Statistics
Statistics
nta_abhyas_2025
Grade 11
Question:
If $\sum_{i=1}^{n}(x_i - 6) = 5$ and $\sum_{i=1}^{n}(x_i - 6)^2 = 25$, then the standard deviation of observations $3x_1 + 2, 3x_2 + 2, 3x_3 + 2, 3x_4 + 2$ and $3x_5 + 2$ is equal to
Step-by-Step Solution
Key Concept: For a linear transformation $aX + b$, the variance becomes $a^2\text{var}(X)$ and standard deviation becomes $|a|\cdot\text{SD}(X)$.
For variance of the linear transformation $(3x_i + 2)$: $\text{var}(3x_i + 2) = 9\text{var}(x_i) = 9\left(\frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2\right) = 9(5 - 1) = 9 \times 4 = 36$. Therefore, the standard deviation is $\sqrt{36} = 6$.
Correct Answer: 6