Question

The mean of $13$ numbers is$24$. If $3$ is added to each number what will the change in new mean be?

Answer 1

First we will calculate the total sum of all the $13$ numbers and then add $3$ in the total for each number to find the new mean. From the new mean we will find the change in mean.

Formula used: Mean ${\text{ = }}\dfrac{{{\text{Sum of the terms}}}}{{{\text{Number of terms}}}}$

Complete step-by-step solution:
Let $N$ be the total number of terms in the distribution, $S$ be the sum of all the terms and $M$ be the mean of the terms.
It is given that $M = 24$ and $N = 13$
Now we use the formula that: $M = \dfrac{S}{N}$
On substituting the given values, we get:
$24 = \dfrac{S}{{13}}$
On cross multiplying we get:
$S = 24 \times 13$
Let us multiply we get,
$S = 312$
Therefore, we know that the total i.e. the sum of all the terms in the distribution is $312$ therefore, $S = 312$.
Now since $3$ was added to all the numbers in the distribution and there are $13$ numbers in the distribution, we will add $3$ total $13$ times to the sum $S$.
The new total $S$ will become:
$\Rightarrow S = 312 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3$
On simplifying the equation, we get:
$\Rightarrow S = 351$
Since the new Sum of terms is found, we find the new mean by using the formula:
$M = \dfrac{S}{N}$
On substituting the values of $S$ and $N$ we get:
$M = \dfrac{{351}}{{13}}$
On simplifying we get:
$M = 27$
Now the older mean was $24$ and the new mean is $27$ therefore we now find the change in mean:
Change in mean $= 27 - 24$
Change in mean $= 3$

$\therefore$The change in mean is $3$

Note: Whenever there is a same number added to all the distributive numbers then it can change in mean will always be the number which is added to all the distributive numbers, in this case the number was $3$.