The mean of the following distribution is 27 and the sum of all the frequencies is 50. Find the missing frequencies x and y.

Given,

Mean = 27 and sum of all the frequencies = 50

Class	Class Mark (xi)	Frequency (fi)	fixi
0 - 10	5	8	40
10 - 20	15	x	15x
20 - 30	25	12	300
30 - 40	35	13	455
40 - 50	45	y	45y
Total		Σfi = 33 + x + y	Σfixi = 795 + 15x + 45y

Given,

Sum of all the frequencies = 50

⇒ 33 + x + y = 50

⇒ x + y = 17 ......(1)

By formula,

Mean = $\frac{∑f_ix_i}{∑f_i}$

Substituting values, we get :

⇒ 27 = $\frac{(795 + 15x + 45y)}{50}$

⇒ 795 + 15x + 45y = 1350

⇒ 15x + 45y = 555

⇒ x + 3y = 37 ......(2)

Subtracting equation (1) from equation (2), we get :

⇒ (x + 3y) − (x + y) = 37 − 17

⇒ 2y = 20

⇒ y = 10

Substituting y = 10 in equation (1), we get :

⇒ x + 10 = 17

⇒ x = 7.

Hence, the missing frequencies are x = 7 and y = 10.