Question

The mean of $n$ numbers ${X_1},{X_2},{X_3},...,{X_n}$ is $M$. If ${X_1}$ is replaced by $'x'$, then the new mean is
A. $M - {X_1} + x$
B. $\dfrac{{(n - 1)M + x}}{n}$
C. $\dfrac{{nM - {X_1} + x}}{n}$
D. $\dfrac{{M + {X_1} + x}}{n}$

Answer 1

Hint: In this statistical problem, we have given the mean of some $n$ numbers. In that $n$ numbers we need to replace the first number by the other given number. Then after replacing the first number by another given number we have to find the new mean. So by using the usual mean formula students need to find the new mean.

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

Complete step by step solution:
Given that the mean of $n$ numbers ${X_1},{X_2},{X_3},...,{X_n}$ is $M$.
Here our aim is if ${X_1}$ is replaced by $'x'$ , then find the new mean.
We know that the mean of $n$ number $M$ is given.
Mean $(m) = \dfrac{{{\text{sum of all observations }}}}{{{\text{No}}{\text{. of observations}}}}$
$M = \dfrac{{{X_1} + {X_2} + {X_3} + ... + {X_n}}}{n}$
$ \Rightarrow Mn = {X_1} + {X_2} + {X_3} + ... + {X_n}$
Now, let us take the first term ${X_1}$ to the left hand side,
$ \Rightarrow Mn - {X_1} = {X_2} + {X_3} + ... + {X_n}$
Here the value of ${X_2} + {X_3} + ... + {X_n}$ is $Mn - {X_1}$.
When ${X_1}$ is replaced by $x$, then the new mean is
New mean $ = \dfrac{{x + {X_2} + {X_3} + ... + {X_n}}}{n}$, here also there are $n$ observations.
Already we know that the value of ${X_2} + {X_3} + ... + {X_n}$ is $Mn - {X_1}$. So substitute this value in the new mean equation, we get
New mean $ = \dfrac{{x + Mn - {X_1}}}{n}$
We can rewrite this equation as, new mean $ = \dfrac{{Mn - {X_1} + x}}{n}$
 Hence this is the required solution.

$\therefore $ The correct answer is option (C).

Note: We have to remember that, the mean is a statistical indicator that can be used to gauge the performance of a company’s stock price over a period of days, months, or year, a company through its earnings over a number of years, a firm by assessing its fundamental such as price to earnings ratio, free cash flow, and liabilities on the balance sheet, and a portfolio by estimating its average returns over a certain period.
In this problem, we found the value of mean before and after replacing the first term in the given numbers. But we followed the same procedure for finding the mean value that is add up all the numbers, then divide by how many numbers there are. In other words it is the sum divided by the count.