Definition:
A standard score is a statistically transformed raw score that indicates how far above or below the mean a person’s score falls, measured in standard deviation units. The most basic form is the z-score (mean = 0, SD = 1), but many tests use rescaled versions like T-scores (mean = 50, SD = 10) or custom scaled scores to avoid negative numbers and decimals.
In-Depth Explanation
The z-score:
z = (X − M) ÷ SD
Where X = raw score, M = group mean, SD = standard deviation
| z-score | Meaning |
|---|---|
| 0 | Exactly at the mean |
| +1.0 | One SD above the mean (top ~16%) |
| −1.0 | One SD below the mean (bottom ~16%) |
| +2.0 | Two SDs above the mean (top ~2.3%) |
Common standard score scales:
| Scale | Mean | SD | Used by |
|---|---|---|---|
| z-score | 0 | 1 | Statistics, research |
| T-score | 50 | 10 | Many psychological tests |
| IQ scale | 100 | 15 | Intelligence tests |
| TOEFL iBT | Variable | — | Total 0–120 (scaled) |
| JLPT | Variable | — | Scaled scores by section |
Why standard scores are useful:
- Comparability: You can compare scores from different test forms or different tests
- Interpretability: A standard score tells you where you stand relative to the group
- Equating: Standard scores account for differences in test difficulty between forms
- Statistical operations: Standard scores (unlike percentile ranks) can be meaningfully averaged
Standard scores vs. percentile ranks:
Standard scores and percentiles both describe relative standing, but they have different mathematical properties:
- Percentile ranks are ordinal — the distance between the 50th and 60th percentile is not the same as between the 90th and 100th
- Standard scores are interval — the distance between z = 0 and z = 1 represents the same amount of ability difference as between z = 1 and z = 2
In language testing:
The JLPT uses scaled scores (not raw scores) to determine pass/fail. The scaled score is derived using Item Response Theory and ensures that the pass/fail threshold represents the same ability level regardless of test form difficulty.
Related Terms
See Also
Research
- Crocker, L., & Algina, J. (2008). Introduction to Classical and Modern Test Theory. Cengage Learning. — Detailed treatment of score transformations and their properties.
- McNamara, T. (2000). Language Testing. Oxford University Press. — Accessible introduction to scoring in language assessment contexts.