GATE 2023 Scores: Mean vs Median Analysis

This solution focuses on the definitional properties of the median and how they relate to the scores of GATE 2023 candidates, determining which statement must be true.

Understanding the Median Definition

The median of a dataset is the value separating the higher half from the lower half. When the scores of all candidates are arranged in ascending order:

• The median is the score such that 50% of the candidates scored at or below it.
• Consequently, 50% of the candidates scored at or above it.

Therefore, it is definitionally true that at most half the candidates can have a score strictly greater than the median. If the median score itself appears multiple times, the proportion scoring strictly above it will be less than 50%.

Evaluating Option Statements

Let's examine why the correct option must be true and others might not:

• Options 1 & 2: Statements comparing the mean and median (e.g., "The median is at least as large as the mean" or "The mean is at least as large as the median") depend entirely on the distribution of the scores. If the score distribution is skewed (asymmetrical), the mean and median will differ. Neither statement is guaranteed to be true in all cases.
• Option 3: "At most half the candidates have a score that is larger than the median." This statement is true by the fundamental definition of the median. It holds irrespective of the score distribution.
• Option 4: "At most half the candidates have a score that is larger than the mean." This is not necessarily true. In cases of left-skewed data (where many candidates score high, but a few score very low, pulling the mean down), more than half the candidates might have scores above the mean.

Final Conclusion

The core property of the median is that it divides the dataset in half. Hence, the statement affirming that at most 50% of candidates score above the median is the only one that must hold true for any set of GATE 2023 scores.