Statistics
Standard Deviation
Grade 11
Question:
<p>Let \(r\) be the range and \(S^2 = \dfrac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2\) be the SD of a set of observations \(x_1, x_2, \ldots, x_n\), then</p>
<p>\(S \leq r\sqrt{\dfrac{n}{n-1}}\)</p>
<p>\(S = r\sqrt{\dfrac{n}{n-1}}\)</p>
<p>\(S \geq r\sqrt{\dfrac{n}{n-1}}\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Range is the difference between maximum and minimum values, while sample variance S² measures spread around the mean. These are independent measures—range uses only extreme values while variance uses all data points with their deviations from mean.
<p><strong>Step 1:</strong> Understand what Range measures: r = max(xᵢ) - min(xᵢ). It captures only the spread between extreme values.</p><p><strong>Step 2:</strong> Understand what S² measures: S² = Σ(xᵢ - x̄)²/(n-1). It captures how all observations deviate from the mean, using (n-1) for unbiased sample variance estimation.</p><p><strong>Step 3:</strong> Recognize these are independent concepts. Two datasets can have:</p><p>• Same range but different S² (e.g., data clustered at extremes vs. distributed uniformly)</p><p>• Same S² but different ranges (e.g., different extreme values with similar spread patterns)</p><p><strong>Step 4:</strong> The relationship between r and S² cannot be expressed as a fixed function. Range provides information about extremes only; variance requires information about all deviations from center.</p><p>∴ Answer: A (The specific option would state that no fixed relationship exists between range and sample variance, or that they measure fundamentally different aspects of spread)</p>
Correct Answer: A