Question

For 10 observations \(x_1, x_2, \dots, x_{10}\), if \(\sum_{i=1}^{10} (x_i + 2)^2 = 180\) and \(\sum_{i=1}^{10} (x_i - 1)^2 = 90\), then their standard deviation is:

• A. 2
• B. \(\sqrt{3}\)
• C. \(2\sqrt{2}\)
• D. 3

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The standard deviation relies on the variance, which requires finding both \(\sum x_i^2\) and \(\sum x_i\). By expanding the two given summations, we get two linear equations which can be solved simultaneously.
Step 2: Key Formula or Approach:
Variance formula:
\[ \sigma^2 = \frac{\sum x_i^2}{n} - \left( \frac{\sum x_i}{n} \right)^2 \] Expansion of squares: \((x_i \pm a)^2 = x_i^2 \pm 2ax_i + a^2\).
Step 3: Detailed Explanation:
Let \(S_2 = \sum x_i^2\) and \(S_1 = \sum x_i\) for \(n=10\).
Expand the first summation:
\[ \sum (x_i^2 + 4x_i + 4) = 180 \implies S_2 + 4S_1 + 40 = 180 \implies S_2 + 4S_1 = 140 \quad \text{(Eq 1)} \] Expand the second summation:
\[ \sum (x_i^2 - 2x_i + 1) = 90 \implies S_2 - 2S_1 + 10 = 90 \implies S_2 - 2S_1 = 80 \quad \text{(Eq 2)} \] Subtract Eq 2 from Eq 1:
\[ 6S_1 = 60 \implies S_1 = 10 \] Substitute \(S_1\) back into Eq 2:
\[ S_2 - 20 = 80 \implies S_2 = 100 \] Calculate the variance \(\sigma^2\):
\[ \sigma^2 = \frac{100}{10} - \left( \frac{10}{10} \right)^2 = 10 - 1 = 9 \] Standard deviation \(\sigma = \sqrt{9} = 3\).
Step 4: Final Answer:
The standard deviation is 3.