Question

Find out the standard deviation of the marks secured by $10$ students:

$\begin{array}{ccccccccccc}\text{S. No}& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ \text{Marks}& 43& 48& 65& 57& 31& 60& 37& 48& 78& 59\end{array}$

Answer 1

$\begin{array}{cccc}\text{S. No}& \text{Marks}\left(x\right)& d_i=x-A& {d_i}^2\\ 1& 43& -5& 25\\ 2& 48->\left(A\right)& 0& 0\\ 3& 65& 17& 289\\ 4& 57& 9& 81\\ 5& 31& -17& 289\\ 6& 60& 12& 144\\ 7& 37& -11& 121\\ 8& 48& 0& 0\\ 9& 78& 30& 900\\ 10& 59& 11& 121\\ & & \sum d_i=46& \sum {d_i}^2=1970\end{array}$

Now,

Standard deviation $\left(\sigma \right)=\sqrt{\frac{\sum {d_i}^2}{N}-(\frac{\sum d_i}{N}{)}^2}$

Here, $N=10$ [Number of students]

$\Rightarrow \left(\sigma \right)=\sqrt{\frac{1970}{10}-(\frac{46}{10}{)}^2}$

$\Rightarrow \left(\sigma \right)=\sqrt{197-21.16}$

$\Rightarrow \left(\sigma \right)=\sqrt{175.84}$

$\therefore \left(\sigma \right)=13.26$