Question

Find the missing frequencies in the following frequency distribution if it is known that the mean of the distribution is 1.46.

Number of accidents \[\left( x \right)\]	0	123	4	3	4	5	Total
Frequency\[\left( f \right)\]	46	?	?	25	10	5	200

Answer 1

Hint:
Here we have to find the missing frequencies. For that, we will first assume two frequencies to be any variable. Then we will find the sum of all frequencies. We will multiply each frequency with its consecutive variables and then we will find its sum. We will put the values calculated in the formula of mean.

Complete step by step solution:
 Let the two frequencies be ${{f}_{1}}\And {{f}_{2}}$
So the sum of frequencies which is denoted by N is:-
$N=\sum{{{f}_{i}}=46+25+10+5+{{f}_{1}}+{{f}_{2}}}=86+{{f}_{1}}+{{f}_{2}}$
Value of N is given in the question which is 200.
Now, we will put the value of N in this equation.
$\begin{align}
  & 200=86+{{f}_{1}}+{{f}_{2}} \\
 & {{f}_{1}}+{{f}_{2}}=114 \\
\end{align}$ ………………$(1)$
Now, we will draw the frequency distribution table.

${{x}_{i}}$	${{f}_{i}}$	${{f}_{i}}{{x}_{i}}$
0	46	$0\times 46=0$
1	${{f}_{1}}$	$1\times {{f}_{1}}={{f}_{1}}$.
2	${{f}_{2}}$	$2\times {{f}_{2}}=2{{f}_{2}}$
3	25	$3\times 25=75$
4	10	$4\times 10=40$
5	5	$5\times 5=25$

Now, we will find the sum of ${{f}_{i}}{{x}_{i}}$which will be equal to $0+{{f}_{1}}+2{{f}_{2}}+75+40+25=140+{{f}_{1}}+2{{f}_{2}}$
We know the formula of mean.
$mean=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{N}$
We will put the value of N and $\sum{{{f}_{i}}{{x}_{i}}}$in the formula of mean.
$mean=\dfrac{140+{{f}_{1}}+2f}{200}$
Now, we will put the value of mean in the formula.
$1.46=\dfrac{140+{{f}_{1}}+2f}{200}$
We will cross multiply the numerator and denominator.
$1.46\times 200=140+{{f}_{1}}+2f$
Simplifying the equation, we get
${{f}_{1}}+2f=152$…….. $\left( 2 \right)$
We will subtract equation 1 from equation 2.
${{f}_{1}}+2{{f}_{2}}-\left( {{f}_{1}}+{{f}_{2}} \right)=152-114$
Subtracting the like terms, we get
${{f}_{2}}=38$
We will put the value of $f_2$ in the equation.
$\begin{align}
  & {{f}_{1}}+38=114 \\
 & {{f}_{1}}=76 \\
\end{align}$

Hence, the missing frequencies are 76 and 38.

Note:
Since we have drawn the frequency distribution table for the given data. We need to know its meaning and proper definition. A frequency distribution is defined as a tabular representation of data that represents the number of observations within a given class interval or for a particular variable. Remember, the class intervals should always be mutually exclusive. Frequency distribution is used for summarizing and for getting overview of categorical data.