GATE ME 2016 Set 2 — Question 6
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Engineering Mathematics → Statistics & Distributions → Mean, Median, Mode & SD
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Question
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?
Correct answer
(C) Most students of group Q scored marks in a narrower range than students in group P.
Solution
Let's analyze each statement:
(A) and (B): In a normal distribution, the tails extend to infinity. While the probability of very low or very high scores is small, it is never zero. Therefore, absolute statements like "No student..." are almost certainly false for large groups.
(C): Standard deviation (SD) is a measure of dispersion or spread. Group Q has an SD of 5, while group P has an SD of 25. A smaller SD means the data points are more clustered around the mean. Thus, most students in group Q will have marks in a much narrower range (e.g., for 99.7% of students) compared to group P (). This statement is highly likely to be true.
(D): For a normal distribution, the mean, median, and mode are all equal. Since the mean of group P is 105, its median must also be 105, not 100.
(A) and (B): In a normal distribution, the tails extend to infinity. While the probability of very low or very high scores is small, it is never zero. Therefore, absolute statements like "No student..." are almost certainly false for large groups.
(C): Standard deviation (SD) is a measure of dispersion or spread. Group Q has an SD of 5, while group P has an SD of 25. A smaller SD means the data points are more clustered around the mean. Thus, most students in group Q will have marks in a much narrower range (e.g., for 99.7% of students) compared to group P (). This statement is highly likely to be true.
(D): For a normal distribution, the mean, median, and mode are all equal. Since the mean of group P is 105, its median must also be 105, not 100.
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