Question

If ${u_i} = \dfrac{{{x_i} - 25}}{{10}}$ , $\sum {{f_i}{u_i}} = 20$ and $\sum {{f_i}} = 100$ , then $\bar x$ is equal to
A.27
B.25
C.30
D.35

Answer 1

Hint: As we know, the mean is the average of the data. In this question, we need to calculate the mean of grouped data. In this question, we can use assumed mean method according to which mean is given by the formula $\overline x = A + \dfrac{{\sum {{f_i}{u_i}} }}{N} \times h$ . We have to put the values in the above formula to get the mean value.

Complete step-by-step answer:
We need to calculate the mean of the given grouped data.
Mean is given by the formula $\overline x = A + \dfrac{{\sum {{f_i}{u_i}} }}{N} \times h$ where
A = Assumed mean
F = frequency
${u_i} = \dfrac{{{x_i} - A}}{h}$
$N$ = total frequency =$\sum {f_i} = 100$
$h$ = height of each interval
According to question,
${u_i} = \dfrac{{{x_i} - 25}}{{10}}$
By comparing we get that,
$A = 25$ ,
$N = 20$
And $h = 10$
Now, to find mean substitute all values in equation $\overline x = A + \dfrac{{\sum {{f_i}{u_i}} }}{N} \times h$
\[
\Rightarrow \overline x = 25 + \dfrac{{20}}{{100}} \times 10 \\
\Rightarrow \overline x = 25 + 2 \\
\Rightarrow \overline x = 27 \\
 \]
So mean is 27
Hence, option a is correct.
Note: Mean is just another name for average. The mean of the given grouped data can also be calculated by direct method and step-deviation method.