1146 LearnersLast updated on June 18th, 2025
Median of Grouped Data
The median of grouped data is the median of the data that is continuous and in the form of frequency distribution. Median is the middle value of any given data or distribution that separates the distribution into two halves, the lower half and the higher half. While calculating the median for grouped data, we calculate the cumulative frequency, the median class and then apply the median of the grouped data formula. Let us now see more about the median of grouped data and how it is calculated.
What is the median of grouped data
The meaning of median is the middle value of a given distribution, and the median of grouped data remains the same as the meaning of median. In grouped data, we have data that is in the form of intervals or classes. We also have a median class to find the value of the median.
We also need to define the cumulative frequencies for each class, and then apply the formula given below to calculate the median of grouped data:
Median = L + (n/2 - cf / f) x w
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How to Calculate the Median of Grouped Data?
To calculate the median of grouped data, we follow the following steps:
Step 1: We have to find the total number of observations by summing up the frequencies that are given.
Step 2: After that, we need to find the median class, which is the class having cumulative frequency just greater than half of the total number of observations.
Step 3: Now we note the values of lower limit of median class (l), frequency of the median class (f), cumulative frequency of the class preceding median class (cf), and the class width (w).
Step 4: We then substitute the values in the given formula and calculate the median of grouped data.
Median = l + ((n/2 - cf)/f) x w
How to Find Median of Grouped Data
Comparison between mean, median, and mode of grouped data
To compare between mean, median, and mode of grouped data let us see the following table:
| Mean | Median | Mode |
| The average of all values in the dataset. | The middle value that divides the dataset into two equal halves. | The most frequently occurring value in the dataset. |
| Uses all values in the dataset. | Uses cumulative frequency to determine the middle class. | Uses frequency of the classes to determine the most repeated class. |
| Affected by extreme values (skewed data can distort the mean). | Less impacted by extreme values. | Not impacted by extreme values. |
| Finding the central tendency when all data points are important. | Finding the central value, especially when there are extreme values. | Finding the most frequent observation. |
| Used in statistics, economics, and finance. | Used in income distribution, exam scores, and hospital stays. | Used in fashion trends, marketing, and sales analysis. |
| The formula is: Mean = Σ(fixi)/Σfi |
The formula is: Median = L + ((n/2 - cf)/f) x h |
The formula is: Mode = L + (f1 - f0 / 2f1 - f0 - f2) x h |
Real life applications on Median of Grouped Data
We use median of grouped data for many things in our day-to-day life. Let us now see the various fields and applications we use in median of grouped data:
Income and Wealth Distribution Analysis:
We use the median of grouped data in income and wealth distribution, where governments and economists use the median income to understand income inequality in a country.
Exam Results:
We use the median of grouped data in exam results, where schools and universities analyze students’ marks using the median to determine a typical student’s performance.
Healthcare and Medical Studies:
We use the median of grouped data in medical studies and healthcare, where hospitals use the median length of hospital stays for specific treatments to measure healthcare efficiency.
Common mistakes and How to Avoid Them in Median of Grouped Data
Students tend to make mistakes when they solve problems related to the median of grouped data. Let us now see the common mistakes they make and the solutions to avoid them:
Mistake 1
Incorrectly Identifying the Median Class:
Students should always calculate the cumulative frequency first. After that, they must find n/2. Then they must locate the class interval where the cumulative frequency just exceeds. That is the median class.
Mistake 2
Using Wrong Formula:
Students should remember the formula to calculate the median:
= Lm + [n/2 - cf / f] x w
Mistake 3
Incorrect Lower Boundary Selection:
Students should remember that when the lower boundary of the class interval is not uniform. We calculate it by taking the lower limit of the median class and subtracting 0.5.
Mistake 4
Rounding Errors:
Students should use at least 2 decimal places before final rounding. Round only at the final step.
Mistake 5
Ignoring Discontinuous Class Intervals:
Students should convert to continuous intervals before calculating.
For example, if given 10-19, 20-29, 30-39, modify them as 9.5-19.5, 19.5-29.5, etc. by adjusting 0.5 on both ends.
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Solved examples on Median of Grouped Data
Problem 1
Find the median of the following frequency distribution:
| Class Interval | Frequency |
| 10 – 20 | 5 |
| 20 – 30 | 8 |
| 30 – 40 | 12 |
| 40 – 50 | 5 |
The median is 31.67
Explanation
Total Frequency (n) = 5 + 8 + 12 + 5 = 30
n/2 = 15
Cumulative Frequency:
10 – 20: 5
20 – 30: 5 + 8 =13
30 – 40: 13 + 12 = 25
40 – 50: 25 + 5 = 30
Identify the median class:
The first class where cumulative frequency is less than or equal to 15 is 30 – 40
Apply the formula:
Median = L + [n/2 - cf / f] x h = 30 + [(15-13)/12] x 10 = 30 +1.67 = 31.67
Problem 2
Given the age distribution below, find the median age:
| Class Interval | Frequency |
| 0 – 10 | 5 |
| 10 – 20 | 8 |
| 20 – 30 | 12 |
| 30 – 40 | 5 |
The median age is 17.5
Explanation
Total Frequency:
n = 7 + 12 + 9 + 4 = 32
n/2 = 16
Cumulative frequency:
0-10: 7
10-20: 7 + 2 = 9
20-30: 19 + 9 = 28
30-40: 28 + 4 = 32
Median class is 10-20
Apply the formula:
L + [n2 - cf / f] x h = 10 + (9/12 x 10)
= 10 + 7.5 = 17.5.
Problem 3
Find the median for the grouped data:
| Class Interval | Frequency |
| 5– 15 | 10 |
| 15 – 25 | 15 |
| 25 – 35 | 5 |
The median is 18.33
Explanation
Total Frequency:
n = 10 + 15 + 5= 30
n/2 = 15
Cumulative frequency:
5-15: 10
15-25: 10 + 15 = 25
25-35: 25 + 5 = 30
Median class is 15-25
Apply the formula:
L + [n2 - cf / f] x h = 15 + (5/15 x 10)
= 15 + 3.33 = 18.33.
Problem 4
Determine the median of the following data:
| Class Interval | Frequency |
| 0 – 5 | 3 |
| 5 – 10 | 7 |
| 20 – 15 | 10 |
| 15 – 20 | 5 |
| 20 – 25 | 2 |
The median class is 11.75.
Explanation
Total Frequency:
n = 3 + 7 + 10 + 5 + 2 = 27
n/2 = 13.5
Cumulative frequency:
0-5: 3
5-10: 3 + 7 = 10
10-15: 10 + 10 = 20
15-20: 20 + 5 = 25
20-25: 25 + 2 = 27
Median class is 10-15
Apply the formula:
L + [n2 - cf / f] x h = 10 + (3.5/10 x 5)
= 10 + 1.75 = 11.75.
Problem 5
An exam score distribution is given below. Find the median score:
| Class Interval | Frequency |
| 40 – 50 | 2 |
| 50 – 60 | 5 |
| 60 – 70 | 12 |
| 70 – 80 | 20 |
| 80 – 90 | 8 |
The median class is 72.25
Explanation
Total Frequency:
n = 2 + 5 + 12 + 20 + 8 = 47
n/2 = 23.5
Cumulative frequency:
40-50: 2
50-60: 2 + 5 = 7
60-70: 7 + 12 = 19
70-80: 19 + 20 = 39
80-90: 39 + 8 = 47
Median class is 70-80
Apply the formula:
L + [n2 - cf / f] x h = 70 + (4.5/20 x 10)
= 70 + 2.25 = 72.25.
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FAQs on Median of Grouped Data
1.What is the median of grouped data?
2.Why do we need to estimate the median for a grouped data?
3.How do you determine the median class?
4.Are there any assumptions made when using the median formula for a grouped data?
5.How accurate is the median obtained from grouped data?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
