Three students were applying to the same graduate school. They came from schools with different grading systems.

**Table 2.74**StudentGPASchool Average GPASchool Standard Deviation

Thuy

2.42.4

2.22.2

0.50.5

Vichet

8585

9696

2020

Kamala

8.58.5

7.97.9

0.80.8
 
How many standard deviations is each student away from hiir school average? If the student GPA is higher than his school average, enter this as a positive number. If the student GPA is lower than his school average, enter this as a negative number.

Thuy is     standard deviations from Thuy's school average.

Vichet is     standard deviations from Vichet's school average.

Kamala is     standard deviations from Kamala's school average.
 
Which student had the best GPA when compared to other students at his school?

Thuy

Vichet

Kamala

To find out how many standard deviations each student's GPA is away from their school average, we will use the formula for the Z-score, which represents the number of standard deviations a value is from the mean. The Z-score is calculated as follows:

\[
Z = \frac{\text{Student's GPA} - \text{School Average GPA}}{\text{School Standard Deviation}}
\]

### Step-by-Step Calculations

#### 1. **Thuy**

- **Student's GPA:** 2.4  
- **School Average GPA:** 2.2  
- **School Standard Deviation:** 0.5  

\[
Z = \frac{2.4 - 2.2}{0.5} = \frac{0.2}{0.5} = 0.4
\]

Thuy is **0.4 standard deviations above** her school average.

#### 2. **Vichet**

- **Student's GPA:** 85  
- **School Average GPA:** 96  
- **School Standard Deviation:** 20  

\[
Z = \frac{85 - 96}{20} = \frac{-11}{20} = -0.55
\]

Vichet is **-0.55 standard deviations below** his school average.

#### 3. **Kamala**

- **Student's GPA:** 8.5  
- **School Average GPA:** 7.9  
- **School Standard Deviation:** 0.8  

\[
Z = \frac{8.5 - 7.9}{0.8} = \frac{0.6}{0.8} = 0.75
\]

Kamala is **0.75 standard deviations above** her school average.

### Final Answers

- **Thuy** is **0.4** standard deviations from Thuy's school average.
- **Vichet** is **-0.55** standard deviations from Vichet's school average.
- **Kamala** is **0.75** standard deviations from Kamala's school average.

#### Which student had the best GPA when compared to other students at their school?

Kamala had the best GPA when compared to other students at her school because her Z-score (0.75) is the highest among the three students, indicating she is the furthest above her school's average.

Would you like more details on these calculations or have any other questions?

Here are 5 related questions to expand your understanding:

1. What does a positive Z-score tell you about a student's GPA in relation to their school average?
2. How might different grading systems affect the interpretation of GPAs across schools?
3. What would a Z-score of 0 indicate about a student's GPA?
4. Why might a lower Z-score still be considered "good" in some contexts?
5. How could you use Z-scores to compare students across different grading systems more fairly?

**Tip:** Z-scores are a great way to standardize scores from different distributions, allowing for a fair comparison between different systems.

Learn how to calculate Z-scores to analyze student GPAs compared to their school averages. Includes detailed calculations for Thuy, Vichet, and Kamala, showing how many standard deviations each GPA is from their respective school averages. Suitable for college-level students studying statistics.

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