Question

If the median of the following distribution is 32.5, then find the values of x and y.

Class	Frequency
0 – 10	x
10 – 20	5
20 – 30	9
30 – 40	12
40 – 50	y
50 – 60	3
60 – 70	2
Total	40

Answer 1

Class	Frequency	Cumulative frequency
0 – 10	x	x
10 – 20	5	x + 5
20 – 30	9	x + 14
30 – 40	12	x + 26
40 – 50	y	x + y + 26
50 – 60	3	x + y + 29
60 – 70	2	x + y + 31
Total	40

Now, sum of all frequency is 40.

x + 5 + 9 + 12 + y + 3 + 2 = 40

x + y + 31 = 40

x + y = 40 – 31

x + y = 9   ...(i)

Median = `l + ((N/2 - cf))/f xx h`

Given median = 32.5

∴ Median class = 30 – 40

l = 30, `N/2 = 40/2 = 20`, cf = x + 14, h = 10, f = 12

∴ Median = `l + ((N/2 - cf))/f xx h`

= `30 + ((20 - (x + 14)))/12 xx 10`

= `30 + ((20 - x - 14))/12 xx 10`

= `32.5 = 30 + ((6 - x)/12) xx 10`

= `32.5 - 30 = ((6 - x)/12) xx 10`

`2.5 = ((6 - x) xx 5)/6`

2.5 × 6 = 5(6 – x)

15 = 30 – 5x

5x = 30 – 15

5x = 15

`x = 15/3`

x = 3

∴ From equation (i)

x + y = 9

3 + y = 9

y = 9 – 3

y = 6