Median of Grouped Data
The median is a measure of central tendency which gives the value of the middle-most observation in the data.
Recall that for finding the median of ungrouped data, we first arrange the data values of the observations in ascending order.
Then, if n is odd, the median is the ` ( (n+1)/2)` th observation. And, if `n` is even, then the median will be the average of the `n/2` th and the ` (n/2 +1 )` th observations.
Suppose, we have to find the median of the following data, which gives the marks, out of 50, obtained by 100 students in a test :
First, we arrange the marks in ascending order and prepare a frequency table as
follows :
Here n = 100, which is even. The median will be the average of the `n/2` th and the
`(n/2 +1)` th observations, i.e., the 50th and 51st observations. To find these
observations, we proceed as follows:
Now we add another column depicting this information to the frequency table
above and name it as cumulative frequency column.
From the table above, we see that:
50th observaton is 28
51st observation is 29
So, Medium ` = (28 +29 )/2 = 28.5`
`text ( Remark : )` The part of Table 14.11 consisting Column 1 and Column 3 is known as Cumulative Frequency Table.
The median marks 28.5 conveys the information that about 50% students obtained marks less than 28.5 and another 50% students obtained marks more than 28.5.
Now, let us see how to obtain the median of grouped data, through the following
situation.
Consider a grouped frequency distribution of marks obtained, out of 100, by 53 students, in a certain examination, as follows:
From the table above, try to answer the following questions:
How many students have scored marks less than 10? The answer is clearly 5.
How many students have scored less than 20 marks? Observe that the number of students who have scored less than 20 include the number of students who have scored marks from 0 - 10 as well as the number of students who have scored marks from 10 - 20.
So, the total number of students with marks less than 20 is 5 + 3, i.e., 8.
We say that the cumulative frequency of the class 10-20 is 8.
Similarly, we can compute the cumulative frequencies of the other classes, i.e., the number of students with marks less than 30, less than 40, . . ., less than 100. We give them in Table 14.13 given below:
The distribution given above is called the cumulative frequency distribution of the less than type. Here 10, 20, 30, . . . 100, are the upper limits of the respective class intervals.
We can similarly make the table for the number of students with scores, more than or equal to 0, more than or equal to 10, more than or equal to 20, and so on.
From Table 14.12, we observe that all 53 students have scored marks more than or equal to 0. Since there are 5 students scoring marks in the interval 0 - 10, this means that there are 53 – 5 = 48 students getting more than or equal to 10 marks.
Continuing in the same manner, we get the number of students scoring 20 or above as 48 – 3 = 45, 30 or above as 45 – 4 = 41, and so on, as shown in Table 14.14.
The table above is called a cumulative frequency distribution of the more than type. Here 0, 10, 20, . . ., 90 give the lower limits of the respective class intervals.
Now, to find the median of grouped data, we can make use of any of these cumulative frequency distributions.
Let us combine Tables 14.12 and 14.13 to get Table 14.15 given below:
Now in a grouped data, we may not be able to find the middle observation by looking at the cumulative frequencies as the middle observation will be some value in a class interval.
It is, therefore, necessary to find the value inside a class that divides the whole distribution into two halves. But which class should this be?
To find this class, we find the cumulative frequencies of all the classes and `n/2` We now locate the class whose cumulative frequency is greater than (and nearest to) `n/2`.
This is called the median class. In the distribution above, n = 53. So `n/2 = 26.5` .
Now 60 – 70 is the class whose cumulative frequency 29 is greater than (and nearest to) `n/2` ,i.e., 26.5 .
Therefore, 60 – 70 is the median class.
After finding the median class, we use the following formula for calculating the
median.
Median `color{orange}{ = l + ( ( n/2 -cf )/f) xx h}` ,
where l = lower limit of median class,
n = number of observations,
cf = cumulative frequency of class preceding the median class,
f = frequency of median class,
h = class size (assuming class size to be equal).
Substituting the values `n/2 = 26.5 , l =60 , cf =22 , f =7 , h =10`
in the formula above, we get
Median ` = 60 + ( (26.5 -22 )/7 ) xx 10`
`= 60 + 45/7`
`= 66.4`
So, about half the students have scored marks less than 66.4, and the other half have
scored marks more than 66.4.
Recall that for finding the median of ungrouped data, we first arrange the data values of the observations in ascending order.
Then, if n is odd, the median is the ` ( (n+1)/2)` th observation. And, if `n` is even, then the median will be the average of the `n/2` th and the ` (n/2 +1 )` th observations.
Suppose, we have to find the median of the following data, which gives the marks, out of 50, obtained by 100 students in a test :
First, we arrange the marks in ascending order and prepare a frequency table as
follows :
Here n = 100, which is even. The median will be the average of the `n/2` th and the
`(n/2 +1)` th observations, i.e., the 50th and 51st observations. To find these
observations, we proceed as follows:
Now we add another column depicting this information to the frequency table
above and name it as cumulative frequency column.
From the table above, we see that:
50th observaton is 28
51st observation is 29
So, Medium ` = (28 +29 )/2 = 28.5`
`text ( Remark : )` The part of Table 14.11 consisting Column 1 and Column 3 is known as Cumulative Frequency Table.
The median marks 28.5 conveys the information that about 50% students obtained marks less than 28.5 and another 50% students obtained marks more than 28.5.
Now, let us see how to obtain the median of grouped data, through the following
situation.
Consider a grouped frequency distribution of marks obtained, out of 100, by 53 students, in a certain examination, as follows:
From the table above, try to answer the following questions:
How many students have scored marks less than 10? The answer is clearly 5.
How many students have scored less than 20 marks? Observe that the number of students who have scored less than 20 include the number of students who have scored marks from 0 - 10 as well as the number of students who have scored marks from 10 - 20.
So, the total number of students with marks less than 20 is 5 + 3, i.e., 8.
We say that the cumulative frequency of the class 10-20 is 8.
Similarly, we can compute the cumulative frequencies of the other classes, i.e., the number of students with marks less than 30, less than 40, . . ., less than 100. We give them in Table 14.13 given below:
The distribution given above is called the cumulative frequency distribution of the less than type. Here 10, 20, 30, . . . 100, are the upper limits of the respective class intervals.
We can similarly make the table for the number of students with scores, more than or equal to 0, more than or equal to 10, more than or equal to 20, and so on.
From Table 14.12, we observe that all 53 students have scored marks more than or equal to 0. Since there are 5 students scoring marks in the interval 0 - 10, this means that there are 53 – 5 = 48 students getting more than or equal to 10 marks.
Continuing in the same manner, we get the number of students scoring 20 or above as 48 – 3 = 45, 30 or above as 45 – 4 = 41, and so on, as shown in Table 14.14.
The table above is called a cumulative frequency distribution of the more than type. Here 0, 10, 20, . . ., 90 give the lower limits of the respective class intervals.
Now, to find the median of grouped data, we can make use of any of these cumulative frequency distributions.
Let us combine Tables 14.12 and 14.13 to get Table 14.15 given below:
Now in a grouped data, we may not be able to find the middle observation by looking at the cumulative frequencies as the middle observation will be some value in a class interval.
It is, therefore, necessary to find the value inside a class that divides the whole distribution into two halves. But which class should this be?
To find this class, we find the cumulative frequencies of all the classes and `n/2` We now locate the class whose cumulative frequency is greater than (and nearest to) `n/2`.
This is called the median class. In the distribution above, n = 53. So `n/2 = 26.5` .
Now 60 – 70 is the class whose cumulative frequency 29 is greater than (and nearest to) `n/2` ,i.e., 26.5 .
Therefore, 60 – 70 is the median class.
After finding the median class, we use the following formula for calculating the
median.
Median `color{orange}{ = l + ( ( n/2 -cf )/f) xx h}` ,
where l = lower limit of median class,
n = number of observations,
cf = cumulative frequency of class preceding the median class,
f = frequency of median class,
h = class size (assuming class size to be equal).
Substituting the values `n/2 = 26.5 , l =60 , cf =22 , f =7 , h =10`
in the formula above, we get
Median ` = 60 + ( (26.5 -22 )/7 ) xx 10`
`= 60 + 45/7`
`= 66.4`
So, about half the students have scored marks less than 66.4, and the other half have
scored marks more than 66.4.
