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:
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:
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Standard deviation measures the "spread" of data. Adding or subtracting a constant to all numbers shifts the whole data set but doesn't change how spread out they are, so the standard deviation stays the same.
Updated On: May 6, 2026
- \( 9 \)
- \( 4 \)
- \( 3 \)
- \( 2 \)
- \( 1 \)
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The Correct Option is D
Solution and Explanation
Concept:
Standard deviation \( \sigma \) is invariant under translation. This means \( \sigma(x_i) = \sigma(x_i - k) \). Let \( d_i = x_i - 5 \). Then the variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\sum d_i^2}{n} - \left( \frac{\sum d_i}{n} \right)^2 \]
Step 1: Identify the given values.
\( n = 9 \) \( \sum d_i = 9 \) \( \sum d_i^2 = 45 \)
Step 2: Calculate the variance.
\[ \sigma^2 = \frac{45}{9} - \left( \frac{9}{9} \right)^2 \] \[ \sigma^2 = 5 - (1)^2 = 5 - 1 = 4 \]
Step 3: Find the standard deviation.
\[ \sigma = \sqrt{4} = 2 \]
Step 1: Identify the given values.
\( n = 9 \) \( \sum d_i = 9 \) \( \sum d_i^2 = 45 \)
Step 2: Calculate the variance.
\[ \sigma^2 = \frac{45}{9} - \left( \frac{9}{9} \right)^2 \] \[ \sigma^2 = 5 - (1)^2 = 5 - 1 = 4 \]
Step 3: Find the standard deviation.
\[ \sigma = \sqrt{4} = 2 \]
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