Find the values of frequencies 𝑥 and 𝑦 in the following frequency distribution table, if 𝑁 = 100 and median is 32. (2019)

 

Find the values of frequencies 𝑥 and 𝑦 in the following frequency distribution table, if 𝑁 = 100 and median is 32. (2019)

Marks	No. of students
0 – 10	10
10 – 20	𝑥
20 – 30	25
30 – 40	30
40 – 50	𝑦
50 – 60	10
Total	100

 

Soln:

First find out the cumulative frequency (cf)

Marks	No. of students (f)	Cumulative frequency (f)
0 – 10	10	10
10 – 20	𝑥	10+x
20 – 30	25	35+x
30 – 40	30	65+x
40 – 50	𝑦	65+x+y
50 – 60	10	75+x+y
Total	100

Now, Given that there are 100 students

          f = 100

          Þ 75+x+y = 100

          Þ x + y = 100 – 75 = 25            …. (1)

Also given that

          Median = 32

          That is, median lies in the range 30-40.

Therefore, 30-40 is the median class.

        

          Here,  L = Lower class containing the median

                   N = Total number of students

                   f  = Frequency of the class containing median

                    cf  = Cumulative frequency before the median class

                   h = class interval = upper limit – lower limit

In our problem

          L = 30

          N = 100

          f = 30

          cf = 35 + x

          h = 10 – 0 = 10

Substitute these values, we get

         

          2 x 3 = 15 – x

          x = 15 - 6

          x = 9

Substituting the value of x in eqn (1) we get

          y = 25 – x

          y = 25 – 9

        y = 16

Hence, the value of x = 9 and value of y = 16.