The mean and standard deviation of 15 observations are found to be 8 and 3 respectively. On rechecking it was found that, in the observations, 20 was misread as 5. Then, the correct variance is equal to _____________.

Solution

Maths sir · 3 steps

Rebuild the two totals, Σ x and Σ x², fix the misread value in both, then recompute.

1. Recover the totals. Mean 8 with 15 observations gives Σ x = 15 × 8 = 120. Variance is 3² = 9, and σ² = (Σ x²)/n - x̄², so (Σ x²)/15 = 9 + 64 = 73. That gives Σ x² = 1095.
2. Correct both totals. Take out the wrong 5 and put in the right 20. Then Σ x = 120 - 5 + 20 = 135 and Σ x² = 1095 - 25 + 400 = 1470.

   σ² = 1470/15 - (135/15)² = 98 - 81

3. So. The new mean is 9 and the correct variance is 98 - 81 = 17.

The common slip: 34. Subtracting the old mean squared, 64, instead of the new one, 81, gives 98 - 64 = 34, not 17.

The idea to keep. For any corrected-data question, go back to Σ x and Σ x². Swap the wrong value for the right one in both totals, and remember the mean changes too.

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