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 frequency= 120

Answer 1

Hint: To find the missing values of frequencies we will generate two equations in terms of frequency. One of the equations by applying the condition of sum of the frequency is given 120 and other by applying the mean of the grouped data is 50. We will get two equations and two variables just solve the two generated equations and then find the value of missing frequencies.

Complete step-by-step answer:
Given sum of frequency =120
So adding all the given frequencies we will get
\[ \Rightarrow 17 + f1 + 32 + f2 + 19 = 120\]
Or,
\[ \Rightarrow f1 + f2 = 52\] …………(i)
The mean of the grouped data is given as 50 so we will calculate the mean and equate it with the given value of mean such that we will get another equation.
We know mean is defined as
Mean \[ = \dfrac{{\sum xifi}}{{\sum fi}}\]
On putting the given values in the above formula we get​​
$\Rightarrow$ mean \[ = \dfrac{{\left( {10 \times 17} \right) + \left( {30 \times f1} \right) + \left( {50 \times 32} \right) + \left( {72 \times f2} \right) + (90 \times 19)}}{{120}}\]
Simplifying the above and equate with the given value of mean,
$\Rightarrow$ mean \[ = \dfrac{{170 + 30f1 + 1600 + 70f2 + 1710}}{{120}} = 50\]
Cross multiplying and simplifying we get
\[ \Rightarrow 3{f_1} + 7{f_2} = 252\] ………………(ii)
Now on solving equation i and equation ii
We get
\[
  { \Rightarrow 3{f_1} + 7(52 - {f_1}) = 252} \\
   { \Rightarrow {f_{1}} = 28} \\
  { \Rightarrow {f_2} = 24}
\]
Hence the value of f1 and f2 are 28 and 24 respectively.

Note: This is solved by a simple mean calculation method. It can be solved by assuming a mean method also. Generally there must at least be two equations to find two unknowns.