Q87 / 180 · COMEDK 2026 (9 May Shift 1) · Mathematics
If for a distribution of 20 items, $\sum(x - 4) = 10$ and $\sum(x - 4)^2 = 85$ then the standard deviation is:
- A$1.41$
- Correct answerB$2$
- C$4$
- D$4.25$
Solution
Let $y_i = x_i - 4$.
Given $n = 20$, $\sum y_i = 10$, and $\sum y_i^2 = 85$.
The standard deviation of $x$ is equal to the standard deviation of $y$ since it is independent of the change of origin.
Variance $\sigma^2 = \dfrac{\sum y_i^2}{n} - \left(\dfrac{\sum y_i}{n}\right)^2$
$\sigma^2 = \dfrac{85}{20} - \left(\dfrac{10}{20}\right)^2$
$\sigma^2 = 4.25 - (0.5)^2$
$\sigma^2 = 4.25 - 0.25 = 4$
Standard deviation $\sigma = \sqrt{4} = 2$.
Answer: $2$
Given $n = 20$, $\sum y_i = 10$, and $\sum y_i^2 = 85$.
The standard deviation of $x$ is equal to the standard deviation of $y$ since it is independent of the change of origin.
Variance $\sigma^2 = \dfrac{\sum y_i^2}{n} - \left(\dfrac{\sum y_i}{n}\right)^2$
$\sigma^2 = \dfrac{85}{20} - \left(\dfrac{10}{20}\right)^2$
$\sigma^2 = 4.25 - (0.5)^2$
$\sigma^2 = 4.25 - 0.25 = 4$
Standard deviation $\sigma = \sqrt{4} = 2$.
Answer: $2$
Hence, the correct answer is option: 'B'.