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

Step 1: Understanding the Concept:
Standard Deviation (\(\sigma\)) is invariant under a shift of origin. Thus, \(SD(x_{i}) = SD(x_{i} - 5)\). We can treat \(d_{i} = x_{i} - 5\) as our primary data.
Step 2: Key Formula or Approach:
\[ \text{Variance } (\sigma^{2}) = \frac{\sum d_{i}^{2}}{n} - \left( \frac{\sum d_{i}}{n} \right)^{2} \]
Step 3: Detailed Explanation:
1. Given \(n = 9, \sum d_{i} = 9, \sum d_{i}^{2} = 45\).
2. Calculate Variance:
\[ \sigma^{2} = \frac{45}{9} - \left( \frac{9}{9} \right)^{2} \]
\[ \sigma^{2} = 5 - (1)^{2} = 5 - 1 = 4 \]
3. Standard Deviation is the positive square root of variance:
\[ \sigma = \sqrt{4} = 2 \]
Step 4: Final Answer:
The standard deviation is 2.