$$\begin{aligned} \because \quad \text { Mean } & =\frac{\Sigma x_i}{n}=\frac{1}{n}\left[\frac{n}{2}(2 a+(n-1) d]\right. \\ & =a+\frac{(n-1)}{2} d \end{aligned}$$

$$\begin{aligned} & \therefore \quad \Sigma\left(x_i-a\right)=d[1+2+3+\ldots+(n-1) d] \\ & =d \frac{(n-1) n}{2} \\ & \text { and } \\ & \Sigma\left(x_i-a\right)^2=d^2\left[1^2+2^2+3^2+\ldots+(n-1)^2\right] \\ & =\frac{d^2(n-1) n(2 n-1)}{6} \\ & \sigma=\sqrt{\frac{\left(x_i-\mathrm{a}\right)^2}{n}-\left(\frac{x_i-\mathrm{a}}{n}\right)^2} \\ & =\sqrt{\frac{d^2(n-1)(n)(2 n-1)}{6 n}-\left[\frac{d(n-1) n}{2 n}\right]^2} \\ & =\sqrt{\frac{d^2(n-1)(2 n-1)}{6}-\frac{d^2(n-1)^2}{4}} \\ & =d \sqrt{\frac{(n-1)(2 n-1)}{6}-\frac{(n-1)^2}{4}} \\ & =d \sqrt{\frac{(n-1)}{2}\left(\frac{2 n-1}{3}-\frac{n-1}{2}\right)} \\ & =d \sqrt{\frac{(n-1)}{2}\left[\frac{4 n-2-3 n+3}{6}\right]} \\ & =d \sqrt{\frac{(n-1)(n+1)}{12}}=d \sqrt{\frac{\left(n^2-1\right)}{12}} \end{aligned}$$

For Ravi,

$$\begin{aligned} \sigma & =\sqrt{\frac{\Sigma d^2 i}{n}-\left(\frac{\Sigma d_i}{n}\right)^2} \\ & =\sqrt{\frac{1699}{10}-\left(\frac{-14}{10}\right)^2}=\sqrt{169.9-0.0196} \\ & =\sqrt{169.88}=13.03 \\ \text{Now,}\quad \bar{x} & =A+\frac{\Sigma d_i}{\Sigma f_i}=45-\frac{14}{10}=43.6 \end{aligned}$$

For Hashina,

$$\begin{aligned} &\begin{array}{ll} \because & \text { Mean }=55 \\ \therefore & \sigma=\sqrt{\frac{6368}{10}}=\sqrt{636.8}=25.2 \\ \text { For Ravi, } & C V=\frac{\sigma}{\bar{x}} \times 100=\frac{13.03}{43.6} \times 100=29.88 \\ \text { For Hashina, } & C V=\frac{\sigma}{\bar{x}} \times 100=\frac{25.2}{55} \times 100=45.89 \end{array}\\ &\text { Hence, Hashina is more consistent and intelligent. } \end{aligned}$$

$$\begin{aligned} &\begin{aligned} \text { Given, }\quad n=100, \bar{x}=40, \sigma & =10 \text { and } \bar{x}=40 \\ \therefore\quad \frac{\Sigma x_i}{n} & =40 \\ \Rightarrow\quad \frac{\Sigma x_i}{100} & =40 \\ \Rightarrow\quad\Sigma x_i & =4000 \\ \text { Corrected } \Sigma x_i & =4000-30-70+3+27 \\ \therefore\quad & =4030-100=3930 \\ \text { Corrected mean } & =\frac{2930}{100}=39.3 \end{aligned} \end{aligned}$$

$$\begin{aligned} \text{Now,}\quad & \sigma^2=\frac{\Sigma x_i^2}{n}-(40)^2 \\ \Rightarrow\quad & 100=\frac{\Sigma x_i^2}{100}-1600 \\ \Rightarrow\quad & \Sigma x_i^2=170000 \\ & \text { Now, } \quad \text { Corrected } \Sigma x_i^2=170000-(30)^2-(70)^2+3^2+(27)^2 \\ & =164939 \\ & \text { Corrected } \sigma=\sqrt{\frac{164939}{100}-(39.3)^2} \\ & =\sqrt{1649.39-39.3 \times 39.3} \\ & =\sqrt{1649.39-1544.49} \\ & =\sqrt{104.9}=10.24 \end{aligned}$$

$$\begin{aligned} & \text { Given, } \quad n=10, \bar{x}=45 \text { and } \sigma^2=16 \\ & \therefore \quad \bar{x}=45 \Rightarrow \frac{\Sigma x_i}{n}=45 \\ & \Rightarrow \quad \frac{\Sigma x_i}{10}=45 \Rightarrow \Sigma x_i=450 \\ & \text { Corrected } \Sigma x_i=450-52+25=423 \\ \therefore\quad & \bar{x}=\frac{423}{10}=42.3 \\ & \sigma^2=\frac{\Sigma x_i^2}{n}-\left(\frac{\Sigma x_i}{n}\right)^2 \\ & 16=\frac{\Sigma x_i^2}{10}-(45)^2 \\ & \Sigma x_i^2=10(2025+16) \\ & \Sigma x_i^2=20410 \\ & \therefore \quad \text { Corrected } \Sigma x_i^2=20410-(52)^2+(25)^2=18331 \\ & \text { and } \quad \text { corrected } \sigma^2=\frac{18331}{10}-(42.3)^2=43.81 \end{aligned} $$