Question

The average height of 30 boys was calculated to be 150 cm. It was detected later that one value of 165 cm was wrongly copied as 135 cm for the computation of the mean. Find the correct mean.

Answer 1

We know that,

$\mathrm{Mean}=\frac{\mathrm{Sum}\mathrm{of}\mathrm{observa}\mathrm{tions}}{\mathrm{Number}\mathrm{of}\mathrm{observations}}$
Mean of height of 30 boys = $\frac{{\displaystyle \sum _{i=1}^{30}{x}_i}}{30}$
$$\begin{gathered} \Rightarrow 150=\frac{{\displaystyle \sum _{i=1}^{30}}{x}_i}{30} \\ \Rightarrow \sum _{i=1}^{30}{x}_i=150\times 30 \\ \Rightarrow \sum _{i=1}^{30}{x}_i=4500...\left(1\right) \end{gathered}$$
It was detected later that one value of 165 cm was wrongly copied as 135 cm for the computation of the mean.
$$\begin{gathered} \therefore \mathrm{The}\mathrm{correct}\mathrm{mean}=\frac{{\displaystyle \underset{i=1}{\overset{30}{\sum x_i}}-135+165}}{30} \\ =\frac{{\displaystyle \underset{i=1}{\overset{30}{\sum x_i}}+30}}{30} \\ =\frac{{\displaystyle 4500+30}}{30}\left(\mathrm{from}\left(1\right)\right) \\ =\frac{{\displaystyle 4530}}{30} \\ =151 \end{gathered}$$
Hence, the correct mean is 151.