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

Concept:
Variance and standard deviation are related to sums of observations using: \[ \sigma^2 = \frac{\sum x_i^2}{N} - \left(\frac{\sum x_i}{N}\right)^2 \] We expand the given expressions to obtain \( \sum x_i \) and \( \sum x_i^2 \).
Step 1: Expand the first expression
\[ \sum_{i=1}^{10}(x_i+2)^2 = 180 \] \[ \sum x_i^2 + 4\sum x_i + \sum 4 = 180 \] \[ \sum x_i^2 + 4\sum x_i + 40 = 180 \] \[ \sum x_i^2 + 4\sum x_i = 140 \] Step 2: Expand the second expression
\[ \sum_{i=1}^{10}(x_i-1)^2 = 90 \] \[ \sum x_i^2 - 2\sum x_i + \sum 1 = 90 \] \[ \sum x_i^2 - 2\sum x_i + 10 = 90 \] \[ \sum x_i^2 - 2\sum x_i = 80 \] Step 3: Solve the two equations
\[ \sum x_i^2 + 4\sum x_i = 140 \] \[ \sum x_i^2 - 2\sum x_i = 80 \] Subtracting, \[ 6\sum x_i = 60 \] \[ \sum x_i = 10 \] Substitute into second equation: \[ \sum x_i^2 - 2(10) = 80 \] \[ \sum x_i^2 = 100 \] Step 4: Calculate variance
\[ \sigma^2 = \frac{\sum x_i^2}{N} - \left(\frac{\sum x_i}{N}\right)^2 \] \[ \sigma^2 = \frac{100}{10} - \left(\frac{10}{10}\right)^2 \] \[ \sigma^2 = 10 - 1 = 9 \] Step 5: Standard deviation
\[ \sigma = \sqrt{9} = 3 \]