Standard Deviation for the marks obtained by a student in monthly test in mathematic (out of $50$) as $30, 35, 25, 20, 15$ is

We are given the marks: $30, 35, 25, 20, 15$. The number of observations is $n = 5$. 1. Calculate the arithmetic mean ($\displaystyle$$\displaystyle \bar{x}$): $$\bar{x}=\frac{30+35+25+20+15}{5}=\frac{125}{5}=25$$$\displaystyle \bar{x} = \frac{30 + 35 + 25 + 20 + 15}{5} = \frac{125}{5} = 25$ 2. Calculate the deviations from the mean $\displaystyle (x_i -$$\displaystyle \bar{x}$) and their squares $\displaystyle (x_i -$$\displaystyle \bar{x}$)^2: - $30 - 25 = 5 \implies 25$ - $35 - 25 = 10 \implies 100$ - $25 - 25 = 0 \implies 0$ - $20 - 25 = -5 \implies 25$ - $15 - 25 = -10 \implies 100$ 3. Calculate the sum of squared deviations: $$\sum (x_i-\bar{x}{)}^2=25+100+0+25+100=250$$$\displaystyle \sum$ (x_i - $\displaystyle \bar{x}$)^2 = 25 + 100 + 0 + 25 + 100 = 250 4. Calculate variance ($\displaystyle$$\displaystyle \sigma^2$) and Standard Deviation ($\displaystyle$$\displaystyle \sigma$): $${\sigma }^2=\frac{250}{5}=50⟹\sigma =\sqrt{50}$$$\displaystyle \sigma^2 = \frac{250}{5} = 50$ \implies $\displaystyle \sigma = \sqrt{50}$ Hence, **Option B** is the correct answer.