Median of Grouped Data

As you have studied in Class IX, the median is a measure of central tendency which gives the value of the middle-mostmost extreme observation in the data.

Recall that for finding the median of ungrouped data, we first arrange the data values of the observations in ascendingdescending order.

Then, if n is , the median is the n+12 th observation.

And, if n is , then the median will be the averagegreater of the n2 th and the n2+1th 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 :

Solution

First, we arrange the marks in ascending order and prepare a frequency table as follows :

Here n = 100, which is .

The median will be the average of the n2th and the n2+1 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.

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From the table above, we see that: 50th observaton is 28

51st observaton is

Median = 28 + 292 = (Upto one decimal place)

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:

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 given below:

Table

The distribution given above is called the cumulative frequency distribution of the less than type. Here 10, 20, 30, . . . 100, are the 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 the above Table, 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 = students getting more than or equal to 10 marks.

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Continuing in the same manner, we get the number of students scoring 20 or above as 48 – 3 = , 30 or above as 45 – 4 = , and so on, as shown in below Table.

The table above is called a cumulative frequency distribution of the more than type. Here 0, 10, 20, . . ., 90 give the 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.

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Let us combine Tables to get Table 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 n2.

We now locate the class whose cumulative frequency is greater than (and nearest to) n2.

This is called the median class. In the distribution above, n = . So, n2 = . (Upto one decimal place)

Now 60 – 70 is the class whose cumulative frequency i.e. is than (and nearest to) n2, i.e., 26.5.

Therefore, 60 – 70 is the median class.

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After finding the median class, we use the following formula for calculating the median.

Substituting the values n2 = , l = , cf = , f = , h = in the formula above, we get

Median = + 26.5227 ×

= 60 + 457 =

So, about half the students have scored marks than 66.4, and the other half have scored marks than 66.4.

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7. The median of the following data is 525. Find the values of x and y, if the total frequency is 100.

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