How to Find Median of Grouped Data Step by Step with Solved Questions
The concept of median of grouped data is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding this concept enables students to find the central tendency of a large dataset organized in class intervals, a crucial skill for board exams and statistics applications.
Understanding Median of Grouped Data
A median of grouped data refers to the value that divides a grouped frequency distribution into two equal halves. In grouped data, observations are presented in class intervals, and you cannot simply pick the middle value. Instead, you must identify the "median class" and use the median formula. This concept is widely used in statistics, measures of central tendency, and is a key chapter in the CBSE Class 10 Maths curriculum.
Formula Used in Median of Grouped Data
The standard formula for finding the median of grouped data is:
Median = l + (n/2 - cf) / f × h
Where:
l = lower class boundary of median class
n = total number of observations
cf = cumulative frequency of class before the median class
f = frequency of the median class
h = class interval width
Step-by-Step Method to Find Median of Grouped Data
Follow these steps to calculate the median of grouped data:
1. Arrange the data in a frequency distribution table, including class intervals and their frequencies.2. Calculate the cumulative frequency (cf) for each class interval.
3. Find the total number of observations (n).
4. Compute n/2 to determine the halfway point.
5. Identify the median class – the class interval whose cumulative frequency is just greater than (or equal to) n/2.
6. Note the lower class boundary (l), frequency of the median class (f), cumulative frequency before median class (cf), and class width (h).
7. Substitute these values into the median formula:
Median = l + [(n/2 - cf)/f] × h
8. Calculate the result. The answer you get is the median for the grouped data.
Worked Example – Solving a Problem
Let’s solve a typical example from board exam practice:
Example: The following table shows the heights (in cm) of 51 girls of Class X. Find the median height.
| Class Interval (Height in cm) | Frequency (f) | Cumulative Frequency (cf) |
|---|---|---|
| Below 140 | 4 | 4 |
| 140–145 | 7 | 11 |
| 145–150 | 18 | 29 |
| 150–155 | 11 | 40 |
| 155–160 | 6 | 46 |
| 160–165 | 5 | 51 |
Stepwise Solution:
1. Total number of observations, n = 512. Calculate n/2: 51 / 2 = 25.5
3. Find the median class: The cumulative frequency just greater than 25.5 is 29, which falls in the class interval 145–150.
4. Values for formula:
cf = 11 (cumulative frequency of previous class)
f = 18 (frequency of median class)
h = 5 (class width)
5. Substitute in the formula:
= 145 + (14.5 / 18) × 5
= 145 + 0.8056 × 5
= 145 + 4.03
= 149.03 cm
Therefore, the median height is 149.03 cm.
Practice Problems
- Find the median of grouped data for the following frequency table:
Class Intervals: 0–10, 10–20, 20–30, 30–40
Frequencies: 5, 8, 12, 5 - If n = 40, and the median class is 15–20, with f = 8, cf = 17, and h = 5, find the median.
- What is the effect if class intervals are not of equal width while using the median of grouped data formula?
Common Mistakes to Avoid
- Failing to calculate the cumulative frequency correctly.
- Picking the frequency instead of cumulative frequency for identifying the median class.
- Using upper instead of lower class boundary for l in the formula.
- Not maintaining consistent class width (h) in all steps.
Real-World Applications
The concept of the median of grouped data appears in fields such as demography, economics, quality control, and when analyzing survey results. It helps in understanding the typical or central value when working with large data sets, making it highly relevant for data science and board exams. Vedantu helps students master these practical applications so they are confident in both theory and real-world scenarios.
Comparison: Median of Grouped vs Ungrouped Data
| Aspect | Grouped Data | Ungrouped Data |
|---|---|---|
| Data Format | Class intervals | Individual values |
| Formula | Median = l + [(n/2 - cf)/f] × h | (n+1)/2th value (odd n), average of n/2 and (n/2+1) values (even n) |
| Calculation step | Need cumulative frequency | Sort and count positions |
| Common Usage | Large datasets in statistics | Small datasets or raw data |
For more on the ungrouped median, see: Median of Ungrouped Data
We explored the idea of median of grouped data, its formula, step-by-step calculation, common mistakes, and real-world applications. Practice more examples and worksheets with Vedantu for quick revision and better exam performance.
Related reading: Mean, Median, Mode | Variance | Central Tendency | Mean Absolute Deviation | Mean | Mode of Grouped Data | Frequency Distribution Grouped | CBSE Class 10 Maths Important Topics | Probability and Statistics | Measures of Central Tendency for Class 10
FAQs on Median of Grouped Data: Formula, Steps & Examples
1. What is median of grouped data?
The median of grouped data is a statistical measure that represents the central or middle value in a frequency distribution where data are grouped into class intervals. It divides the entire dataset into two equal halves by calculating the value that lies within the median class.
2. How to find median in a grouped frequency table?
To find the median in a grouped frequency table, follow these steps:
1. Calculate the total number of observations (n) by summing all frequencies.
2. Find n/2 to locate the position of the median.
3. Identify the median class where the cumulative frequency just exceeds n/2.
4. Use the median formula:
Median = l + [(n/2 - cf) / f] × h
where l = lower boundary of the median class, cf = cumulative frequency before median class, f = frequency of the median class, and h = class width.
3. What is the formula for median of grouped data?
The standard formula for finding the median of grouped data is:
Median = l + [(n/2 - cf) / f] × h
where:
- l = lower class boundary of the median class
- n = total frequency (number of observations)
- cf = cumulative frequency before the median class
- f = frequency of the median class
- h = class width (size of the class interval)
4. Can you give a solved example of finding median of grouped data?
Yes. Consider a frequency distribution of girls' heights:
Class Intervals: 140-145, 145-150, 150-155
Frequencies: 7, 18, 11
Steps:
1. Calculate total frequency, n = 7 + 18 + 11 = 36.
2. Find n/2 = 18.
3. Determine cumulative frequency:
- Below 140-145: 7
- Below 145-150: 7 + 18 = 25
4. Median class is 145-150 as its cumulative frequency crosses 18.
5. Using formula:
l = 145, cf = 7, f = 18, h = 5
Median = 145 + [(18 - 7) / 18] × 5 = 145 + (11 / 18) × 5 = 145 + 3.06 = 148.06 cm
This example shows the stepwise calculation of the median of grouped data.
5. What’s the difference between median of grouped and ungrouped data?
The key differences are:
• In ungrouped data, the median is the middle value after sorting the dataset.
• In grouped data, raw data aren’t available; hence, the median is estimated using the median class within a frequency distribution.
• Grouped median uses class intervals and cumulative frequencies, whereas ungrouped median directly uses data points.
These distinctions are crucial for applying the correct method per dataset type.
6. Is there an online calculator for median of grouped data?
Yes, there are many online median of grouped data calculators available that allow students to input class intervals and frequencies to automatically compute the median. These tools help reduce calculation errors and are especially useful for quick revision and practice.
7. Why do students often misidentify the median class in frequency tables?
Students frequently misidentify the median class because they:
• Fail to correctly calculate the cumulative frequencies.
• Don’t properly find the value of n/2 or misunderstand its role.
• Incorrectly pick the class whose cumulative frequency just exceeds n/2.
Careful cumulative frequency calculation and clear understanding of median class definition prevent this common error.
8. What happens if class intervals are unequal while calculating median?
If class intervals are unequal, the standard median formula assumes equal width and may give inaccurate results. In such cases:
• Adjustments or alternative methods are required.
• Use class boundaries carefully and consider proportional allocation.
• Some advanced statistical software can handle unequal intervals precisely.
Therefore, verify if class intervals are equal before applying the formula.
9. Why is the median less affected by outliers than the mean?
The median is less influenced by outliers because it depends on the position of values rather than their magnitude. As the median is the middle value dividing data into halves, extreme values at either end do not skew it, unlike the mean, which averages all data points and shifts towards outliers.
10. How do errors in cumulative frequency impact the median calculation?
Errors in calculating cumulative frequency can lead to identifying the wrong median class, resulting in an incorrect median value. Since the median class is based on where cumulative frequency first exceeds n/2, miscalculated cumulative frequencies distort this critical step. Accurate cumulative frequency calculation is essential for a precise median.
11. Can the median of grouped data ever be one of the raw data points?
Typically, the median of grouped data is an estimated value and may not match any exact raw data point because it falls within a class interval. Unlike ungrouped data where median is an actual observation or average of two observations, grouped median is derived by interpolation within the median class.
