Question

Find the median of the given data graphically.

x	2-4	4-6	6-8	8-10
frequency	1	3	5	7

(a) 4
(b) 7
(c) 8
(d) 10

Answer 1

Hint: We start solving the problem by checking whether the given frequency distribution is continuous or not. If it is continuous, we proceed further into the problem to find median otherwise, we make corrections at the lower and upper boundaries to make it continuous. We then find the cumulative frequencies in ascending and descending order. We then plot these curves and find the value of x at the intersection of both curves which is the required value of median.

Complete step by step answer:
According to the problem, we need to find the median of the given data in the frequency distribution table graphically.

x	2-4	4-6	6-8	8-10
frequency	1	3	5	7

We can see that the upper limit of the previous interval of x is equal to the lower limit of the next interval of x, which makes the given frequency distribution table continuous.
Let us find the sum of the given frequencies.

x	frequency
2-4	1
4-6	3
6-8	5
8-10	7
Total	16

Now, let us find the cumulative frequency for each interval of x in the ascending way.
So, we get

x	Cumulative frequency
Less than 2	0
Less than 4	1
Less than 6	4
Less than 8	9
Less than 10	16

Let us plot the graph between x and the cumulative frequency which we can see from the frequency table.

Now, let us find the cumulative frequency for each interval of x in the descending way.
So, we get

x	Cumulative frequency
more than 2	16
more than 4	15
more than 6	11
more than 8	2
more than 10	0

Let us plot the graph between x and the cumulative frequency which we can see from the frequency table in the same graph we plotted before.

We know that the median of the given frequency distribution lies at the intersection of both cumulative frequency curves.
From the plot, we can see that the cumulative frequency curves intersect at $x=7$, which is the median.
So, we have found the median as 7.
The correct option for the given problem is (b).
Note:
We should know that the median obtained need not be absolutely correct. We can plot one curve and check at the cumulative frequency $\dfrac{16}{2}=8$, but the values of the median will be different in both the plots. This is the reason why we need to plot both curves in a single graph. Similarly, we can expect problems to find the median without graphical methods.