Median of Grouped Data is the value of the middlemost data point in any dataset when dataset is grouped. For ungrouped data, it is easy to find the median but finding the Median of Grouped Data is slightly complex. When we have any data in statistics, we try to find some basic parameters related to it which provide ease in data interpretation and making further predictions related to data. To measure the central tendency of data, i.e., a single value that can be used to represent an entire distribution, we generally have three parameters: mean, median, and mode.
In this article, we will discuss what is meant by Grouped Data and Median of Grouped Data. We will also learn about the Formula for Median of Grouped Data and steps to calculate the Median of Grouped Data, solved examples, and some frequently asked questions as well.
What is Grouped Data?
Data can be classified mainly into two types on basis of how it is organized, i.e. Grouped Data and Ungrouped Data. Ungrouped data is the raw data which consists of a simple list of values where each value corresponds to a distinct observation or measurement.
For example, a list of marks of students in a classroom (e.g. 95, 94, 96, 92, 98, 99, …). This kind of representation is used when we have a smaller dataset and we need to deal with individual data points.
But, when we have a larger dataset, it is preferable to group similar data values in form of an interval and assign a value to that interval corresponding to the frequency of data points in that range. The interval should have a uniform length defined by its upper limits and lower limits.
The interval so formed is also referred to as class or class interval, and the table containing classes (intervals) and their frequencies is known as the frequency distribution table. For example, a frequency distribution table showing marks of students in a class in various ranges,
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0 – 10
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3
|
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10 – 20
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5
|
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20 – 30
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7
|
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30 – 40
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12
|
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40 – 50
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3
|
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Total Students
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30
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Grouped data is useful when we have large datasets and when our analysis is concerned to study specific patterns and relationships with respect to ranges of values rather than individual data points.
What is Median of Grouped Data?
The genric meaning of median, i.e. the middle value corresponding to a given distribution, remains same in this case too. As we have data in form of intervals (classes) in this case, we have a corresponding median class to find the value of median.
Also, we need to define cumulative frequencies for each class, which is a kind of prefix sum of frequencies of classes taken in order. The median value lies between the lower limit and upper limit of the median class. This value can be used by using a specified formula discussed as follows.
What is Median?
Median is a value corresponding to the middlemost data point in a dataset, when arranged in ascending order. The value of median helps one to know about center of a dataset. On comparing the value of median with that of mean, one can get idea of distribution of values in a dataset.
To find median of ungrouped data, one can simply sort the data points in ascending order. In case of odd number of observations, the middle value would be the median. On the other hand , for even number of observations, one can take mean of the two middle values to find the median. But there is a different method to find median of grouped data discussed later in this article.
Median of Grouped Data Formula
We can use the following formula to calculate median of grouped data:
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).
The steps listed below illustrate the procedure to find median of grouped data.
How to Calculate Median of Grouped Data?
The steps followed to calculate 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
Below are some examples that will help you understand above mentioned steps to find the Median of Grouped Data in a better way.
Mean, Median and Mode of Grouped Data
A comparison between Mean, Median and Mode of Grouped Data has been discussed in the table below:
| Mean is the average value of all the data points in a given dataset. |
Median is the value of middlemost data point when the given dataset is arranged in ascending order. |
Mode is the most frequent data point in a dataset. |
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Formula to find mean of grouped data:
Mean = ∑(fi.xi)/∑fi
Where,
- xi is the mean of upper limit and lower limit of ith class interval.
- fi is the frequency of the ith class interval.
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Formula to find median of grouped data:
Median = l + ((n/2-cf)/f)×h
Where,
- l is the lower limit of median class.
- h is class size.
- f is frequency of median class,
- cf is the cumulative frequency of the class preceding the median class.
- N = ∑fi
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Formula to find mode of grouped data:
Mode = xk + h{(fk – fk-1)/(2fk – fk-1 – fk+1)}
Where,
- xk is lower limit of the modal class.
- fk is frequency of the modal class.
- fk-1 is the frequency of the class preceding the modal class.
- fk+1 is the frequency of the class succeeding the modal class.
- h is class size.
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Also,Check
Solved Examples on Median of Grouped Data
Example 1: Calculate the value of median for the following data distribution:
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0-10
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10-20
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20-30
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30-40
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40-50
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5
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7
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12
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10
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6
|
Solution:
To find the median of given data, we build a table containing cumulative frequencies for each class interval along with the frequencies.
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0-10
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5
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0+5 = 5
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10-20
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7
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5+7 = 12
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20-30
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12
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12+12 = 24
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30-40
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10
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24+10 = 34
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40-50
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6
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34+6 = 40
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Here, the total number of observations 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:
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25-30
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30-35
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35-40
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40-45
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45-50
|
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8
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12
|
10
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5
|
3
|
Solution: To find median of the given grouped data, first of all we form a frequency distribution table as follows:
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25-30
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8
|
0+8=8
|
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30-35
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12
|
8+12=20
|
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35-40
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10
|
20+10=30
|
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40-45
|
5
|
30+5=35
|
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45-50
|
5
|
35+5=40
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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:
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80-100
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100-120
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120-140
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140-160
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160-180
|
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3
|
7
|
4
|
4
|
2
|
Solution: Let us create a frequency distribution table for the given data,
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80-100
|
3
|
0+3=3
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100-120
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6
|
3+6=9
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120-140
|
4
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9+4=13
|
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140-160
|
4
|
13+4=17
|
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160-180
|
3
|
17+3=20
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Here, total number of observations (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.
Practice Problems on Median of Grouped Data
Q1. Find the value of median for following grouped data distribution:
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0-20
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20-40
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40-60
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60-80
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80-100
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5
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20
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12
|
18
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15
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Q2. Find the median salary of employees working at an ABC organisation:
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10-20
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20-30
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30-40
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40-50
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50-60
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15
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20
|
10
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10
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5
|
Q3. Find the median height of students in a class based upon following data:
Median of Grouped Data – FAQs
1. What do you mean by Median in Statistics?
Median refers to the middle value of the given dataset when arranged in ascending order.
2. 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.
3. 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.
4. 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.
5. 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).
6. 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.
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