Question

The mean of the series ${x_1},{x_2},{x_3},....,{x_n}$ is $\bar X$. If ${x_2}$ is replaced by $\lambda $, then what is the new mean?
A) $\bar X - {x_2} + \lambda $
B) $\dfrac{{\bar X - {x_2} - \lambda }}{n}$
C) $\dfrac{{\left( {n - 1} \right)\bar X + \lambda }}{n}$
D) $\dfrac{{n\bar X - {x_2} + \lambda }}{n}$

Answer 1

Hint: In this question we are given certain numbers along with their mean, and we have been asked the new mean if one of the given numbers is replaced by another number. First, find the sum of the given observations in the terms of $n$ and $\bar X$. Then subtract the term which was to be replaced from the old mean and add the new term. Put this new sum in the formula and you will have your answer.

Formula used: Mean = $\dfrac{{{\text{Sum of observations}}}}{{{\text{Total observations}}}}$

Complete step-by-step solution:
We are given $n$ numbers and we are also given the mean of these numbers. We have been asked the new mean when one number- ${x_2}$ is replaced by another number -$\lambda $. Let us put the information that we are given in the question in the formula,
Mean = $\dfrac{{{\text{Sum of observations}}}}{{{\text{Total observations}}}}$
We are given that, mean = $\bar X$
Total observation = $n$
$ \Rightarrow $$\bar X = \dfrac{{{x_1} + {x_2} + {x_3} + ........... + {x_n}}}{n}$
Shifting n to the other side to find the sum of the observations,
$ \Rightarrow n\bar X = {x_1} + {x_2} + {x_3} + ...... + {x_n}$
Since an observation ${x_2}$ has to be replaced, we will subtract ${x_2}$ from both the sides.
$ \Rightarrow n\bar X - {x_2} = {x_1} + {x_2} + {x_3} + ...... + {x_n} - {x_2}$
Simplifying RHS,
$ \Rightarrow n\bar X - {x_2} = {x_1} + {x_3} + ...... + {x_n}$
${x_2}$ Had to be replaced by $\lambda $, so now we will add $\lambda $to both the sides.
$ \Rightarrow n\bar X - {x_2} + \lambda = \lambda + {x_1} + {x_3} + ...... + {x_n}$ ….. (1)
Now, we have our new sum of observations. Let us put the sum in the formula.
$ \Rightarrow $New Mean = $\dfrac{{\lambda + {x_1} + {x_3} + .... + {x_n}}}{n}$ …. (2)
From (1), we know that $\lambda + {x_1} + {x_3} + ..... + {x_n}$ = $nM - {x_1} + a$. We will put this in equation (2).
$ \Rightarrow $New Mean = $\dfrac{{n\bar X - {x_2} + \lambda }}{n}$

Therefore, our new mean is option (D) $\dfrac{{n\bar X - {x_2} + \lambda }}{n}$.

Note: We find the sum of the given observations in terms of $n$ and $\bar X$ because the answer given in the options is in that terms. We could have skipped the step where we substitute $\lambda + {x_1} + {x_3} + ..... + {x_n}$ = $nM - {x_1} + a$ but since the options are in those terms.