The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If σ is the standard deviation of the data after omitting the two wrong observations from the data, then 38 σ² is equal to ___________.

Solution

Maths sir · 4 steps

Turn mean and SD back into Σ x and Σ x², remove the two bad values, then rebuild the variance.

1. Recover the sums. With n = 40 and mean 30, Σ x = 40 × 30 = 1200. Variance is 25, so (Σ x²)/40 - 30² = 25, which gives Σ x² = 40 × 925 = 37000.
2. Remove 12 and 10. Now n = 38 and Σ x = 1200 - 22 = 1178, so the new mean is 1178/38 = 31. Also Σ x² = 37000 - 144 - 100 = 36756.

   38σ² = Σ x² - 38 x̄² = 36756 - 38 × 31²

3. Put in numbers. 38 × 961 = 36518, so 38σ² = 36756 - 36518 = 238.
4. So. 38σ² = 238.

The common slip: 2556. Keeping the old mean 30 instead of the new mean 31 gives 36756 - 38 × 900 = 2556, not 238.

The idea to keep. To add or remove observations, never work with mean and SD directly. Convert them to Σ x and Σ x², adjust those sums, then compute the new mean and variance.

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