Statistics
Variance and Standard Deviation
Grade 11
Question:
<p>For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is</p>
<p>\(\dfrac{11}{2}\)</p>
<p>6</p>
<p>\(\dfrac{13}{2}\)</p>
<p>\(\dfrac{5}{2}\)</p>
Step-by-Step Solution
Key Concept: The variance of combined data sets requires using the formula: Var(combined) = [n₁Var₁ + n₂Var₂]/N + [n₁(m₁ - m_combined)² + n₂(m₂ - m_combined)²]/N, where the second term accounts for deviation of individual means from the overall mean.
<p><strong>Step 1:</strong> Find the combined mean.</p><p>m_combined = (n₁m₁ + n₂m₂)/(n₁ + n₂) = (5·2 + 5·4)/(5 + 5) = (10 + 20)/10 = 3</p><p><strong>Step 2:</strong> Apply the combined variance formula.</p><p>Var(combined) = [n₁Var₁ + n₂Var₂ + n₁(m₁ - m_combined)² + n₂(m₂ - m_combined)²]/(n₁ + n₂)</p><p><strong>Step 3:</strong> Substitute values.</p><p>Var(combined) = [5(4) + 5(5) + 5(2-3)² + 5(4-3)²]/10</p><p>= [20 + 25 + 5(1) + 5(1)]/10</p><p>= [20 + 25 + 5 + 5]/10</p><p>= 55/10 = 5.5</p><p><strong>∴ Answer: A (5.5)</strong></p>
Correct Answer: A