---
title: "Mean and SD of GPA and SAT scores"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:hs-gpa-college-gpa-sat-stats"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/hs-gpa-college-gpa-sat-stats
site: BCT Engineering
---

# Mean and SD of GPA and SAT scores

Probability and Statistics · Chapter 1: Descriptive Statistics and Basic Probability

- Asked in: 2068 Chaitra, 2×4 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/hs-gpa-college-gpa-sat-stats
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

The admission staff of a university, concerned with the success of the students it selects for admission wishes to compare the students' college performances with high school grades and test scores. The high school and college grade-point average (GPA) and student's average test (SAT) scores of 20 sampled students are as follows:
(H.S. GPA, College GPA, SAT score): (3.6, 2.5, 1100), (2.6, 2.7, 940), (2.7, 2.2, 950), (3.7, 3.2, 1160), (4.0, 3.8, 1340), (3.5, 3.6, 1180), (3.5, 3.8, 1250), (2.2, 3.5, 1040), (3.9, 3.7, 1310), (4.0, 3.9, 1330), (3.4, 3.6, 1180), (2.9, 3.0, 1010), (3.9, 4.0, 1330), (3.2, 3.5, 1150), (2.1, 2.5, 940), (2.2, 2.8, 960), (3.4, 3.4, 1170), (3.6, 3.0, 1100), (2.6, 1.9, 860), (2.4, 3.2, 1070)
a) Find, for each of the HS GPA, college GPA and SAT scores, the mean and standard deviation.
b) What is your conclusion about variability and uniformity from the analysis?

## Short answer

HS GPA: mean 3.17, SD 0.648; College GPA: mean 3.19, SD 0.598; SAT: mean 1118.5, SD 146.98. By CV: HS GPA most variable (20.4%), SAT most uniform (13.1%). College GPA slightly higher and less variable than HS GPA.

## Answer

For each variable n = 20. Mean x̄ = Σx/n and sample SD s = √[(Σx² − n·x̄²)/(n − 1)]. The coefficient of variation CV = (s / x̄) × 100 is used to compare the variability of variables with different units.

#### a) Mean and standard deviation

| Variable | Σx | Σx² | Mean | s | CV |
|---|---|---|---|---|---|
| H.S. GPA | 63.4 | 208.96 | 3.170 | 0.648 | 20.4 % |
| College GPA | 63.8 | 210.32 | 3.190 | 0.598 | 18.8 % |
| SAT score | 22,370 | 25,431,300 | 1118.5 | 146.98 | 13.1 % |

Working for H.S. GPA:

```text
x̄ = 63.4 / 20 = 3.17
Sxx = 208.96 − 20 (3.17)² = 208.96 − 200.978 = 7.982
s² = 7.982 / 19 = 0.4201,  s = 0.648
CV = 0.648 / 3.17 × 100 = 20.4 %
```

(With divisor n the SDs would be 0.632, 0.583 and 143.26.)

#### b) Conclusion on variability and uniformity
- The SDs cannot be compared directly, as the SAT is on a scale of about 1000 and the GPAs on a scale of 4.
- By CV, **H.S. GPA is the most variable** (20.4 %), then college GPA (18.8 %). **SAT score is the most uniform** (13.1 %).
- College GPA (mean 3.19) is slightly higher and less variable than H.S. GPA (mean 3.17), so students' college results are a little more uniform than their school grades.
- All three show moderate variation (CV 13 to 20 %). The sample contains students from low to high performers, so none of the three is very uniform.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why can the SDs of GPA and SAT not be compared directly?
   A. Different scales and units; use CV
   B. They have the same mean
   C. SDs are negative
   D. SAT has no SD
   Answer: A. SAT is about 1000 scale while GPA is about 4, so CV is used. (Key point: Why CV)

2. Which variable is most variable by CV?
   A. H.S. GPA (20.4%)
   B. SAT score
   C. College GPA
   D. None
   Answer: A. CVs: H.S. GPA 20.4%, college GPA 18.8%, SAT 13.1%. (Key point: Most variable)

3. Which variable is most uniform?
   A. H.S. GPA and SAT tie
   B. SAT score (13.1%)
   C. College GPA
   D. H.S. GPA
   Answer: B. The smallest CV means most uniform; SAT has 13.1%. (Key point: Most uniform)

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