Question

Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is 50.
x .
10
30
50
70
90
f:
17
f1
32
f2
19
Total 120

Answer 1

ARITHMETIC MEAN OR MEAN OR AVERAGE :  

The arithmetic mean of a set of observations is obtained by dividing the sum of the values of all observations by the total number of observations .

Mean = Sum of all the observations / Total number of observations .

MEAN = Σfixi / Σfi

[‘Σ’ Sigma means ‘summation’ ]

FREQUENCY DISTRIBUTION TABLE IS IN THE ATTACHMENT  

From the table : Σfixi = 3480 + 30f1 + 70f2  , Σfi = 68 + f1 + f2

Given : Σfi = 120 , Mean = 50

Σfi = 68 + f1 + f2

120 = 68 + f1 + f2

f1 + f2 = 120 - 68

f1 + f2 = 52  

f1 = 52 - f2 ………….(1)  

MEAN = Σfixi / Σfi

50 = (3480 + 30f1 + 70f2) /120

50 × 120 = (3480 + 30f1 + 70f2)

6000 = 3480 + 30f1 + 70f2

6000 - 3480 = 30f1 + 70f2

2520 = 30f1 + 70f2

30f1 + 70f2 = 2520

10 ( 3f1 + 7f2) = 2520

3f1 + 7f2 = 2520/10

3f1 + 7f2 = 252 ……………(2)

3(52 - f2) + 7f2 = 252  

[From eq 1]

156 - 3f2 + 7f2 = 252

156 + 4f2 = 252  

4f2 = 252 - 156  

4f2 = 96

f2 = 96/4

f2 = 24

Putting this value of f2 in eq 1  

f1 = 52 - f2

f1 = 52 - 24

f1 = 28  

Hence, the value of f1 is 28 & f2 is 24 .

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

Answer:
$f_{1} = 28\\\\f_{2} = 24$

Solution:

x______f_________xf

10_____17________170

30_____f1________30f1

50_____32______1600

70_____f2_______70f2

90_____19_______1710

We know that

$\bar x = \frac{\Sigma x_{i}f_{i}}{\Sigma f _{i}}$
$\Sigma x_{i}f_{i} = 170 + 30f_{1} + 1600 + 70f_{2} + 1710 \\ \\ = 3480 + 30 \: f_{1} + 70 \: f_{2}$
and

$f =120 \\ \\ f = 17 + f_{1} + 32 + f_{2} + 19 \\ \\ = > 68 + f_{1} + f_{2} = 120 \\ \\ f_{1} + f_{2} = 52 \: \: \: eq1$

Mean = 50

$50 = \frac{3480 + 30f_{1} + 70f_{2}}{120} \\ \\ 600 = 348 + 3f_{1} + 7f_{2} \\ \\ 3f_{1} + 7f_{2} = 252 \: \: \: eq2$
Multiply eq 1 by 3 and subtract from eq 2

$3f_{1} + 3f_{2} = 156 \\ \\ 3f_{1} + 7f_{2} = 252 \\ - \: \: \: \: \: - \: \: \: \: \: \: \: \: \: - \\ - 4f_{2} = - 96 \\ \\ f_{2} = \frac{96}{4} \\ \\ f_{2} = 24 \\ \\ so \\ \\ f_{1} = 52 - 24 \\ \\ f_{1} = 28$
Hope it helps you.