Statistics
Variance and Standard Deviation
Grade 11
Question:
<p>Suppose a population A has 100 observations 101, 102, …, 200, and another population B has 100 observations 151, 152, …, 250. If \(V_A\) and \(V_B\) represent the variances of the two populations, respectively, then \(\dfrac{V_A}{V_B}\) is</p>
<p>1</p>
<p>\(\dfrac{9}{4}\)</p>
<p>\(\dfrac{4}{9}\)</p>
<p>\(\dfrac{2}{3}\)</p>
Step-by-Step Solution
Key Concept: Variance is invariant under translation (shifting all values by a constant doesn't change variance), so both populations have the same spread pattern. The ratio of variances depends only on the relative spacing of observations, which is identical for both populations.
<p><strong>Step 1:</strong> Identify the structure of both populations.</p><p>Population A: 101, 102, 103, ..., 200 (100 consecutive integers)</p><p>Population B: 151, 152, 153, ..., 250 (100 consecutive integers)</p><p><strong>Step 2:</strong> Observe that Population B = Population A + 50 (each element shifted by 50).</p><p><strong>Step 3:</strong> Apply the translation invariance property of variance: If Y = X + c (constant shift), then Var(Y) = Var(X).</p><p>Therefore: V_B = Var(A + 50) = Var(A) = V_A</p><p><strong>Step 4:</strong> Calculate the ratio.</p><p>$$\frac{V_A}{V_B} = \frac{V_A}{V_A} = 1$$</p><p>∴ Answer: A (which is 1)</p>
Correct Answer: A