Understanding Z-Scores, Standard Scores, and Percentiles
How to compare measurements from two completely different distributions using normalized z-scores.
What Is a Z-Score?
A z-score (also called a standard score) describes the position of a raw score in terms of its distance from the mean, measured in standard deviation units. A z-score of 0 is exactly at the mean; positive z-scores are above the mean, and negative z-scores are below the mean.
Where x = raw value, μ = population mean, and σ = population standard deviation.
Comparing Apples to Oranges: SAT vs. ACT Scores
Z-scores allow fair comparison between completely different scales.
Student A scored 1400 on the SAT. Student B scored 32 on the ACT. Who scored relatively higher?
• SAT distribution: Mean μ = 1060, Standard Deviation σ = 210.
Student A: z = (1400 − 1060) ÷ 210 = +1.62.
• ACT distribution: Mean μ = 21, Standard Deviation σ = 5.6.
Student B: z = (32 − 21) ÷ 5.6 = +1.96.
Conclusion: Student B performed better relative to their peer group (+1.96σ vs. +1.62σ).
Converting Z-Scores to Percentiles
Once you have a z-score, standard normal distribution tables give the cumulative probability (percentile rank):
• z = 0.00 → 50th percentile (exact median/mean)
• z = +1.00 → 84.1th percentile
• z = +1.96 → 97.5th percentile
• z = +2.58 → 99.5th percentile