How to Find the Median of Grouped Data?
Written by Neeraj Anand | Published by ANAND TECHNICAL PUBLISHERS
Introduction:
In statistics, the Median is the middle value that divides a dataset into two equal parts when arranged in ascending or descending order. For grouped data, we cannot directly determine the median by simply arranging values. Instead, we use a mathematical formula that considers class intervals and cumulative frequencies.
Finding the median of grouped data involves estimating the middle value when the data is organized into intervals (or classes).
Table of Contents
How to Calculate Median of Grouped Data?
Steps to Find the Median of Grouped Data are discussed as follows:
Step 1: First, we find out the total number of observations by summing up all the frequencies.
Step 2: Then, we need to find the median class, i.e. the class having cumulative frequency just greater than half of 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 class size (h).
Step 4: Next, we can substitute these values in the formula to calculate median of grouped data, i.e.
Median = l + ((N/2-cf)/f)×h
Formula for Finding the Median of Grouped Data
After finding the median class, use the below formula to find the median value.
\(\begin{array}{l}Median = l+ \left ( \frac{\frac{N}{2}-cf}{f} \right )\times h\end{array} \)
Where
l is the lower limit of the median class
N is the summing up all frequencies
f is the frequency of median class
h is the class size
CF is the cumulative frequency of class preceding the median class.
Solved Examples on Median of Grouped Data
Example 1: Calculate the value of median for the following data distribution:
| Class Interval | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|---|---|---|---|---|
| Frequency | 5 | 7 | 12 | 10 | 6 |
Solution:
To find the median of given data, we build a table containing cumulative frequencies for each class interval along with the frequencies.
| Class Interval | Frequency (f) | Cumulative Frequency (cf) |
|---|---|---|
| 0-10 | 5 | 0+5 = 5 |
| 10-20 | 7 | 5+7 = 12 |
| 20-30 | 12 | 12+12 = 24 |
| 30-40 | 10 | 24+10 = 34 |
| 40-50 | 6 | 34+6 = 40 |
Here, the total summation of all frequencies are 40, i.e. N = 40. We have, N/2 = 20, now the class having cumulative frequency just greater than or equal to 20 is the class interval 20-30 (cf = 24).
Thus, the median class is 20-30. Also, here the value of class size (h) is 10 (upper limit – lower limit). The lower limit (l) and frequency (f) of the median class are 20 and 12 respectively. And, the cumulative frequency (cf) of class preceding the median class is 12. Now, we can substitute these values in the formula to calculate value of median,
Median = l + ((N/2-cf)/f)×h
= 20 + ((20-12)/12)×10
= 20 + (8/12)×10
= 20 + 6.67
Median = 26.67
Thus, the value of median corresponding to the given grouped data comes out to be 26.67.
Example 2: Find the median age of employees working at XYZ organisation, based on the following data:
| Ages (in years) | 25-30 | 30-35 | 35-40 | 40-45 | 45-50 |
|---|---|---|---|---|---|
| No. of Employees | 8 | 12 | 10 | 5 | 3 |
Solution: To find median of the given grouped data, first of all we form a frequency distribution table as follows:
| Class Interval | Frequency | Cumulative Frequency |
|---|---|---|
| 25-30 | 8 | 0+8=8 |
| 30-35 | 12 | 8+12=20 |
| 35-40 | 10 | 20+10=30 |
| 40-45 | 5 | 30+5=35 |
| 45-50 | 5 | 35+5=40 |
Here, we have total number of employees, N = 40. So, the median class is the class having cumulative frequency just greater than or equal to 20 (i.e. N/2). Thus, median class is 35-40.
Now, we have,
Lower limit of median class, l = 35.
Class size, h = 5.
Cumulative frequency of the class preceding the median class, cf = 20
Frequency of median class, f = 10
On substituitng these values in the formula, i.e.
Median = l + ((N/2-cf)/f)×h
we get,
Median = 35 + ((20-20)/10)×5
Median = 35
Thus, median age of employees based upon given distribution comes out to be 35 years.
Example 3: Find the median score of a cricket team in past 20 matches based on the following data:
| Scores | 80-100 | 100-120 | 120-140 | 140-160 | 160-180 |
|---|---|---|---|---|---|
| No. of matches | 3 | 7 | 4 | 4 | 2 |
Solution: Let us create a frequency distribution table for the given data,
| Class Interval | Frequency | Cumulative Frequency |
|---|---|---|
| 80-100 | 3 | 0+3=3 |
| 100-120 | 6 | 3+6=9 |
| 120-140 | 4 | 9+4=13 |
| 140-160 | 4 | 13+4=17 |
| 160-180 | 3 | 17+3=20 |
Here, total number of matches (N) are 20.
Now, the class having cumulative frequency just greater than or equal to N/2, i.e. 10, is the class 120-140. Thus, it is the median class for the given distribution.
Lower limit of the median class, l = 120,
Frequency of the median class, f = 4,
Cumulative frequency of the class preceding median class, cf = 9,
Class size (upper limit – lower limit), h = 20,
On substituting these values in formula to find median of grouped data, i.e.
Median = l + ((n/2-cf)/f)×h
we get,
Median = 120 + ((10-9)/4)×20
Median = 120 + 5 = 125
Thus, the median score of team comes out to be 125.
Example.4 : The following data represents the survey regarding the heights (in cm) of 51 girls of Class x. Find the median height.
| Height (in cm) | Number of Girls |
| Less than 140 | 4 |
| Less than 145 | 11 |
| Less than 150 | 29 |
| Less than 155 | 40 |
| Less than 160 | 46 |
| Less than 165 | 51 |
Solution:
To find the median height, first, we need to find the class intervals and their corresponding frequencies.
The given distribution is in the form of being less than type,145, 150 …and 165 gives the upper limit. Thus, the class should be below 140, 140-145, 145-150, 150-155, 155-160 and 160-165.
From the given distribution, it is observed that,
4 girls are below 140. Therefore, the frequency of class intervals below 140 is 4.
11 girls are there with heights less than 145, and 4 girls with height less than 140
Hence, the frequency distribution for the class interval 140-145 = 11-4 = 7
Likewise, the frequency of 145 -150= 29 – 11 = 18
Frequency of 150-155 = 40-29 = 11
Frequency of 155 – 160 = 46-40 = 6
Frequency of 160-165 = 51-46 = 5
Therefore, the frequency distribution table along with the cumulative frequencies are given below:
| Class Intervals | Frequency | Cumulative Frequency |
| Below 140 | 4 | 4 |
| 140 – 145 | 7 | 11 |
| 145 – 150 | 18 | 29 |
| 150 – 155 | 11 | 40 |
| 155 – 160 | 6 | 46 |
| 160 – 165 | 5 | 51 |
Here, N= 51.
Therefore, N/2 = 51/2 = 25.5
Thus, the observations lie between the class interval 145-150, which is called the median class.
Therefore,
Lower class limit l = 145
Class size, h = 5
Frequency of the median class, f = 18
Cumulative frequency of the class preceding the median class, cf = 11.
We know that the formula to find the median of the grouped data is:
\(\begin{array}{l}Median = l+ \left ( \frac{\frac{N}{2}-cf}{f} \right )\times h\end{array} \)
Now, substituting the values in the formula, we get
\(\begin{array}{l}Median = 145+ \left ( \frac{25.5-11}{18} \right )\times 5\end{array} \)
Median = 145 + (72.5/18)
Median = 145 + 4.03
Median = 149.03.
Therefore, the median height for the given data is 149. 03 cm.
Practice Problems on Median of Grouped Data
Q1. Find the value of median for following grouped data distribution:
| Class Interval | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |
|---|---|---|---|---|---|
| Frequency | 5 | 20 | 12 | 18 | 15 |
Q2. Find the median salary of employees working at an ABC organisation:
| Salary (in thousands) | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |
|---|---|---|---|---|---|
| No. of Employees | 15 | 20 | 10 | 10 | 5 |
Q3. Find the median height of students in a class based upon following data:
| Heights (in cms) | 152-156 | 156-160 | 160-164 | 164-168 | 168-172 |
|---|---|---|---|---|---|
| No. of students | 8 | 7 | 12 | 2 | 1 |
Frequently Asked Questions on Median of Grouped Data
Q1
What is meant by the median in statistics?
In statistics, the median is the middle value of the given dataset.
Q2
How to find the median value if the number of observations is odd?
If the number of observations (n) is odd, the median is the (n+1)/2th observation.
Q3
How to find the median value if the number of observations is even?
If the number of observations (n) is even, the median is the average of n/2th and (n/2)+1th observation.
Q4
What is the formula to find the median of grouped data?
The formula to find the median of grouped data is:
Median = l+ [((N/2) – cf)/f] × h
Where l = lower limit of median class, n = number of observations, h = class size, f = frequency of median class, cf = cumulative frequency of class preceding the median class.
Q5
What is the median class?
The median class is the class interval whose cumulative frequency is greater than (and nearest to) N/2.
Q6
What is difference between Grouped Data and Ungrouped Data?
Ungrouped data is the data presented in form of discrete individual points. Each data point corresponds to a single observation in this case.
In grouped data, we represent the data in form of ranges or intervals, and the observations having values corresponding to that range are counted against them named as frequency of that interval.
Q7
What is Median Class in Grouped Data?
Median class is the class having cumulative frequency just greater than or equal to half of the total number of observations.
Q8
Why is Median is also called as Positional Average in Statistics?
Median is the middle value of the given data distribution when data points are arranged in ascending order. As it depends upon the position of data values when arranged in a specific order, so it is also called as positional average.
Q9
What is the formula to Calculate Median of Grouped Data in statistics?
We use the following formula to calculate median of grouped data in statistics,
Median = l + ((N/2-cf)/f)×h
Where,
- l is the lower limit of the median class,
- N is the total number of observations,
- cf is the cumulative frequency of the class preceding median class,
- f is the frequency of the median class, and
- h is the class size (upper limit – lower limit).
Q10
What are the steps involved in finding Median of Grouped Data?
Below are the steps involved in finding median of a grouped data:
- Arrange data in groups or classes.
- Find the midpoint of the data in each group.
- Calculate the cumulative frequency.
- Determine the group containing the median.
- Apply the formula
- Median = l + ((N/2-cf)/f)×h
- Calculate the median using the formula.
Why is the Median Important?
✔️ Less affected by extreme values (outliers) than the mean.
✔️ Provides a better central value for skewed distributions.
✔️ Used in real-world applications like income distribution analysis and percentile calculations.
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