Understanding the Median of Grouped Data
A comprehensive guide to calculating median for grouped frequency distributions
Introduction
In statistics, the median is a measure of central tendency that represents the middle value of a dataset. While finding the median of ungrouped data is straightforward, calculating it for grouped data requires a different approach. In this blog post, we'll explore how to calculate the median for grouped data step by step.
Median of Ungrouped Data
Before we dive into grouped data, let's recall how to find the median for ungrouped data:
When number of observations (n) is odd
The median is the middle value, which is at position:
Example: Find the median of: 78, 85, 92, 64, 80, 72, 90
Step 1: Arrange in ascending order: 64, 72, 78, 80, 85, 90, 92
Step 2: n = 7 (odd)
Step 3: Position = (7+1)/2 = 4
Step 4: The 4th value is 80
Median = 80
When number of observations (n) is even
The median is the average of the two middle values:
Example: Find the median of: 78, 85, 92, 64, 80, 72
Step 1: Arrange in ascending order: 64, 72, 78, 80, 85, 92
Step 2: n = 6 (even)
Step 3: Positions = 3 and 4
Step 4: Values are 78 and 80
Median = (78 + 80)/2 = 79
Median of Grouped Data
When data is grouped, we can't identify the exact median, but we can estimate it using this formula:
Where:
- I = Lower limit of the median class
- n = Total number of observations
- cf = Cumulative frequency of the class before the median class
- f = Frequency of the median class
- h = Class size (interval width)
Example: Test Scores
Let's find the median for the following test score data:
| Marks Range | Number of Students (f) | Cumulative Frequency (cf) |
|---|---|---|
| 0 - 20 | 4 | 4 |
| 20 - 40 | 10 | 14 |
| 40 - 60 | 14 | 28 |
| 60 - 80 | 8 | 36 |
| 80 - 100 | 4 | 40 |
\( I = 40 \), \( \frac{n}{2} = 20 \), \( cf = 14 \), \( f = 14 \), \( h = 20 \)
\[ \text{Median} = 40 + \frac{(20 - 14)}{14} \times 20 \] \[ = 40 + \frac{6}{14} \times 20 \] \[ = 40 + \frac{120}{14} \] \[ = 40 + 8.57 = 48.57 \]
Interpretation: The median test score is approximately 48.57, meaning half the students scored below this value and half scored above.
Practice Problems
Problem 1: Find the median for the following data:
| Height (cm) | Number of Students |
|---|---|
| 150-155 | 5 |
| 155-160 | 8 |
| 160-165 | 15 |
| 165-170 | 10 |
| 170-175 | 2 |
Solution:
Total students (N) = 5+8+15+10+2 = 40
\( \frac{N}{2} = \frac{40}{2} = 20 \)
| Height (cm) | Frequency | Cumulative Frequency |
|---|---|---|
| 150-155 | 5 | 5 |
| 155-160 | 8 | 13 |
| 160-165 | 15 | 28 |
| 165-170 | 10 | 38 |
| 170-175 | 2 | 40 |
Median class: 160-165 (where CF first exceeds 20)
\( I = 160 \), \( cf = 13 \), \( f = 15 \), \( h = 5 \)
\[ \text{Median} = 160 + \frac{(20 - 13)}{15} \times 5 \]
\[ = 160 + \frac{7}{15} \times 5 \]
\[ = 160 + \frac{35}{15} \]
\[ = 160 + 2.33 = 162.33 \text{ cm} \]
Key Points to Remember
- Median is the middle value of a dataset
- For ungrouped data: find the middle value directly
- For grouped data, we estimate the median using a formula
- Always calculate cumulative frequencies first
- The median class is where the cumulative frequency first exceeds N/2
- Median is not affected by extreme values
Conclusion
Calculating the median for grouped data is an essential statistical skill. By following the steps outlined in this post and practicing with various examples, you can master this concept. Remember that the median provides a robust measure of central tendency that is less affected by outliers than the mean.
If you have any questions or would like to see more examples, feel free to leave a comment below!