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:

The median splits the data so that {no more than half} the observations lie on either side when using “>” or “<”. The mean has no such split-by-count guarantee—it depends on magnitudes.

Answer 1

The Correct Option is C

Step 1: Recall the definition of median.
For any dataset, a median $m$ satisfies: at least half of the observations are $\le m$ and at least half are $\ge m$.
Step 2: Translate to a statement about “strictly larger”.
Because at least half are $\le m$, the number that are {strictly greater} than $m$ cannot exceed half of the observations.
Hence: “At most half the candidates have a score larger than the median” is always true.
Step 3: Check the remaining options.
(A) and (B): Mean vs. median ordering depends on skewness; either can be larger, so neither is guaranteed.
(D): There is no universal bound that {at most half} exceed the {mean}; e.g., a highly right-skewed set can have far less than half below the mean or more than half above it. Thus (D) is not guaranteed. Final Answer:\fbox{(C)}