The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is 70. If the correct mean is 16, then the correct variance is equal to :

Solution

Maths sir · 4 steps

Find which number was wrong first, then fix both the sum and the sum of squares.

1. Fix the sum. The old sum is 50 × 15 = 750 and the correct sum is 50 × 16 = 800. So the correct observation is 50 more than the incorrect one.
2. Find both values. With c - w = 50 and c + w = 70, the correct value is c = 60 and the incorrect value is w = 10.

   σ² = (Σ x²)/n - x̄² ⇒ Σ x² = 50(4 + 15²) = 11450

3. Fix the squares. Remove 10² = 100 and add 60² = 3600. The correct Σ x² = 11450 - 100 + 3600 = 14950.
4. So. Correct variance = 14950/50 - 16² = 299 - 256 = 43, which is option (c).

The idea to keep. When an observation is corrected, update both Σ x and Σ x², then use the new mean in (Σ x²)/n - x̄².

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