Question

The marks obtained by $15$ students of a class test are $12,14,07,09,23,11,08,13,11,19,16,24,17,03,20$. Find
(i) The mean of their marks.
(ii) The mean of their marks when the marks of each student are increased by $4$.

Answer 1

In this problem, we have given the marks obtained by $15$ students of a class test. First we are asked to find the mean of the marks which is obtained by the students of a class test. Next we are asked to find the mean of their marks when the marks of each student are increased by $4$. For this we have to add $4$ to each given mark.

Formula used: $m = \dfrac{{{\text{sum of the marks }}}}{{{\text{number of the students}}}}$, where m is the mean.

Complete step by step solution:
Given that, the marks obtained by $15$ students of a class test are $12,14,07,09,23,11,08,13,11,19,16,24,17,03,20$.
For our convenient let us arrange the marks obtained by $15$ students in an ascending order, $03,07,08,09,11,11,12,13,14,16,17,19,20,23,24$
(i) Mean of the marks obtained by $15$ students of the class test $(m) = \dfrac{{{\text{sum of the marks}}}}{{{\text{number of the students}}}}$
$\Rightarrow m = \dfrac{{03 + 07 + 08 + 09 + 11 + 11 + 12 + 13 + 14 + 16 + 17 + 19 + 20 + 23 + 24}}{{15}}$
$\Rightarrow m = \dfrac{{207}}{{15}}$, solving this fraction, we get
Hence,
$\Rightarrow m = 13.8$
$\therefore$ Mean of the marks obtained by $15$ students of the class test $= 13.8$
(ii) The new marks after increasing each student mark by $4$ are,
$07,11,12,13,15,15,16,17,18,20,21,24,23,27,28$
Mean of the new marks after increasing each student mark by $4$ $(m) = \dfrac{{{\text{sum of the marks}}}}{{{\text{number of the students}}}}$
$\Rightarrow m = \dfrac{{07 + 11 + 12 + 13 + 15 + 15 + 16 + 17 + 18 + 20 + 21 + 23 + 24 + 28}}{{15}}$
$\Rightarrow m = \dfrac{{267}}{{15}}$, solving this fraction, we get
Hence,
$\Rightarrow m = 17.8$
$\therefore$ Mean of the new marks after increasing each student mark by $4$ $= 17.8$

Note: In the first sub division we found the mean value for the given marks obtained by $15$ student of the class test. In the second sub division we found the mean value for the student marks when the marks of each student are increased by $4$, that is in simple we can add $(15 \times 4 = )60$ to the sum of the marks in the first subdivision.