Question

The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations were omitted.

Answer 1

$$\begin{gathered} n=100 \\ \mathrm{Mean}=\overline{X}\mathit{}=20 \\ \mathrm{SD}=\sigma =3 \\ \mathrm{M}\mathrm{isread}\mathrm{values}\mathrm{are}21,21\mathrm{and}18. \\ \underset{}{\frac{\mathit{1}}{n}\mathit{\sum }}{x}_{\mathrm{i}}=\overline{X}\mathit{} \\ \Rightarrow \frac{1}{100}\underset{}{\sum x_i=20} \\ \Rightarrow \underset{}{\sum x_i\mathit{}=20\times 100=2000}\left[\mathrm{This}\mathrm{sum}\mathrm{is}\mathrm{incorrect}\mathrm{due}\mathrm{to}\mathrm{misread}\mathrm{values}.\right]....\left(1\right) \\ \mathrm{If}\mathrm{three}\mathrm{misread}\mathrm{values}\mathrm{are}\mathrm{to}\mathrm{be}\mathrm{omitted},\mathrm{the}\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{enteries}\mathrm{will}\mathrm{be}97. \\ \mathrm{Also},\sum x_i=2000-\left(21-21-18\right)=1940 \\ \mathrm{Corrected}\overline{\mathit{}X}\mathit{}=\frac{1940}{97}=20....\left(2\right) \end{gathered}$$

$$\begin{gathered} \sigma =3 \\ \Rightarrow \mathrm{Variance}={\sigma }^2=9 \\ =\mathrm{Variance}=\frac{1}{n}\sum _{}{x_i}^2-{\left(\overline{X}\right)}^2 \\ \Rightarrow \frac{1}{100}\sum _{}{x_i}^2-{20}^2=9 \\ \Rightarrow \frac{1}{100}\sum _{}{x_i}^2=9+400 \\ \Rightarrow \frac{1}{100}\sum _{}{x_i}^2=409 \\ \Rightarrow \mathit{\sum }_{}{x_i}^2=409\times 100=40900\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left(\mathrm{This}\mathrm{is}\mathrm{an}\mathrm{incorrect}\mathrm{sum}\mathrm{due}\mathrm{to}\mathrm{misread}\mathrm{values}.\right)...\left(2\right) \\ \mathrm{Corrected}\sum _{}{x_i}^2=40900-\left({21}^2+{21}^2+{18}^2\right) \\ =40900-441-441-324 \\ =39694\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}...\left(3\right) \\ \mathrm{From}\mathrm{equations}\left(2\right)\mathrm{and}\left(3\right),\mathbf{}\mathrm{we}\mathrm{get}: \\ \mathrm{Corrected}\mathrm{variance}=\frac{1}{n}\sum _{}{x_i}^2-{\left(\overline{X}\right)}^2 \\ =\frac{1}{97}\times 39694-{\left(20\right)}^2 \\ =409.216-400 \\ =9.216 \\ \mathrm{Corrected}\mathrm{SD}=\sqrt{\mathrm{Corrected}\mathrm{variance}} \\ =\sqrt{9.216} \\ =3.0357 \end{gathered}$$

Thus, after omitting three values, the mean would be 20 and SD would be 3.0357.