Statistics
Variance
Grade 11
Question:
<p>5 students of a class have an average height 150 cm and variance 18 cm². A new student, whose height is 156 cm, joined them. The variance (in cm²) of the height of these six students is ______.</p>
Step-by-Step Solution
Key Concept: When adding a new data point, recalculate variance using the formula Var = (Σx²/n) - (mean)². The key is computing the new sum of squares and new mean correctly.
<p><strong>Step 1:</strong> Find initial sum of squares for 5 students.</p><p>Given: n₁ = 5, mean₁ = 150 cm, variance₁ = 18 cm²</p><p>Variance = (Σx²/n) - (mean)²</p><p>18 = (Σx²/5) - 150²</p><p>18 = (Σx²/5) - 22500</p><p>Σx² = 5(18 + 22500) = 5(22518) = 112590</p><p><strong>Step 2:</strong> Add new student and recalculate.</p><p>New sum of heights = 5(150) + 156 = 750 + 156 = 906 cm</p><p>New mean = 906/6 = 151 cm</p><p>New sum of squares = 112590 + 156² = 112590 + 24336 = 136926</p><p><strong>Step 3:</strong> Calculate new variance.</p><p>Variance₂ = (136926/6) - 151²</p><p>Variance₂ = 22821 - 22801</p><p>Variance₂ = 20 cm²</p><p>∴ Answer: 20</p>
Correct Answer: 20