Question

The mean of a data is $p$. If each observation is multiplied by $3$ and then $1$ is added to each result, then the mean of the new observations so obtained is
A. $p$
B. $3p$
C. $p + 1$
D. $3p + 1$

Answer 1

In this problem, the mean of the data is given. We are asked to find the new mean of the observation when each observation is multiplied by $3$ and then $1$ is added to each result. But here we don’t have the number of terms in data so we need to assume some value for the number of terms in data.

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

Complete step by step solution:
Let us take the number of terms in data be $n$.
Given that, mean $= p$
Since the number of terms in data be $n$, so the sum of data $= pn$
If each term is multiplied by $3$, then the new sum of data becomes thrice, that is $3pn$
Also given that $1$ is added in each term, since there are $n$ terms so if we add $1$ for $n$ times then $n$ is added in sum, that is, New sum $= 3pn + n$
Mean of new observations $m = \dfrac{{{\text{sum of the terms}}}}{{{\text{number of terms}}}}$
$\Rightarrow m = \dfrac{{3pn + n}}{n}$
Take $n$ common in the numerator of above equation, we get
$\Rightarrow m = \dfrac{{n(3p + 1)}}{n}$
In the numerator and denominator $n$ get cancel each other,
$\Rightarrow m = 3p + 1$
Hence, the mean of new observation is $3p + 1$.

$\therefore$ The answer is option (D)

Note: The way how we calculate mean in this problem is, the mean is the average of the numbers. It is easy to calculate: add up all the numbers, then divide by how many numbers there are. In other words, it is the sum divided by the count. Here the count is $n$ and the sum is $3pn + n$.