Z-Scores
You scored 720 on the biology section and 510 on the psychology section. Which performance was better? You cannot compare raw scores directly because the two sections have different means and different spreads. You need a common yardstick - a way to say “how far above or below average was this score, relative to everyone else?” That yardstick is the z-score.
The Z-Score Formula
A z-score tells you how many standard deviations a data point is above or below the mean.
z = (x - μ) / σ
where x is the individual value, μ is the population mean, and σ is the standard deviation.
Interpreting Z-Scores
- z = 0: The value equals the mean. Perfectly average.
- z = +1: The value is 1 SD above the mean.
- z = +2: The value is 2 SDs above the mean.
- z = -1: The value is 1 SD below the mean.
- z = -1.5: The value is 1.5 SDs below the mean.
Positive z-scores are above average. Negative z-scores are below average. The magnitude tells you how far from average.
Z-Scores to Percentiles
By combining z-scores with the 68-95-99.7 rule, you can quickly estimate percentiles.
| Z-Score | Percentile | Reasoning |
|---|---|---|
| -3 | 0.15th | 99.7% is below +3 SD; 0.15% is below -3 SD |
| -2 | 2.5th | 95% is within 2 SD; 2.5% is below -2 SD |
| -1 | 16th | 68% within 1 SD; 16% below -1 SD |
| 0 | 50th | The mean is the midpoint |
| +1 | 84th | 50% + half of 68% = 50% + 34% = 84% |
| +2 | 97.5th | 50% + half of 95% = 50% + 47.5% = 97.5% |
| +3 | 99.85th | 50% + half of 99.7% = 50% + 49.85% |
Worked Example
A medical school class has a mean Step 1 score of 230 with SD = 20. A student scored 270.
z = (270 - 230) / 20 = 40 / 20 = 2
This student is exactly 2 SDs above the mean. From the table above, z = +2 corresponds to the 97.5th percentile. This student scored higher than 97.5% of the class.
Comparing Across Different Scales
Z-scores let you compare values from completely different distributions. Suppose:
- Biology exam: your score = 82, mean = 75, SD = 5. Your z = (82 - 75)/5 = 1.4
- Chemistry exam: your score = 68, mean = 60, SD = 4. Your z = (68 - 60)/4 = 2.0
Even though your raw biology score (82) was higher, your chemistry performance (z = 2.0) was relatively better compared to the class. You were further above average in chemistry.