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:

Remember the core definitions. The median's definition is about position (the middle value), which guarantees a split of the dataset into halves. The mean's definition is about value (the average), which gives no guarantee about how many data points are above or below it.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question tests the fundamental definitions of two measures of central tendency: mean and median.
- Mean: The arithmetic average of a dataset (sum of all scores divided by the number of scores).
- Median: The middle value of a dataset when it is sorted in ascending order. If there is an even number of observations, the median is the average of the two middle values.
Step 3: Detailed Explanation:
Let's evaluate each statement:
(A) The median is at least as large as the mean. This is not always true. In a right-skewed distribution (e.g., a few candidates scoring very high), the mean is pulled higher than the median. For example, scores {10, 20, 30, 40, 100} have a median of 30 and a mean of 40. Here, mean > median.
(B) The mean is at least as large as the median. This is not always true. In a left-skewed distribution (e.g., a few candidates scoring very low), the mean is pulled lower than the median. For example, scores {1, 60, 70, 80, 90} have a median of 70 and a mean of 60.2. Here, median > mean.
Since we have no information about the distribution of GATE scores, we cannot make any definitive statement about the relationship between the mean and median.
(C) At most half the candidates have a score that is larger than the median. This is true by the definition of the median. The median is the value that divides the dataset into two equal halves. - 50% of the data points are less than or equal to the median.
- 50% of the data points are greater than or equal to the median.
This means that the number of candidates with a score strictly *larger* than the median can be at most 50%. It could be less than 50% if multiple candidates have a score equal to the median. Therefore, the statement "at most half" is always correct.
(D) At most half the candidates have a score that is larger than the mean. This is not always true. Consider the scores {1, 60, 70, 80, 90}. The mean is 60.2. Three scores (70, 80, 90) are larger than the mean. This is 3/5 or 60% of the candidates, which is more than half.
Step 4: Final Answer:
The only statement that must be true, regardless of the score distribution, is that at most half the candidates have a score larger than the median.
Step 5: Why This is Correct:
This statement is a direct consequence of the definition of the median as the 50th percentile of a distribution. It is the only option that holds true for any dataset.