If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and 20/3, respectively, then the value of | a - b | is equal to :

Solution

Maths sir · 4 steps

Measure everything from the mean 10, and you need just the sum and the squares of two small deviations.

1. Use the mean. Six observations with mean 10 add up to 60. The known four give 7 + 10 + 11 + 15 = 43, so a + b = 17.
2. Use the variance. Variance 20/3 means the squared deviations from 10 add up to 6 × 20/3 = 40. The known four give 9 + 0 + 1 + 25 = 35. So a and b must give the remaining 5.
3. Set up deviations. Let p = a - 10 and q = b - 10. Then p + q = 17 - 20 = -3 and p² + q² = 5. Also p - q = a - b, because the 10s cancel.

   (p - q)² = 2(p² + q²) - (p + q)² = 10 - 9 = 1

4. So. |a - b| = |p - q| = 1, which is option (d). As a check, a and b are 9 and 8, and 81 + 64 = 145 = 640 - 495, as required.

The idea to keep. When two observations are unknown, the mean gives their sum and the variance gives the sum of their squares. Then (a-b)² = 2(a²+b²) - (a+b)² finishes the job.

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