If the mean and variance of five observations are 24/5 and 194/25 respectively and the mean of the first four observations is 7/2, then the variance of the first four observations in equal to

Solution

Maths sir · 3 steps

Turn mean and variance into a total and a sum of squares, then remove the fifth observation.

1. Totals for five. Sum of all five is 5 × 24/5 = 24. Variance is the mean of squares minus the square of the mean, so (Σ xᵢ²)/5 = 194/25 + 576/25 = 770/25, which gives Σ xᵢ² = 154.
2. Find the fifth. The first four add up to 4 × 7/2 = 14, so x₅ = 24 - 14 = 10. Then the squares of the first four add up to 154 - 10² = 54.

   σ² = 54/4 - (7/2)² = 54/4 - 49/4 = 5/4

3. So. The variance of the first four observations is 5/4, which is option (a).

The idea to keep. Whenever observations are added or removed, work with the raw sums Σ xᵢ and Σ xᵢ². Change those, then rebuild the mean and the variance.

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