Question

Mean of twenty observations is 15. If two observations 3 and 14 are replaced by 8 and 9 respectively, then what will be the new mean.
(a) 14
(b) 15
(c) 16
(d) 17

Answer 1

Hint: To solve this problem we need to know how to calculate the mean of the n given terms. Mean of the n terms, ${{x}_{1}},{{x}_{2}},{{x}_{3}}.....,{{x}_{n}}$ is given by the ratio of sum of the all term to the total number of terms i.e. $\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+.....+{{x}_{n}}}{n}$. In the given question we have mean of the 20 observations as 15 but two terms 3 and 14 are replaced by 8 and 9 respectively so we will first find the extra sum that is added to the sum of all terms then divide it by the number of terms i.e. 20 to find the answer.

Complete step-by-step answer:
We are given that there are a total of 20 observations and their mean is 15. Now the two terms 3 and 14 out of 20 terms are replaced by 8 and 9 and we have to find the new mean. To solve this, we should know about the mean first,
Mean of n terms, ${{x}_{1}},{{x}_{2}},{{x}_{3}}.....,{{x}_{n}}$ is equal to $\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+.....+{{x}_{n}}}{n}$,
Now if we suppose the 20 terms as ${{x}_{1}},{{x}_{2}},{{x}_{3}}.....,{{x}_{20}}$ then we are given,
$\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+.....+{{x}_{20}}}{20}=15\,\,.....\left( 1 \right)$
Now suppose ${{x}_{1}}\,and\,{{x}_{2}}$ are equal to 3 and 14 respectively. According to question now ${{x}_{1}}\,and\,{{x}_{2}}$ are replaced by 8 and 9 respectively,
$\begin{align}
  & {{x}_{1}}=3 \\
 & {{x}_{2}}=14 \\
\end{align}$
We will get new mean as,
\[\operatorname{New}\,Mean=\dfrac{8+9+{{x}_{3}}+.....+{{x}_{20}}}{20}\]
Now we can write 8 and 14 as,
$\begin{align}
  & 8={{x}_{1}}+5\,\,and \\
 & 9={{x}_{2}}-5 \\
\end{align}$
Putting these values in the new mean we get,
\[\begin{align}
  & \operatorname{New}\,Mean=\dfrac{\left( {{x}_{1}}+5 \right)+\left( {{x}_{2}}-5 \right)+{{x}_{3}}+.....+{{x}_{20}}}{20} \\
 & \operatorname{New}\,Mean=\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+.....+{{x}_{20}}}{20} \\
\end{align}\]
Using the equation 1 we get,
\[\operatorname{New}\,Mean=15\]
Hence we get option (b) as the correct answer.

Note:
You can also solve this question by comparing the sum of initial two terms and the changed two terms like initially 3 + 14 = 17 and 8 + 9 = 17, both are equal so there will be no change in the mean otherwise it will change. In the above solution the trick is we need to express the new terms in the form of old terms so that we can find the new Mean .