If the mean and variance of the following data : 6, 10, 7, 13, a, 12, b, 12 are 9 and 37/4
respectively, then (a - b)² is equal to :

Solution

Maths sir · 3 steps

The mean gives you a + b and the variance gives you a² + b². One identity then gives (a-b)².

1. Use the mean. There are 8 values, so their sum is 8 × 9 = 72. The known values add to 6 + 10 + 7 + 13 + 12 + 12 = 60, so a + b = 12.
2. Use the variance. Variance = (Σ x²)/8 - 9² = 37/4, so (Σ x²)/8 = 81 + 9.25 = 90.25 and Σ x² = 722. The known squares add to 36 + 100 + 49 + 169 + 144 + 144 = 642, so a² + b² = 80.

   (a-b)² = 2(a²+b²) - (a+b)² = 2 × 80 - 12²

3. So. (a-b)² = 160 - 144 = 16, which is option (d).

The idea to keep. With two unknowns in the data, the mean always gives their sum and the variance gives their sum of squares. Then (a-b)² = 2(a²+b²) - (a+b)² finishes it without solving for a and b.

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