The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and σ² respectively. If the variance of all the 30 numbers in the two sets is 13 , then σ² is equal to :

Solution

Maths sir · 4 steps

Do not just average the two variances; turn each into a sum of squares, add them, and solve.

1. Combined mean. Both sets have 15 numbers, so the mean of all 30 is the plain average of the two means: (12 + 14)/2 = 13.
2. Sum of squares. Variance is (Σ x²)/n - x̄², so Σ x² = n(σ² + x̄²). For the first set, Σ x² = 15(14 + 144) = 2370. For the second set, Σ y² = 15(σ² + 196).

   (2370 + 15(σ² + 196))/30 - 13² = 13

3. Solve. The top must be 30 × 182 = 5460. So 2370 + 15σ² + 2940 = 5460, which gives 15σ² = 150.
4. So. σ² = 10, which is option (c).

The idea to keep. To combine variances, work with Σ x² for each group, or add the extra term (n₁ n₂ (x̄₁ - x̄₂)²)/(n₁+n₂)². Different means always spread the combined data more.

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