The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. if α and √β are the mean and standard deviation respectively for correct data, then (α, β) is :

Solution

Maths sir · 4 steps

Rebuild the sum and the sum of squares, fix the one wrong value in each, then recompute.

1. Fix the sum. Wrong sum = 20 × 10 = 200. Replacing 25 by 35 adds 10, so the correct sum is 210 and α = 210/20 = 10.5.
2. Recover sum of squares. Variance = 2.5² = 6.25. Use σ² = (Σ x²)/n - x̄², so (Σ x²)/20 = 6.25 + 100 = 106.25. That gives a wrong Σ x² = 2125.
3. Fix the squares. Remove 25² = 625 and add 35² = 1225. The correct Σ x² = 2125 - 625 + 1225 = 2725.

   β = 2725/20 - (10.5)² = 136.25 - 110.25 = 26

4. So. α = 10.5 and β = 26, which is option (d).

The idea to keep. When a data value is misread, do not correct the variance directly. Get back Σ x and Σ x², swap the wrong value for the right one in both, then recompute with the new mean.

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