Median
If you line up all the students in your class from shortest to tallest, the person standing in the exact middle has the median height. The median is the middle value of a data set when the values are arranged in order.
Unlike the mean, the median is not affected by extremely large or small values. This makes it very useful when the data has outliers.
In Class 7 NCERT Maths, you will learn how to find the median for both odd and even numbers of observations.
What is Median - Grade 7 Maths (Data Handling)?
Definition: The median is the middle value of a data set when the observations are arranged in ascending (or descending) order.
- If the number of observations is odd, the median is the middle value.
- If the number of observations is even, the median is the average of the two middle values.
Median Formula
Formula:
Step 1: Arrange the data in ascending order.
For odd number of observations (n is odd):
Median = Value at position (n + 1) / 2
For even number of observations (n is even):
Median = Average of values at positions n/2 and (n/2 + 1)
Where:
- n = total number of observations
Types and Properties
Two cases for finding the median:
- Case 1: Odd number of observations. There is exactly one middle value. For example, in the data 3, 5, 7, 9, 11 (5 values), the median is the 3rd value = 7.
- Case 2: Even number of observations. There are two middle values. Take their average. For example, in the data 2, 4, 6, 8 (4 values), the two middle values are the 2nd and 3rd values: 4 and 6. Median = (4 + 6) / 2 = 5.
Important:
- Always arrange the data in order before finding the median.
- The median divides the data into two equal halves — half the values are below it and half are above it.
Solved Examples
Example 1: Median of an Odd Number of Values
Problem: Find the median of: 12, 7, 15, 9, 20.
Solution:
Step 1: Arrange in ascending order: 7, 9, 12, 15, 20
Step 2: Number of observations (n) = 5 (odd)
Step 3: Position of median = (5 + 1) / 2 = 3rd value
Step 4: The 3rd value is 12.
Answer: The median is 12.
Example 2: Median of an Even Number of Values
Problem: Find the median of: 18, 22, 14, 30, 25, 10.
Solution:
Step 1: Arrange in ascending order: 10, 14, 18, 22, 25, 30
Step 2: n = 6 (even)
Step 3: Two middle values are at positions 3 and 4: 18 and 22
Step 4: Median = (18 + 22) / 2 = 40 / 2 = 20
Answer: The median is 20.
Example 3: Median of Test Scores
Problem: The marks of 7 students are: 45, 78, 56, 89, 34, 67, 72. Find the median.
Solution:
Step 1: Ascending order: 34, 45, 56, 67, 72, 78, 89
Step 2: n = 7 (odd). Median position = (7 + 1) / 2 = 4th value
Step 3: The 4th value is 67.
Answer: The median score is 67.
Example 4: Median with Repeated Values
Problem: Find the median of: 5, 3, 5, 7, 3, 5, 9.
Solution:
Step 1: Ascending order: 3, 3, 5, 5, 5, 7, 9
Step 2: n = 7 (odd). Median position = 4th value
Step 3: The 4th value is 5.
Answer: The median is 5.
Example 5: Median of Even Data Set with Decimals
Problem: Find the median of: 2.5, 3.1, 1.8, 4.6.
Solution:
Step 1: Ascending order: 1.8, 2.5, 3.1, 4.6
Step 2: n = 4 (even). Middle values are 2nd and 3rd: 2.5 and 3.1
Step 3: Median = (2.5 + 3.1) / 2 = 5.6 / 2 = 2.8
Answer: The median is 2.8.
Example 6: Median of Heights
Problem: The heights of 9 students (in cm) are: 140, 135, 150, 145, 138, 155, 142, 148, 136. Find the median height.
Solution:
Step 1: Ascending order: 135, 136, 138, 140, 142, 145, 148, 150, 155
Step 2: n = 9 (odd). Median position = (9 + 1) / 2 = 5th value
Step 3: The 5th value is 142.
Answer: The median height is 142 cm.
Example 7: Median vs Mean Comparison
Problem: The ages of 5 children are: 10, 11, 12, 12, 40. Find the mean and median. Which gives a better idea of the typical age?
Solution:
Mean: (10 + 11 + 12 + 12 + 40) / 5 = 85 / 5 = 17
Median: Data in order: 10, 11, 12, 12, 40. Median = 3rd value = 12.
The mean (17) is pulled up by the outlier 40. The median (12) better represents the typical age.
Answer: Mean = 17, Median = 12. The median is more representative here.
Example 8: Median of Two Values
Problem: Find the median of: 8, 15.
Solution:
Step 1: Already in order: 8, 15
Step 2: n = 2 (even). Median = (8 + 15) / 2 = 23 / 2 = 11.5
Answer: The median is 11.5.
Real-World Applications
Real-life uses of the median:
- Income reports: Governments report median income because a few very rich people can raise the mean, making it misleading. The median gives a better picture of what a typical person earns.
- House prices: Median house price is reported instead of mean to avoid distortion by a few very expensive houses.
- Test performance: When a class has a few very low or very high scores, the median shows the typical performance better than the mean.
- Sports: Median scores or timings help compare athletes without the influence of one unusually good or bad result.
Key Points to Remember
- The median is the middle value of data arranged in order.
- Always arrange the data in ascending or descending order first.
- For odd n: Median = value at position (n + 1) / 2.
- For even n: Median = average of the two middle values.
- The median is not affected by outliers (extreme values).
- The median divides the data into two equal halves.
- The median may or may not be one of the values in the data set.
- Median is one of the three measures of central tendency along with mean and mode.
Practice Problems
- Find the median of: 23, 15, 31, 8, 19.
- Find the median of: 44, 52, 36, 60, 48, 55.
- The ages of 7 children are: 8, 12, 6, 10, 14, 9, 11. Find the median age.
- The weights of 6 bags (in kg) are: 5.2, 3.8, 4.5, 6.1, 3.2, 5.5. Find the median weight.
- The mean of 5 numbers is 20 and the median is 18. Is this possible? Give an example.
- Arrange the data 50, 42, 38, 47, 55, 43 in ascending order and find the median.
Frequently Asked Questions
Q1. What is the median of a data set?
The median is the middle value when all observations are arranged in ascending or descending order. For an odd number of values, it is the exact middle value. For an even number, it is the average of the two middle values.
Q2. Why do we need to arrange data in order before finding the median?
The median is the middle value. You can only identify the middle value when the data is in order. Without ordering, you would pick a random value, not the actual middle.
Q3. Can the median be a decimal even if all values are whole numbers?
Yes. When the number of observations is even, the median is the average of two middle values, which can be a decimal. For example, median of 3 and 8 is (3 + 8) / 2 = 5.5.
Q4. When is the median better than the mean?
The median is better when the data has outliers (very large or very small values). For example, in incomes or house prices, a few extreme values can distort the mean, but the median stays in the middle.
Q5. Is the median always one of the data values?
Not always. For an odd number of observations, the median is one of the data values. For an even number, the median is the average of two values, which may not be in the original data.
Q6. What is the median of a single number?
If there is only one observation, that value is the median. For example, the median of {7} is 7.
