The mean and standard deviation of 20 observations are found to be 10 and 2 , respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

Solution

Maths sir · 3 steps

Go back to the totals: fix Σ x and Σ x² first, then rebuild the variance from them.

1. Recover the totals. With n = 20 and mean 10, Σ x = 20 × 10 = 200. The variance is 2² = 4, and σ² = (Σ x²)/n - x̄², so Σ x² = 20 × (4 + 100) = 2080.
2. Swap 8 for 12. Correct sum: 200 - 8 + 12 = 204, so the new mean is 204/20 = 10.2. Correct sum of squares: 2080 - 64 + 144 = 2160.

   σ² = 2160/20 - (10.2)² = 108 - 104.04 = 3.96

3. So. The correct standard deviation is √3.96, which is option (b).

The idea to keep. When a data entry is wrong, never patch the mean or SD directly. Rebuild Σ x and Σ x², correct both, then use σ² = (Σ x²)/n - x̄² with the new mean.

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