GATE IN 2023 | GA Question: 4

Question

Which one among the following statements must be $\text{TRUE}$ about the mean and the median of the scores of all candidates appearing for $\text{GATE }2023?$

• 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. 

Answer 1

Let's analyze each option:

• A. "The median is at least as large as the mean."
  This statement does not necessarily have to be true. The median is the middle value when the data is sorted, and the mean is the average of all values. The median can be less than, equal to, or greater than the mean, depending on the distribution of scores.
   
• B. "The mean is at least as large as the median."
  This statement does not necessarily have to be true for the same reasons mentioned in option A. The relationship between the mean and median depends on the distribution of scores.
   
• C. "At most half the candidates have a score that is larger than the median."
  This statement is always true when you have a set of scores. By definition, the median is the middle value when the data is sorted. Since half of the data is below the median and half is above it, at most half the candidates can have scores larger than the median.
   
• D. "At most half the candidates have a score that is larger than the mean."
  This statement is not necessarily true. The mean is the average of all scores, and there can be scenarios where more than half of the candidates have scores larger than the mean, especially if the distribution is skewed with a few very high scores.

Correct Answer: C