Question

Let in a series of 2n observations, half of them are equal to a and remaining half are equal to -a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of a² + b² is equal to :

• A. 925
• B. 425
• C. 650
• D. 250

Hint:

Standard deviation $(\sigma)$ is {invariant} under translation (adding/subtracting a constant) but {variant} under scaling.

Answer 1

The Correct Option is B

Step 1: Original observations: $n$ values of $a$ and $n$ values of $-a$. Original Mean $\bar{x} = \frac{n(a) + n(-a)}{2n} = 0$. New Mean $= \bar{x} + b = 0 + b = 5 \implies b = 5$.
Step 2: Standard deviation is unaffected by adding a constant. $SD = \sqrt{\frac{\sum x_i^2}{N} - \bar{x}^2} \implies 20 = \sqrt{\frac{n(a^2) + n(-a)^2}{2n} - 0^2} = \sqrt{\frac{2na^2}{2n}} = \sqrt{a^2} = |a|$. So, $a^2 = 20^2 = 400$.
Step 3: $a^2 + b^2 = 400 + 25 = 425$.