Question

Consider the frequency distribution of the given number

Value	$1$	$2$	$3$	$4$
Frequency	$5$	$4$	$0$	$f$

If mean of the distribution is equal to $3$, then the value of $f$ is
A. $7$
B. $9$
C. $12$
D. $14$

Answer 1

Hint: In this question, a frequency distribution table is given and the mean of the distribution is given. You have to find the value of the missing frequency $f$. Mean is nothing but an average of the given data. It is obtained by adding all the values of the given data and then dividing it by the total number of observations. So, at first multiply all the given values with corresponding frequencies to get the sum of all the observations and then add all the frequencies to get the total number of observations. After that, divide the sum of observations by the total number of observations to get the mean. Also, the mean of the distribution is given in the question. So, make an equation by equating the obtained mean and the given mean and solve it to find the value of the missing frequency $f$.

Formula Used:
Mean of the numbers ${x_1},{x_2},{x_3},...,{x_n}$ having frequencies ${f_1},{f_2},{f_{3,}}.....,{f_n}$ is defined by $\bar x = \dfrac{{\sum\limits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }} = \dfrac{{{f_1}{x_1} + {f_2}{x_2} + {f_3}{x_3} + ... + {f_n}{x_n}}}{{{f_1} + {f_2} + {f_3} + ... + {f_n}}}$, where $n$ is total number of given data.

Complete step by step solution:
In this question, four values are given with frequencies.
The frequency of $1$ is $5$
The frequency of $2$ is $4$
The frequency of $3$ is $0$
The frequency of $4$ is $f$

Value $\left( {{x_i}} \right)$	Frequency $\left( {{f_i}} \right)$	$\left( {{f_i}{x_i}} \right)$
$1$	$5$	$5$
$2$	$4$	$8$
$3$	$0$	$0$
$4$	$f$	$4f$

Here, number of the observations is $n = 4$
Sum of the observations is $\sum\limits_{i = 1}^4 {{f_i}{x_i}} = 5 + 8 + 0 + 4f = 13 + 4f$
Sum of the frequencies is $\sum\limits_{i = 1}^4 {{f_i}} = 5 + 4 + 0 + f = 9 + f$
$\therefore $ Mean of the given numbers is $\bar x = \dfrac{{\sum\limits_{i = 1}^4 {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^4 {{f_i}} }} = \dfrac{{13 + 4f}}{{9 + f}}$
The given mean is $3$
So, $\dfrac{{13 + 4f}}{{9 + f}} = 3$
Solve this equation to find the value of $f$.
$\dfrac{{13 + 4f}}{{9 + f}} = 3$
$\begin{array}{l} \Rightarrow 13 + 4f = 3\left( {9 + f} \right)\\ \Rightarrow 13 + 4f = 27 + 3f\\ \Rightarrow 4f - 3f = 27 - 13\\ \Rightarrow f = 14\end{array}$

Option ‘D’ is correct

Note: For ungrouped data, the formula for finding the mean of the data is $\bar x = \dfrac{{{x_1} + {x_2} + {x_3} + ... + {x_n}}}{n}$
For grouped data, the formula for finding the mean of the data is $\bar x = \dfrac{{\sum\limits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }} = \dfrac{{{f_1}{x_1} + {f_2}{x_2} + {f_3}{x_3} + ... + {f_n}{x_n}}}{{{f_1} + {f_2} + {f_3} + ... + {f_n}}}$