Question

The mean of the following frequency distribution is 57.6 and the sum of observations is 50. Find the values of f₁ and f₂.

Class	0 – 20	20 – 40	40 – 60	60 – 80	80 – 100	100 – 120
Frequency	7	f1	12	f2	8	5

Answer 1

1. Organize the data

First, let’s find the class marks (x_i) and the products (f_ix_i) for each class interval.

The class mark is the midpoint of the interval:

`x_i = ("Lower Limit" + "Upper Limit")/2`

Class	Frequency (fi)	Class Mark (xi)	fixi
0 – 20	7	10	70
20 – 40	f1	30	30f1
40 – 60	12	50	600
60 – 80	f2	70	70f2
80 – 100	8	90	720
100 – 120	5	110	550
Total	50		1940 + 30f1 + 70f2

2. Formulate the equations

Equation 1: From the sum of frequencies

The problem states the sum of observations is 50.

7 + f₁ + 12 + f₂ + 8 + 5 = 50

32 + f₁ + f₂ = 50

f₁ + f₂ = 18   ...(1)

Equation 2: From the mean

The mean formula is `barx = (sumf_ix_i)/(sumf_i)`.

Given `barx = 57.6`:

`57.6 = (1940 + 30f_1 + 70f_2)/50`

Multiply both sides by 50:

2880 = 1940 + 30f₁ + 70f₂

940 = 30f₁ + 70f₂

Divide the entire equation by 10 to simplify:

3f₁ + 7f₂ = 94   ...(2)

3. Solve the system of equations

Multiply equation (1) by 3:

3f₁ + 3f₂ = 54   ...(3)

Subtract equation (3) from equation (2):

(3f₁ + 7f₂) – (3f₁ + 3f₂) = 94 – 54

4f₂ = 40

f₂ = 10

Substitute f₂ = 10 into equation (1):

f₁ + 10 = 18

f₁ = 8

The values of the missing frequencies are f₁ = 8 and f₂ = 10.