Question

The mean of the following frequency distribution is 14.5. Find the values of ${f_1}$ and ${f_2}$.

CLASS	FREQUENCY
0-50	8
50-100	${f_1}$
100-150	32
150-200	26
200-250	${f_2}$
250-300	7
TOTAL	100

Answer 1

We are already given the total of frequency and the mean of the data. We will find the values of the frequencies with the help of two equations. One equation we will form from the total of frequency and other from the formula of mean. So let’s start!

Complete step by step solution:
Given that mean of the frequency distribution is 14.5
We know that
$mean = \dfrac{{\sum {fx} }}{{\sum f }}$
Now let’s tabulate x that is midpoint of the class. And product of frequency(f) and midpoint(x).

CLASS	FREQUENCY(f)	MIDPOINT(x)	$f \times x$
0-50	8	25	$8 \times 25 = 200$
50-100	${f_1}$	75	75${f_1}$
100-150	32	125	$32 \times 125 = 4000$
150-200	26	175	$26 \times 175 = 4550$
200-250	${f_2}$	225	225${f_2}$
250-300	7	275	$7 \times 275 = 1925$
TOTAL	100

Now we know that,
$\sum {f \times x = 200 + 75{f_1} + 4000 + } 4550 + 225{f_2} + 1925 = 10675 + 75f + 225{f_2}$
Using the formula and substituting the values in it,
$mean = \dfrac{{\sum {fx} }}{{\sum x }}$
$\Rightarrow 14.5 = \dfrac{{10675 + 75{f_1} + 225{f_2}}}{{100}}$
Cross multiplying,

$$\Rightarrow 14.5 \times 100 = 10675 + 75{f_1} + 225{f_2} \\\ \Rightarrow 1450 = 10675 + 75{f_1} + 225{f_2} \\\ \Rightarrow 1450 - 10675 = 75{f_1} + 225{f_2} \\\$$

Subtracting the numbers on left side
$\Rightarrow - 9225 = 75{f_1} + 225{f_2}$
Dividing both sides by 25 we get,
$\Rightarrow - 369 = 3{f_1} + 9{f_2}$
Dividing both sides again by 3 we get,
$\Rightarrow - 123 = {f_1} + 3{f_2}.......... \to equation1$
Now we know that the total of all frequencies is 100.
Thus,
$\sum {f = 8 + {f_1} + } 32 + 26 + {f_2} + 7$

$$\Rightarrow 100 = 73 + {f_1} + {f_2} \\\ \Rightarrow 100 - 73 = {f_1} + {f_2} \\\$$

Subtracting numbers on left side
$\Rightarrow 27 = {f_1} + {f_2}........ \to equation2$
Now performing subtraction as equation2-equations1
$\Rightarrow 27 - ( - 123) = {f_1} + {f_2} - {f_1} - 3{f_2}$
Performing the necessary operations and cancelling ${f_1}$

$$\Rightarrow 27 + 123 = - 2{f_2} \\\ \Rightarrow 150 = - 2{f_2} \\\ \Rightarrow \dfrac{{150}}{2} = - {f_2} \\\ \Rightarrow - 75 = {f_2} \\\$$

Since we found the value of ${f_2}$ we will get value of ${f_1}$ from any of the above equations
From equation2 we get,

$$\Rightarrow 27 = {f_1} + ( - 75) \\\ \Rightarrow 27 + 75 = {f_1} \\\ \Rightarrow 102 = {f_1} \\\$$

Thus values of the frequencies are ${f_1} = 102$ and ${f_2} = - 75$.

Note:
Remember one thing students, whenever you solve these types of problems you should first work at the frequencies and midpoints of the class. If you find them very difficult to multiply with other numbers, then switch the formula you are using to find the mean immediately because it will take too much of your time.