Question

If \( \sum_{i=1}^{10} (x_i + 2)^2 = 180 \) and \( \sumᵢ=1¹0 (xᵢ - 1)² = 90 , then the Standard Deviation is equal to

• A. \(3\)
• B. \(2\)
• C. \(4\)
• D. \(5\)

Hint:

When expressions like \((x_i+a)^2\) or \((x_i-a)^2\) are given, expand them to form equations in \(\sum x_i\) and \(\sum x_i^2\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The variance (\( \sigma^2 \)) of a data set is given by the formula \( \sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2 \). The Standard Deviation is the square root of the variance. We can expand the given summations to find the required values.
Step 2: Key Formula or Approach:
1. Expand \( (x_i + 2)^2 \) and \( (x_i - 1)^2 \). 2. Subtract the two equations to find \( \sum x_i \). 3. Use \( \sum x_i \) to find \( \sum x_i^2 \). 4. Calculate \( \sigma = \sqrt{\frac{\sum x_i^2}{10} - \left(\frac{\sum x_i}{10}\right)^2} \).
Step 3: Detailed Explanation:
1. Expand the equations: (i) \( \sum (x_i^2 + 4x_i + 4) = 180 \implies \sum x_i^2 + 4\sum x_i + 40 = 180 \implies \sum x_i^2 + 4\sum x_i = 140 \) (ii) \( \sum (x_i^2 - 2x_i + 1) = 90 \implies \sum x_i^2 - 2\sum x_i + 10 = 90 \implies \sum x_i^2 - 2\sum x_i = 80 \) 2. Subtract (ii) from (i): \( (4\sum x_i) - (-2\sum x_i) = 140 - 80 \implies 6\sum x_i = 60 \implies \sum x_i = 10 \). 3. Find \( \sum x_i^2 \) using (ii): \( \sum x_i^2 - 2(10) = 80 \implies \sum x_i^2 = 100 \). 4. Calculate Variance: \( \sigma^2 = \frac{100}{10} - \left(\frac{10}{10}\right)^2 = 10 - 1 = 9 \). 5. Standard Deviation \( \sigma = \sqrt{9} = 3 \).
Step 4: Final Answer:
The Standard Deviation is 3.