To determine which student performed the best relative to their department, we need to standardize each student's score by calculating their z-scores. A z-score measures how many standard deviations a data point (in this case, a student’s score) is from the mean of the distribution. The formula for the z-score is:

$z = \frac{x - \mu}{\sigma}$

where:

• $x$ is the individual score,
• $\mu$ is the mean score of the department,
• $\sigma$ is the standard deviation of the department.

Let's calculate the z-scores for each student.

1. Zephrya's z-score:
   • Score: 92
   • Mean: 72
   • Standard deviation: 15

$z = \frac{92 - 72}{15} = \frac{20}{15} \approx 1.33$

2. Marisa's z-score:
   • Score: 75
   • Mean: 55
   • Standard deviation: 20

$z = \frac{75 - 55}{20} = \frac{20}{20} = 1.00$

3. Brady's z-score:
   • Score: 90
   • Mean: 70
   • Standard deviation: 10

$z = \frac{90 - 70}{10} = \frac{20}{10} = 2.00$

Comparison of z-scores:

• Zephrya: 1.33
• Marisa: 1.00
• Brady: 2.00

Conclusion: Brady performed the best relative to his department, as he has the highest z-score of 2.00.

Would you like more details on z-scores, or do you have any other questions?

Here are five related questions to consider:

1. How would the z-scores change if the standard deviations were higher in each department?
2. What is the meaning of a negative z-score in this context?
3. Could we rank students differently using raw scores instead of z-scores?
4. How could the z-score concept apply to other academic comparisons across departments?
5. What would happen if the means of all departments were the same, but the standard deviations varied?

Tip: Z-scores are useful for comparing values from different distributions because they normalize the scores, making them independent of the original units.