What it shows
Two diagrams for one worked example. Student A scored 1800 on the SAT and Student B scored 24 on the ACT; converting each score to a z-score puts them on the same scale, so they can be compared.
On each normal curve the area to the left of the student’s score is shaded: under the normal model, the share of test-takers who scored lower. Each axis carries the z-score and the test’s own scale.
How to read it
Subtract the mean, then divide by the standard deviation. For Student A, z = (1800 − 1500) / 300 = 1.
For Student B, z = (24 − 21) / 5 = 0.6.
The shaded areas give the share of test-takers below each student under the normal model: 0.8413 (84.13%) for Student A and 0.7257 (72.57%) for Student B.
Student A sits further above the mean in standard-deviation terms, so Student A’s score is the stronger result relative to other test-takers.
The numbers behind it
- SAT (Student A): mean 1500, standard deviation 300, score 1800, z = 1, area to the left 0.8413 (84.13%).
- ACT (Student B): mean 21, standard deviation 5, score 24, z = 0.6, area to the left 0.7257 (72.57%).
- The SAT axis uses the pre-2016 scale, from 600 to 2400; the ACT axis runs from 6 to 36.
Ways to use it in teaching
Before showing the curves, ask which student did better; the raw scores alone cannot answer it.
Have students add a third student on another test, with its own mean and standard deviation, and place them on the same scale.
Note that the percentiles rely on the normal model, which real score distributions only approximate.
For the summary statistics behind this kind of chart, see our plain-language guide to descriptive statistics.
Sources
- Diez, D. M., Barr, C. D., & Çetinkaya-Rundel, M. (2012). OpenIntro statistics (2nd ed.). OpenIntro.
- Standard score. (n.d.). In Wikipedia. Retrieved October 18, 2026, from https://en.wikipedia.org/wiki/Standard_score
