The median of the following data is 525. Find the value of x and y, if the sum of frequency is 100.

Class	0-100	100-200	200-300	300-400	400-500	500-600	600-700	700-800	800-900	900-1000
Frequency	3	4	x	12	17	20	9	y	8	3

Here, we have grouped data given in the table. We need to find cumulative frequency of the data, which will predict our answer.

So,

We have added up all the values of the frequency in the second column and have got,

Total = n = 76 + x + y …(i)

But given is, sum of frequencies, ∑f = n = 100 …(ii)

⇒

⇒

Comparing equations (i) and (ii), we get

76 + x + y = 100

⇒ x + y = 100 – 76

⇒ x + y = 24 …(iii)

Also, given that, median = 525

Corresponding to this value of median, we can say that median lies in the class is 500 – 600. [∵ 525 lies between 500 – 600]

⇒ Median class = 500 – 600

Median is given by,

Where,

l = lower limit of the median class = 500

n = Total number of observation (sum of frequencies) = 100

cf = cumulative frequency of the class preceding the median class = 36 + x

f = frequency of the median class = 20

c = class size (class sizes are equal) = 100

Putting the values, l = 500, n/2 = 50, cf = 36 + x, f = 20 and c = 100 in the given formula of median, we get

⇒ [∵ given that, median = 525]

⇒

⇒

⇒ (14 – x) × 5 = 25

⇒ 14 – x = 5

⇒ x = 14 – 5

⇒ x = 9

Substituting the value, x = 9 in equation (iii), we get

x + y = 24

⇒ 9 + y = 24

⇒ y = 24 – 9

⇒ y = 15

Thus, x = 9 and y = 15.