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 is a positional measure that always divides the dataset into two halves. Unlike the mean, it is not affected by extreme values.

Answer 1

The Correct Option is C

Step 1: Recall the definition of the median.
The median is the middle value (or average of two middle values) when all scores are arranged in order. By definition, at least half the candidates have scores less than or equal to the median, and at least half have scores greater than or equal to the median.

Step 2: Recall the definition of the mean.
The mean (average) can be influenced heavily by extreme values (outliers). Hence, no fixed relation (greater or smaller) must always hold between the mean and the median.

Step 3: Eliminate incorrect options.
- (A) Median ≥ Mean → Not always true. In skewed distributions, mean can be larger. - (B) Mean ≥ Median → Not always true either; depends on skewness. - (D) At most half the candidates have a score larger than the mean → Not necessarily true. For example, if many students score below the mean, more than half could be above the mean.

Step 4: Verify correct option.
(C) "At most half the candidates have a score that is larger than the median" → This is always true because by definition, the median divides the data into two equal halves.

Final Answer: \[ \boxed{\text{At most half the candidates have a score larger than the median.}} \]