Question

If the mean of the following distribution is \[2.6\], then find the value of \[y\].

A. 3
B. 8
C. \[13\]
D. \[24\]

Answer 1

Hint:
First, calculate the product of the variable and frequency. Then calculate the sum of the frequency and the sum of the product of the variable and frequency. After that, substitute the values in the formula of the mean and solve the equation to get the required answer.



Formula Used:
\[\text{Mean} = \dfrac{{\sum {{f_i}{x_i}} }}{n}\] , where \[n\] is the total number of observations.


Complete step-by-step answer:
The given data distribution is,

The mean of the distribution is \[2.6\].

Let’s calculate the product of the variable and frequency.



Now calculate the sum of the frequency and the sum of the product of the variable and frequency.
We get,
The sum of the frequency,
\[\sum {{f_i}} = 4 + 5 + y + 1 + 2\]
\[ \Rightarrow \sum {{f_i}} = 12 + y\]

The sum of the product of the variable and frequency,
\[\sum {{f_i}{x_i}} = 4 + 10 + 3y + 4 + 10\]
\[ \Rightarrow \sum {{f_i}{x_i}} = 28 + 3y\]

We know that the sum of frequency is the total number of observations.
So, \[n = \sum {{f_i}} = 12 + y\]
Now use the formula of the mean
\[\text{Mean} = \dfrac{{\sum {{f_i}{x_i}} }}{n}\].
\[2.6 = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}\]
Substitute the values of the sums in the above equation.
\[2.6 = \dfrac{{28 + 3y}}{{12 + y}}\]
Solve the above equation.
\[2.6\left( {12 + y} \right) = 28 + 3y\]
\[ \Rightarrow 31.2 + 2.6y = 28 + 3y\]
\[ \Rightarrow 31.2 - 28 = 3y - 2.6y\]
\[ \Rightarrow 3.2 = 0.4y\]
Divide both sides by \[0.4\].
 \[y = \dfrac{{3.2}}{{0.4}}\]
\[y = 8\]
Hence the correct option is B.



Note:
Students often get confused and calculate the mean directly by considering the number of variables as the total number of observations and the sum of the frequency as the sum of elements of the observation.