The mean and variance of 7 observations are 8 and 16, respectively. If five observations are 2, 4, 10, 12, 14, then the absolute difference of the remaining two observations is :

Solution

Maths sir · 3 steps

Mean gives you x + y, variance gives you x² + y², and those two together give x − y.

1. Use the mean. Call the missing observations x and y. The total of all 7 is 7 × 8 = 56, and the five known ones add to 42, so x + y = 14.
2. Use the variance. Variance is (Σ xᵢ²)/n - x̄², so (Σ xᵢ²)/7 - 64 = 16 and Σ xᵢ² = 560. The known squares add to 4 + 16 + 100 + 144 + 196 = 460, so x² + y² = 100.

   (x - y)² = 2(x² + y²) - (x + y)² = 200 - 196 = 4

3. So. |x - y| = √4 = 2. The two observations are 8 and 6, and indeed 64 + 36 = 100. The answer is (a).

The idea to keep. When two observations are missing, the mean gives their sum and the variance gives their sum of squares. Then (x-y)² = 2(x²+y²) - (x+y)² finishes the job without solving for each one.

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