The statistics of runs scored in a series by four batsmen are provided in the table below. | Batsman | Average | Standard-deviation | |---------|---------|--------------------| | K | 31.2 | 5.21 | | L | 46.0 | 6.35 | | M | 54.6 | 6.22 | | N | 17.5 | 5.90 | Which one among the following statements must be TRUE about the mean and the median of the scores of all candidates appearing for GATE 2020? 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.

GATE 2023 · General Aptitude · Statistics · medium

Answer: Option C must be TRUE: at most half the candidates have a score larger than the median (by definition of median).

1. Evaluate options A and B (mean vs median ordering): Options A (median >= mean) and B (mean >= median) depend on the direction of skewness. Without knowing whether GATE scores are left or right skewed, neither can be guaranteed. Both A and B can be false for some distributions.
2. Evaluate option C (at most half exceed median): Option C states: at most half the candidates have a score larger than the median. This is true by the definition of median for any dataset - the median is chosen precisely so that no more than half the values lie above it.
3. Evaluate option D (at most half exceed mean): Option D claims at most half score above the mean. This is false in general: in a right-skewed distribution, more than half the values can lie below the mean (e.g., most GATE candidates may score below the mean if a few score very high).