Question

The mean of 24 numbers is 35. If 3 is added to each number, what will be the new mean?

Answer 1

Let the numbers be x₁, x₂,...x₂₄.

We know:

$\mathrm{Mean}=\frac{\mathrm{Sum}\mathrm{of}\mathrm{observa}\mathrm{tions}}{\mathrm{Number}\mathrm{of}\mathrm{observations}}$

Thus, we have:

$35=\frac{x_1+x_2+.........+x_{24}}{24}$

_{$\Rightarrow 840=x_1+x_2+......+x_{24}.......\left(\mathrm{i}\right)$}


After addition, the new numbers become (x₁+3), (x₂+3),...(x₂₄+3).
New mean:

$$\begin{gathered} =\frac{\left(x_1+3\right)+\left(x_2+3\right)+..........+\left(x_{24}+3\right)}{24} \\ =\frac{(x_1+x_2+.........+x_{24})+\left(24\times 3\right)}{24} \\ =\frac{840+72}{24}[\mathrm{From}(\mathrm{i}\left)\right] \\ =\frac{912}{24} \\ =38 \end{gathered}$$