mode | The most frequent value | Count which value appears most |
| Range | How spread out the data is | Highest value - Lowest value |
Types and Properties
Let us understand when to use each measure and how they behave differently:
Mean (When to Use):
- Use the mean when the data does not have extreme values (outliers).
- The mean is affected by very large or very small values. For example, if 4 friends get Rs. 50, Rs. 60, Rs. 55, Rs. 45 as pocket money, the mean is Rs. 52.50. But if one friend gets Rs. 500, the mean jumps to Rs. 142, which does not represent the typical pocket money well.
- Best for: test scores, temperatures, heights, weights, and other data without extreme outliers.
Median (When to Use):
- Use the median when the data has extreme values or outliers, because the median is not affected by them.
- For the pocket money example above (50, 60, 55, 45, 500), the median is 55, which is much more representative than the mean of 142.
- Best for: income data, house prices, and any skewed data.
Mode (When to Use):
- Use the mode for categorical data or to find the most popular or common item.
- Best for: favourite colours, shoe sizes, most ordered dish, most watched TV show.
- A data set can have no mode (all values appear once), one mode, or multiple modes.
Comparison Table:
| Feature | Mean | Median | Mode |
|---|---|---|---|
| Uses all values? | Yes | No (only middle values) | No (only most frequent) |
| Affected by outliers? | Yes (heavily) | No | No |
| Always unique? | Yes | Yes | No (can have 0, 1, or more) |
| Works for categorical data? | No | No | Yes |
| Must arrange data first? | No | Yes (ascending order) | No (but counting helps) |
Solved Examples
Example 1: Finding the Mean of Test Scores
Problem: Riya scored 72, 85, 68, 91, and 74 in five maths tests. Find her mean score.
Solution:
Step 1: Add all scores: 72 + 85 + 68 + 91 + 74 = 390
Step 2: Count the number of observations: n = 5
Step 3: Mean = 390 / 5 = 78
Answer: Riya's mean score is 78.
Example 2: Finding the Median (Odd Number of Values)
Problem: Find the median of: 23, 15, 37, 42, 18, 30, 25
Solution:
Step 1: Arrange in ascending order: 15, 18, 23, 25, 30, 37, 42
Step 2: Count values: n = 7 (odd)
Step 3: Median position = (7 + 1) / 2 = 4th observation
Step 4: The 4th value is 25.
Answer: The median is 25.
Example 3: Finding the Median (Even Number of Values)
Problem: Find the median of: 10, 25, 15, 30, 20, 35
Solution:
Step 1: Arrange in ascending order: 10, 15, 20, 25, 30, 35
Step 2: Count values: n = 6 (even)
Step 3: Middle positions = 3rd and 4th observations = 20 and 25
Step 4: Median = (20 + 25) / 2 = 45 / 2 = 22.5
Answer: The median is 22.5.
Example 4: Finding the Mode
Problem: Find the mode of: 4, 7, 3, 7, 2, 8, 7, 5, 3
Solution:
Step 1: Count the frequency of each value:
2 appears 1 time, 3 appears 2 times, 4 appears 1 time, 5 appears 1 time, 7 appears 3 times, 8 appears 1 time.
Step 2: The value with the highest frequency is 7 (appears 3 times).
Answer: The mode is 7.
Example 5: Data with No Mode
Problem: Find the mode of: 12, 15, 18, 21, 24
Solution:
Step 1: Count frequencies: each value appears exactly once.
Step 2: No value repeats more than others.
Answer: There is no mode for this data set.
Example 6: Data with Two Modes (Bimodal)
Problem: Find the mode of: 5, 8, 3, 5, 8, 2, 1
Solution:
Step 1: Count frequencies: 1 appears 1 time, 2 appears 1 time, 3 appears 1 time, 5 appears 2 times, 8 appears 2 times.
Step 2: Both 5 and 8 have the highest frequency (2 each).
Answer: The data is bimodal with modes 5 and 8.
Example 7: Finding All Three: Mean, Median, and Mode
Problem: The runs scored by a batsman in 7 innings are: 45, 32, 45, 67, 45, 28, 50. Find the mean, median, and mode.
Solution:
Mean: Sum = 45 + 32 + 45 + 67 + 45 + 28 + 50 = 312. Mean = 312 / 7 = 44.57 (approximately).
Median: Arrange in order: 28, 32, 45, 45, 45, 50, 67. n = 7 (odd). Median = 4th value = 45.
Mode: 45 appears 3 times, which is the highest frequency. Mode = 45.
Answer: Mean ≈ 44.57, Median = 45, Mode = 45.
Example 8: Word Problem: Pocket Money
Problem: Five friends receive weekly pocket money of Rs. 100, Rs. 120, Rs. 80, Rs. 150, and Rs. 100. Find the mean, median, and mode.
Solution:
Mean: (100 + 120 + 80 + 150 + 100) / 5 = 550 / 5 = Rs. 110.
Median: Arrange: 80, 100, 100, 120, 150. n = 5 (odd). Median = 3rd value = Rs. 100.
Mode: Rs. 100 appears twice, all others appear once. Mode = Rs. 100.
Answer: Mean = Rs. 110, Median = Rs. 100, Mode = Rs. 100.
Example 9: Word Problem: Effect of an Outlier on Mean
Problem: The ages of 5 students in a group are 12, 13, 12, 11, and 13. Their teacher (age 35) joins. Find the mean age before and after the teacher joins. What do you notice?
Solution:
Before: Mean = (12 + 13 + 12 + 11 + 13) / 5 = 61 / 5 = 12.2 years.
After: Mean = (12 + 13 + 12 + 11 + 13 + 35) / 6 = 96 / 6 = 16 years.
Observation: The mean jumped from 12.2 to 16 because of the teacher's age (an outlier). The median before was 12 and after is (12 + 13)/2 = 12.5, which barely changed. This shows that the mean is heavily affected by outliers, while the median is resistant to outliers.
Example 10: Word Problem: Finding a Missing Value Using Mean
Problem: The mean of 5 numbers is 24. Four of the numbers are 20, 25, 22, and 28. Find the fifth number.
Solution:
Step 1: Mean = Sum / n, so Sum = Mean x n = 24 x 5 = 120
Step 2: Sum of known numbers = 20 + 25 + 22 + 28 = 95
Step 3: Fifth number = 120 - 95 = 25
Answer: The fifth number is 25.
Real-World Applications
Mean, median, and mode are used widely in real life:
School and Education: Teachers use the mean to calculate average marks. The median is used when a few students score extremely high or low. The mode helps identify the most common score or grade.
Sports: In cricket, a batsman's batting average is the mean of runs scored per innings. The mode of a bowler's deliveries might tell you the most common speed. In basketball, the median score across games gives a better picture of typical performance than the mean if there are outlier games.
Weather: Meteorologists report average (mean) temperature, the median rainfall, and the most common (mode) weather condition for a month.
Business: Companies calculate the mean sales per day, the median income of customers (to avoid the effect of a few very rich customers), and the mode of product sizes ordered (to know which size to stock more).
Health: Doctors look at average blood pressure readings (mean), the median weight of children in a class for growth charts, and the most common blood type (mode) in a population.
Daily Life: When deciding where to eat, the most commonly suggested restaurant (mode) is often chosen. When comparing prices, the median price gives a better idea of what to expect than the mean if some items are very expensive.
Key Points to Remember
- The mean (average) is the sum of all observations divided by the number of observations.
- The median is the middle value after arranging data in ascending order. For odd n, it is the middle value. For even n, it is the average of the two middle values.
- The mode is the value that occurs most frequently. A data set can have no mode, one mode, or multiple modes.
- The range = maximum value - minimum value; it measures the spread of data.
- The mean is affected by outliers (extreme values). The median and mode are not affected by outliers.
- Use the mean for data without outliers (test scores, heights).
- Use the median for data with outliers (income, house prices).
- Use the mode for categorical or non-numerical data (favourite colour, shoe size).
- To find a missing value when the mean is known: Missing value = (Mean x n) - Sum of known values.
- All three measures together give a complete picture of the data.
Practice Problems
- Find the mean of: 15, 20, 25, 30, 35, 40.
- Find the median of: 48, 52, 33, 67, 41, 55, 29.
- Find the mode of: 8, 5, 3, 8, 2, 5, 8, 1, 5.
- The mean of 4 numbers is 18. Three of them are 15, 20, and 12. Find the fourth number.
- A student's marks in 6 subjects are: 78, 82, 78, 90, 65, 78. Find the mean, median, and mode.
- Find the median of: 12, 18, 24, 30, 36, 42, 48, 54.
- The heights (in cm) of 7 plants are: 22, 18, 25, 22, 30, 22, 28. Which measure of central tendency would best describe the most common height?
- Explain with an example why the median is better than the mean when data has outliers.
Frequently Asked Questions
Q1. What is the difference between mean, median, and mode?
The mean is the sum of all values divided by the number of values (the 'average'). The median is the middle value when data is arranged in order. The mode is the value that appears most often. For example, for the data 3, 5, 5, 7, 10: mean = 30/5 = 6, median = 5 (the 3rd value), and mode = 5 (appears twice).
Q2. How do you find the median when there is an even number of values?
When there is an even number of values, find the two middle values and take their average. For example, for 4, 7, 12, 15 (n=4), the two middle values are 7 and 12. Median = (7 + 12) / 2 = 9.5.
Q3. Can a data set have more than one mode?
Yes! If two values share the highest frequency, the data is bimodal (two modes). If three or more share the highest frequency, it is multimodal. For example, in 2, 3, 3, 5, 5, 7, both 3 and 5 appear twice, so the data is bimodal with modes 3 and 5.
Q4. What happens if no value repeats in the data?
If every value appears exactly once, there is no mode. For example, the data set 10, 20, 30, 40, 50 has no mode because no value is repeated.
Q5. Why is the mean affected by outliers?
The mean uses every value in its calculation (sum of all values / count). So an extremely large or small value pulls the mean toward it. For example, the data 10, 12, 11, 13, 100 has mean = 146/5 = 29.2, which is much higher than the typical values (10-13) because of the outlier 100. The median (12) is unaffected.
Q6. When should you use the mode instead of the mean?
Use the mode for non-numerical (categorical) data or to find the most common item. For example, if you survey favourite ice cream flavours (chocolate, vanilla, strawberry, chocolate, chocolate), the mode is 'chocolate'. You cannot calculate a mean or median for flavour names.
Q7. How do you find a missing value when the mean is given?
Use the formula: Sum = Mean x n. First calculate the total sum using the given mean and number of values. Then subtract the sum of the known values. The difference is the missing value. For example, if the mean of 4 numbers is 20, the total sum = 80. If three numbers are 15, 22, and 18 (total = 55), the missing number = 80 - 55 = 25.
Q8. What is the range and how is it different from mean, median, and mode?
The range is the difference between the highest and lowest values in a data set. It measures the spread of the data, not the centre. For example, for the data 5, 10, 15, 20, 25, the range = 25 - 5 = 20. While mean, median, and mode describe the centre of the data, the range describes how spread out the data is.
