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Question: The mean of \(n\) numbers \({X_1},{X_2},{X_3},...,{X_n}\) is \(M\). If \({X_1}\) is replaced by \('x...

The mean of nn numbers X1,X2,X3,...,Xn{X_1},{X_2},{X_3},...,{X_n} is MM. If X1{X_1} is replaced by ′x′'x', then the new mean is
A. M−X1+xM - {X_1} + x
B. (n−1)M+xn\dfrac{{(n - 1)M + x}}{n}
C. nM−X1+xn\dfrac{{nM - {X_1} + x}}{n}
D. M+X1+xn\dfrac{{M + {X_1} + x}}{n}

Explanation

Solution

In this statistical problem, we have given the mean of some nn numbers. In that nn 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=sum of the termsnumber of termsm = \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 nn numbers X1,X2,X3,...,Xn{X_1},{X_2},{X_3},...,{X_n} is MM.
Here our aim is if X1{X_1} is replaced by ′x′'x' , then find the new mean.
We know that the mean of nn number MM is given.
Mean (m)=sum of all observations No. of observations(m) = \dfrac{{{\text{sum of all observations }}}}{{{\text{No}}{\text{. of observations}}}}
M=X1+X2+X3+...+XnnM = \dfrac{{{X_1} + {X_2} + {X_3} + ... + {X_n}}}{n}
⇒Mn=X1+X2+X3+...+Xn\Rightarrow Mn = {X_1} + {X_2} + {X_3} + ... + {X_n}
Now, let us take the first term X1{X_1} to the left hand side,
⇒Mn−X1=X2+X3+...+Xn\Rightarrow Mn - {X_1} = {X_2} + {X_3} + ... + {X_n}
Here the value of X2+X3+...+Xn{X_2} + {X_3} + ... + {X_n} is Mn−X1Mn - {X_1}.
When X1{X_1} is replaced by xx, then the new mean is
New mean =x+X2+X3+...+Xnn = \dfrac{{x + {X_2} + {X_3} + ... + {X_n}}}{n}, here also there are nn observations.
Already we know that the value of X2+X3+...+Xn{X_2} + {X_3} + ... + {X_n} is Mn−X1Mn - {X_1}. So substitute this value in the new mean equation, we get
New mean =x+Mn−X1n = \dfrac{{x + Mn - {X_1}}}{n}
We can rewrite this equation as, new mean =Mn−X1+xn = \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.