Question

The standard deviation of the following distribution}

is

• A. $5\sqrt{2}$
• B. $\sqrt{5}$
• C. $2\sqrt{5}$
• D. $20$

Hint:

Logic Tip: The standard deviation is simply the Root Mean Square (RMS) deviation from the mean. Always remember the order of operations: mean of squares minus square of the mean. $\text{Variance} = E(X^2) - [E(X)]^2$.

Answer 1

The Correct Option is C

Concept:
To find the standard deviation for grouped data, we use: \[ \sigma = \sqrt{\frac{\sum f_i x_i^2}{N} - \left(\frac{\sum f_i x_i}{N}\right)^2} \] where $x_i$ is the class mark, $f_i$ is the frequency, and $N = \sum f_i$. 
Step 1: Construct the frequency table.
Class marks: \[ x_i = 3,\ 9,\ 15 \]

C.I.	$f_i$	$x_i$	$x_i^2$	$f_i x_i$	$f_i x_i^2$
0–6	2	3	9	6	18
6–12	4	9	81	36	324
12–18	6	15	225	90	1350
Total	$N=12$			$132$	$1692$

Step 2: Calculate variance.
\[ V(X) = \frac{1692}{12} - \left(\frac{132}{12}\right)^2 \] \[ V(X) = 141 - 121 = 20 \] 
Step 3: Standard deviation.
\[ \sigma = \sqrt{20} = 2\sqrt{5} \] 
Final Answer:
\[ \boxed{2\sqrt{5}} \]