Q.6 Students taking an exam are divided into two groups, P and Q such that each group has the same
number of students. The performance of each of the students in a test was evaluated out of 200
marks. It was observed that the mean of group P was 105, while that of group Q was 85. The
standard deviation of group P was 25, while that of group Q was 5. Assuming that the marks were
distributed on a normal distribution, which of the following statements will have the highest
probability of being TRUE?
(A) No student in group Q scored less marks than any student in group P.
(B) No student in group P scored less marks than any student in group Q.
(C) Most students of group Q scored marks in a narrower range than students in group P.
(D) The median of the marks of group P is 100.
Two student groups, P and Q, share equal sizes with normally distributed marks out of 200: P has mean 105 and SD 25, Q has mean 85 and SD 5. Lower SD means tighter clustering around the mean, making option C most probable.
Option A Analysis
Option A claims no Q student scored below any P student, implying all Q marks exceed P’s lowest. P’s range spans roughly 105 – 3×25 = 30 to 105 + 3×25 = 180 (covering 99.7% of students), so lowest P scores near 30. Q’s tight range is 85 ± 15 (70-100), overlapping P’s low end heavily. Overlap probability exceeds 50%, so A is unlikely.
Option B Analysis
Option B states no P student scored below any Q student, or all P exceed Q’s highest. Q’s range reaches ~100, while P drops to ~30, creating massive overlap. With P’s wide spread, many P students score below Q’s mean of 85. This has near-zero probability.
Option C Analysis
Option C says most Q students scored in a narrower range than P. SD directly measures spread: Q’s SD 5 vs P’s 25 means ~68% of Q within 80-90, ~68% of P within 80-130. For normal distributions, lower SD guarantees most data clusters tighter around its mean, holding regardless of means. This is virtually certain (~100% probability).
Option D Analysis
Option D claims P’s median is 100. Normal distributions set median = mean = 105 exactly. No variation occurs, making D false always.
Correct Answer
C has the highest probability of being true, as standard deviation defines data dispersion in normal distributions.
