Question

Mean of a certain number of observation is $\overline{x}$. If each observation is divided by m(m ≠ 0) and increased by n, then the mean of new observation is

(a) $\frac{\overline{x}}{m}+n$
(b) $\frac{\overline{x}}{n}+m$
(c) $\overline{x}+\frac{n}{m}$
(d) $\overline{x}+\frac{m}{n}$

Answer 1

Let $y_1,y_2,y_3,...,y_k$ be k observations.

Mean of the observations = $\overline{x}$

$$\begin{gathered} \Rightarrow \frac{y_1+y_2+y_3+...+y_k}{k}=\overline{x} \\ \Rightarrow y_1+y_2+y_3+...+y_k=k\overline{x}.....\left(1\right) \end{gathered}$$

If each observation is divided by m and increased by n, then the new observations are

$\frac{y_1}{m}+n,\frac{y_2}{m}+n,\frac{y_3}{m}+n,...,\frac{y_k}{m}+n$

∴ Mean of new observations

$$\begin{gathered} =\frac{\left({\displaystyle \frac{y_1}{m}}+n\right)+\left({\displaystyle \frac{y_2}{m}}+n\right)+...+\left({\displaystyle \frac{y_k}{m}}+n\right)}{k} \\ =\frac{\left({\displaystyle \frac{y_1}{m}+\frac{y_2}{m}+...+\frac{y_k}{m}}\right)+\left({\displaystyle n+n+...+n}\right)}{k} \\ =\frac{y_1+y_2+...+y_k}{mk}+\frac{nk}{k} \\ =\frac{k\overline{x}}{mk}+\frac{nk}{k} \\ =\frac{\overline{x}}{m}+n \end{gathered}$$

Hence, the correct option is (a).