The mean and standard deviation of 100 observations are 40 and 5.1 , respectively. By mistake one observation is taken as 50 instead of 40 . If the correct mean and the correct standard deviation are μ and σ respectively, then 10(μ+σ) is equal to

Solution

Maths sir · 4 steps

Rebuild the raw totals Σ x and Σ x², swap 50 for 40 in both, then recompute.

1. Correct the sum. The wrong sum is 100 × 40 = 4000. Remove 50 and put in 40: the correct sum is 3990, so μ = 39.9.
2. Recover sum of squares. Variance is (Σ x²)/n - x̄², so the wrong (Σ x²)/100 = 5.1² + 40² = 26.01 + 1600 = 1626.01. That gives the wrong Σ x² = 162601.
3. Correct sum of squares. Remove 50² = 2500 and add 40² = 1600: the correct Σ x² = 162601 - 2500 + 1600 = 161701.

   σ² = 161701/100 - (39.9)² = 1617.01 - 1592.01 = 25

4. So. σ = 5 and μ = 39.9, so 10(μ + σ) = 10 × 44.9 = 449, option (c).

The idea to keep. Mean and SD cannot be corrected directly. Go back to Σ x and Σ x², fix the wrong entry in both, and use the corrected mean when you compute the new variance.

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