Statistics
Standard Deviation
Grade 11
Question:
<p>If \(\displaystyle\sum_{i=1}^{9}(x_i - 5) = 9\) and \(\displaystyle\sum_{i=1}^{9}(x_i - 5)^2 = 45\), then the standard deviation of the nine items \(x_1, x_2, ..., x_9\) is</p>
<p>4</p>
<p>2</p>
<p>3</p>
<p>9</p>
Step-by-Step Solution
Key Concept: Use the given constraints to find the mean first: since Σ(xᵢ - 5) = 9, the mean is 5 + (9/9) = 6. Then apply the variance formula: σ² = Σ(xᵢ - 5)²/n - [Σ(xᵢ - 5)/n]².
<p><strong>Step 1:</strong> Find the mean using the first condition.</p><p>Σ(xᵢ - 5) = 9, so Σxᵢ = 9n + 45 where n = 9.</p><p>Mean = Σxᵢ/9 = (9 + 45)/9 = 54/9 = 6</p><p><strong>Step 2:</strong> Use the variance formula for shifted data.</p><p>If we denote yᵢ = xᵢ - 5, then Σyᵢ = 9 and Σyᵢ² = 45.</p><p>Variance of x = Σ(xᵢ - mean)²/n = Σ(xᵢ - 6)²/9</p><p><strong>Step 3:</strong> Expand using the identity: Σ(yᵢ - ȳ)² = Σyᵢ² - nȳ²</p><p>where yᵢ = xᵢ - 5, so ȳ = Σyᵢ/9 = 9/9 = 1</p><p>Σ(xᵢ - 6)² = Σ(yᵢ - 1)² = Σyᵢ² - 9(1)² = 45 - 9 = 36</p><p>Variance σ² = 36/9 = 4</p><p><strong>Step 4:</strong> Standard deviation σ = √4 = 2</p><p>∴ Answer: B</p>
Correct Answer: B