Question

If \( \sum_{i=1}^{9} (x_i - 5) = 9 \) and \( \sum_{i=1}^{9} (x_i - 5)^2 = 45 \), then the standard deviation of the 9 items \(x_1, x_2, \ldots, x_9\) is:

• A. 9
• B. 4
• C. 3
• D. 2

Hint:

Standard deviation does not change with shifting (\(x - a\)), only with scaling.

Answer 1

The Correct Option is D

Concept: Standard deviation is independent of change of origin. Let: \[ y_i = x_i - 5 \] Then: \[ \sigma_x = \sigma_y \]

Step 1: Given data \[ \sum y_i = 9, \quad \sum y_i^2 = 45, \quad n = 9 \]

Step 2: Mean \[ \bar{y} = \frac{9}{9} = 1 \]

Step 3: Variance \[ \sigma^2 = \frac{1}{n}\sum y_i^2 - \bar{y}^2 = \frac{45}{9} - (1)^2 = 5 - 1 = 4 \]

Step 4: Standard deviation \[ \sigma = \sqrt{4} = 2 \] Final: 2