The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to :

Solution

Maths sir · 4 steps

One change in one mark gives you n, then work with the sum of squares, not the variance directly.

1. Find n. Raising one mark from 8 to 12 adds 4 to the total, so the mean rises by 4/n. That rise is 10.2 - 10 = 0.2, so 4/n = 0.2 and n = 20.
2. Old sum of squares. Use σ² = (Σ x²)/n - x̄². Then 4 = (Σ x²)/20 - 100, so Σ x² = 104 × 20 = 2080.
3. Swap the mark. Remove 8² = 64 and add 12² = 144. The new sum of squares is 2080 - 64 + 144 = 2160.

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

4. So. The new variance is 3.96, which is option (c).

The idea to keep. When a data value is changed, update Σ x and Σ x² by removing the old value and adding the new one. Then rebuild the mean and variance from those sums.

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