Question

The mean of 100 observation was found to be 40. Later on, it was discovered that tow items were wrongly taken as 30 and 27 instead of 3 and 72. Find correct mean.

Answer 1

Given:

Mean, $\overline{\mathbf{X}}$=40
Number of observations, N=100
Incorrect observations= 30 and 27
Correct observations = 3 and 72
Wrong X=Wrong ΣXNor, 89.4=Wrong ΣX20⇒Wrong ΣX=1
$$\begin{gathered} (\overline{\mathrm{X}}{)}_{wrong}=\frac{(\mathrm{\Sigma X}{)}_{wrong}}{N} \\ or,40=\frac{(\mathrm{\Sigma X}{)}_{wrong}}{100} \\ (\mathrm{\Sigma X}{)}_{wrong}=40\times 100=4000 \end{gathered}$$

Thus, wrong summation of X, i.e., $\mathbf{(}\mathit{\Sigma }\mathit{X}{\mathbf{)}}_{\mathbf{w}\mathbf{r}\mathbf{o}\mathbf{n}\mathbf{g}}$=4000
Now, correct mean is calculated using the following formula.

$$\begin{gathered} \mathrm{Correct}\overline{X}=\frac{\Sigma X_{\left(wrong\right)}+\mathrm{correct}\mathrm{values}-\mathrm{incorrect}\mathrm{values}}{N} \\ \mathrm{Substituting}\mathrm{the}\mathrm{given}\mathrm{values}\mathrm{in}\mathrm{the}\mathrm{formula}. \\ Correct\overline{X}=\frac{4000-30-27+3+72}{100} \\ \mathrm{Correct}\overline{\mathrm{X}}=\frac{4018}{100}=40.18 \end{gathered}$$

Hence, the correct mean is 40.18