Question

Which one among the following statements must be TRUE about the mean and the median of the scores of all candidates appearing for GATE 2023?

• A. The median is at least as large as the mean.
• B. The mean is at least as large as the median.
• C. At most half the candidates have a score that is larger than the median.
• D. At most half the candidates have a score that is larger than the mean.

Hint:

For datasets with skewed distributions, the mean and median can be quite different. The median divides the data into two equal parts, while the mean is influenced by the extreme values.

Answer 1

The Correct Option is C

Let's break down the given options and analyze them: - Option (A): "The median is at least as large as the mean."
- This is not necessarily true. In a skewed distribution (such as one with many lower scores and a few higher scores), the mean can be larger than the median. Thus, this statement is not universally true.
- Option (B): "The mean is at least as large as the median."
- This is also not necessarily true. In cases where the distribution is skewed to the left (more higher scores, few lower scores), the median can be larger than the mean. Hence, this statement is also not always true.
- Option (C): "At most half the candidates have a score that is larger than the median."
- This statement is true. By definition, the median divides the dataset into two equal halves. So, exactly half of the candidates will have scores smaller than the median, and the other half will have scores larger than the median. Therefore, at most half the candidates have a score larger than the median, which is always true.
- Option (D): "At most half the candidates have a score that is larger than the mean."
- This statement is not necessarily true. The mean is the average of all scores, and if the distribution is skewed, more than half of the candidates could have a score larger than the mean, especially in right-skewed distributions (where the mean is greater than the median). Hence, this is not always true.
Therefore, the correct and always true statement is (C): At most half the candidates have a score that is larger than the median.