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: Relative standing of a single observation compared to the mean is most commonly expressed through standardization, converting the raw score into a z-score.

Step 2: The formula for a z-score is $z = \dfrac{X - \mu}{\sigma}$, where $\mu$ is the population mean and $\sigma$ is the standard deviation; here $\mu = 2.77$ and $X = 2.0$.

Step 3: Checking the alternatives: the median and interquartile range are built around the middle value of the distribution and do not directly connect to the given mean of 2.77, while the number of students at the college has no bearing on an individual score's position.

Step 4: Only the standard deviation supplies the missing piece $\sigma$ needed to compute how far, in standardized units, the GPA of 2.0 lies from the mean of 2.77.

\[\boxed{\text{Standard deviation}}\]