Question

Find the values of a and b, if the sum of all the frequencies is 120 and the median of the following data is 55.

Marks	30 – 40	40 – 50	50 –60	60 – 70	70 –80	80 – 90
Frequency	a	40	27	b	15	24

Answer 1

Given: Median of observations = 55

And sum of frequencies = 120

Marks `(f_i)`	Frequency `(c.f.)`	Cumulative frequency
30 – 40	a	a
40 – 50	40	a + 40
50 – 60	27	a + 67
60 – 70	b	a + b + 67
70 – 80	15	a + b + 82
80 – 90	24	a + b + 106

Now, `sumf_i` = 120  ......(Given)

⇒ a + b + 106 = 120

⇒ a + b = 14  ......(i)

Also, Median = 55

∴ Median class = 50 – 60

So, L = 50, h = 10, f = 27, c.f. = a + 40

Median = `L + ((N/2 - c.f.)/f) xx h`

55 = `50 + ((120/2 - (a + 40))/27) xx 10`

55 – 50 = `((60 - (a + 40))/27) xx 10` 

5 = `((20 - a)/27) xx 10`

5 × 27 = 200 – 10a

135 = 200 – 10a

10a = 65

a = 6.5

Putting the value of a in equation (i), we get

6.5 + b = 14

b = 14 – 6.5 = 7.5

Hence, a = 6.5 and b = 7.5.