Passage: The frequency distribution of marks obtained by 100...

The total number of students is $100$.

$10 + 20 + f + 40 = 100 \Rightarrow f = 30$

The mid-points of the class intervals are $x_i = 5, 15, 25, 35$.

The mean $\bar{x}$ is given by:

$\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i} = \dfrac{10(5) + 20(15) + 30(25) + 40(35)}{100}$

$\bar{x} = \dfrac{50 + 300 + 750 + 1400}{100} = \dfrac{2500}{100} = 25$

The variance $\sigma^2$ is given by:

$\sigma^2 = \dfrac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}$

$\sigma^2 = \dfrac{10(5 - 25)^2 + 20(15 - 25)^2 + 30(25 - 25)^2 + 40(35 - 25)^2}{100}$

$\sigma^2 = \dfrac{10(-20)^2 + 20(-10)^2 + 30(0)^2 + 40(10)^2}{100}$

$\sigma^2 = \dfrac{10(400) + 20(100) + 0 + 40(100)}{100}$

$\sigma^2 = \dfrac{4000 + 2000 + 0 + 4000}{100} = \dfrac{10000}{100} = 100$

The standard deviation is $\sigma = \sqrt{100} = 10$.

Answer: $10$