Here n = 100, Incorrect mean $\left(\overset{-}{x}\right)$ = 20 and Incorrect S.D. (σ) = 3 $\overset{-}{x}$ = $\frac{1}{n} \sum x_i$ ⇒ $\sum x_i$ = n × $\overset{-}{x}$ = 100 × 20 = 2000 ∴ Incorrect $\sum x_i$ = 2000 Now $\frac{1}{n} \sum x_i^2 - \left(\overset{-}{x}\right)^2$ = 9 ⇒ $\frac{1}{100} \sum x_i^2 - (20)^2$ = 9 ⇒ $\sum x_i^2$ = 40900 ∴ Incorrect $\sum x_i^2$ = 40900 When wrong items 21, 21 and 18 are omitted from the data then we have 97 observations. ∴ Correct $\sum x_i$ = Incorrect $\sum x_i$ – 21 – 21 – 18 = 2000 – 21 – 21 – 18 = 1940 ∴ Correct mean = $\frac{1940}{97}$ = 20 Also correct $\sum x_i^2$ = Incorrect $\sum x_i^2$ - $(21)^2-(21)^2-(18)^2$ = 40900 - 441 - 441 - 324 = 39694 ∴ Correct variance = $\frac{1}{97} \left(\text{correct} \sum x_i^2\right) - (\text{correct mean})^2$ = $\frac{1}{97} \times 39694 - (20)^2$ = 409.22 - 400 = 9.22 Correct S.D. = $\sqrt{9.22}$ = 3.036