The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation √13.44, then the standard deviation of the second sample is :

Solution

Maths sir · 4 steps

Find the second sample's mean first, then use the sum of squares of the whole group.

1. Second sample's mean. The second sample has 250 - 100 = 150 items. The total of all items is 250 × 15.6 = 3900 and the first sample gives 100 × 15 = 1500. So the second sample's mean is (3900 - 1500)/150 = 16.
2. Use sum of squares. For any set, Σ x² = n(σ² + x̄²). The sum of squares of the whole group is the sum of the two samples' sums of squares.

   250(13.44 + 15.6²) = 100(3² + 15²) + 150(σ₂² + 16²)

3. Put in numbers. Left side: 250 × (13.44 + 243.36) = 250 × 256.8 = 64200. First sample: 100 × 234 = 23400. So 150(σ₂² + 256) = 40800, which gives σ₂² + 256 = 272.
4. So. σ₂² = 16, and the standard deviation of the second sample is σ₂ = 4, option (c).

The idea to keep. To combine or split groups, you cannot add variances directly. Means add as totals, and squares add as n(σ² + x̄²), so work with those totals.

Answer from the NTA final answer key. Solution by PrepWiser.