Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their variance is 20. A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of students can fail is _________.

Solution

Maths sir · 4 steps

Turn the variance into a total of squared deviations, then check if even one failure fits.

1. Total squared deviation. Variance is the mean of the squared deviations from the mean. With 7 students, Σ (xᵢ - 62)² = 7 × 20 = 140.
2. One failing student. A student who fails has x < 50, so the student is more than 62 - 50 = 12 marks below the mean. That one student's squared deviation is more than 12² = 144.

   (xₖ - 62)² ≤ Σᵢ₌₁⁷ (xᵢ - 62)² = 140 < 144

3. Contradiction. Every squared deviation is at least 0, so no single term can be larger than the whole sum of 140. A failing student would need more than 144 on their own, which is impossible.
4. So. Even in the worst case, no student can score below 50. The number of students who can fail is 0.

The idea to keep. Multiply the variance by n to get the total of squared deviations. No single student's squared deviation can exceed that total, which caps how far anyone can sit from the mean.

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