Question

The mean of $n$ numbers ${x_1},{x_2},{x_3},.....,{x_n}$ is M. If ${x_1}$ is replaced by $'a'$, then what is the new mean?
A) $\dfrac{{nM - {x_1} + a}}{n}$
B) $\dfrac{{M - {x_1} + a}}{n}$
C) $\dfrac{{nM - a + {x_1}}}{n}$
D) None of these.

Answer 1

In this question we are given certain numbers along with their mean. And we have been asked the new meaning 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 $M$. Then subtract the term which was to be replaced 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- ${x_1}$ is replaced by another number - $a$. Let us put the information that we are given in the question in the formula,
$\Rightarrow$ Mean = $\dfrac{{{\text{Sum of observations}}}}{{{\text{Total observations}}}}$
We are given that, mean = $M$
Total observations = $n$
Putting in the formula,
$\Rightarrow$ $M = \dfrac{{{x_1} + {x_2} + {x_3} + ........... + {x_n}}}{n}$
Shifting n to the other side to find the sum of the observations,
$\Rightarrow nM = {x_1} + {x_2} + {x_3} + ...... + {x_n}$
Since an observation ${x_1}$ has to be replaced, we will subtract ${x_1}$ from both the sides.
$\Rightarrow nM - {x_1} = {x_1} + {x_2} + {x_3} + ...... + {x_n} - {x_1}$
Simplifying RHS,
$\Rightarrow nM - {x_1} = {x_2} + {x_3} + ...... + {x_n}$
${x_1}$ had to be replaced by $a$, so now we will add $a$to both the sides.
$\Rightarrow nM - {x_1} + a = a + {x_2} + {x_3} + ...... + {x_n}$ ….. (1)
Now, we have our new sum of observations. Let us put the sum in the formula.
$\Rightarrow$New Mean = $\dfrac{{a + {x_2} + {x_3} + .... + {x_n}}}{n}$ …. (2)
From (1), we know that $a + {x_2} + {x_3} + ..... + {x_n}$ = $nM - {x_1} + a$. We will put this in equation (2).
$\Rightarrow$New Mean = $\dfrac{{nM - {x_1} + a}}{n}$

Option A is the correct answer.

Note: We have to remember that there are several kinds of the mean in mathematics, especially in statistics. For a data set, the arithmetic mean, also called the expected value or average, is the central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ${x_1}$, ${x_2}$, ${x_3}$, ….. ${x_n}$ is typically denoted by the $\overline x$. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (denoted $\overline x$) to distinguish it from the mean of the underlying distribution, the population mean (denoted $\mu$ or $\mu_s$).