Question

The mean GPA for all students in Statistics at a certain college in the odd semester was 2.77. A student with a GPA of 2.0 wants to know her relative standing relation to the mean GPA. A numerical summary that would be useful for this purpose is the:

• A. Standard deviation
• B. Median
• C. Interquartile range
• D. Number of students at the college

Hint:

To compare an individual value to the mean, you need a z-score, which requires the standard deviation.

Answer 1

The Correct Option is A

Step 1: The student already knows the mean (2.77) and her own GPA (2.0). To describe her relative standing compared to the mean, we need a measure of spread that tells us how far a typical score lies from the mean, not another measure of central tendency.

Step 2: The standard deviation measures the typical distance of individual scores from the mean and lets us compute a standardized z-score: \[z = \frac{X - \text{Mean}}{\text{SD}} = \frac{2.0 - 2.77}{\text{SD}}\]

Step 3: This z-score directly expresses how many standard deviations below or above the mean her GPA of 2.0 falls, which is exactly the relative standing being asked for.

Step 4: The median and interquartile range describe the distribution relative to the middle value, not relative to the mean, and the number of students is irrelevant to this comparison.

Final answer: \(\boxed{\text{Standard deviation}}\)