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:
Expand the given summation terms to find \(\sum x_i\) and \(\sum x_i^2\). Standard deviation is the square root of variance, and variance is independent of the choice of origin for mean.

Step 2: Key Formula or Approach:
1. Variance \(\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2\).
2. Expand \(\sum (x_i + a)^2\) as \(\sum x_i^2 + 2a \sum x_i + n a^2\).

Step 3: Detailed Explanation:
Let \(\sum x_i^2 = A\) and \(\sum x_i = B\).
1) \(A + 4B + 10(4) = 180 \implies A + 4B = 140\) ...(i)
2) \(A - 2B + 10(1) = 90 \implies A - 2B = 80\) ...(ii)
(i) - (ii) \(\implies 6B = 60 \implies B = 10 \implies \bar{x} = 1\).
From (ii), \(A - 20 = 80 \implies A = 100\).
\(\sigma^2 = \frac{100}{10} - (1)^2 = 10 - 1 = 9\).
\(\sigma = \sqrt{9} = 3\).

Step 4: Final Answer:
The standard deviation is 3.