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Statistical approaches to normality
“Normal” has several meanings. In assessment it usually means statistically typical for a reference group. Understanding that idea, and its limits, helps you interpret test scores sensibly.
Different meanings of “normal”
- Statistical: close to the average of a reference group.
- Developmental: what is expected at a given age.
- Functional: whether the difference affects everyday life, learning or wellbeing.
A score can be statistically unusual without causing any difficulty, and a person can struggle even when their score is in the typical range. That is why clinicians combine numbers with observation and the person’s own account.
The normal distribution
Many measurements cluster around an average with fewer cases further away, forming a symmetrical bell shape. Two numbers describe it:
- Mean: the average.
- Standard deviation (SD): how spread out the scores are.
In a normal distribution about 68% of people fall within 1 SD of the mean, about 95% within 2 SD and about 99.7% within 3 SD.
z-scores
A z-score says how many standard deviations a score is above or below the mean:
z = (score − mean) ÷ SD
For example, a score of 85 on a test with mean 100 and SD 15 gives z = (85 − 100) ÷ 15 = −1.0.
Percentiles
A percentile rank shows the percentage of the reference group scoring at or below a given score. If scores are normally distributed, z-scores convert to percentiles like this:
| z-score | Approx. percentile | Standard score (mean 100, SD 15) |
|---|---|---|
| −2.0 | 2nd | 70 |
| −1.5 | 7th | 77.5 |
| −1.0 | 16th | 85 |
| 0 | 50th | 100 |
| +1.0 | 84th | 115 |
| +2.0 | 98th | 130 |
Other scales are common too: scaled scores (mean 10, SD 3) and T-scores (mean 50, SD 10).
Cut-offs are conventions
Services and tests often flag scores at −1, −1.5 or −2 SD below the mean. These are agreed conventions, not natural boundaries. A lower cut-off finds fewer people and misses more mild difficulties; a higher cut-off finds more, but flags more people who turn out to be fine. Always use the cut-off recommended by the test manual and your service.
Try it: z-score calculator
Things that can mislead
- Wrong reference group. Norms for one age group, language, dialect or culture may not suit another. This matters in multilingual settings.
- Skewed data. Measures such as counts of disfluencies or reaction times are often skewed, so means and SDs mislead; percentiles or medians fit better.
- Measurement error. Every score has a standard error of measurement. A score is best reported with a confidence interval rather than as an exact figure.
- Small or old norm samples. Norms built on small or outdated samples are less trustworthy.
- Statistically unusual is not the same as disordered. Interpretation needs context.
Rule of thumb. Use statistics to describe where a score sits, then ask what it means for this person in daily life.
Sources and further reading
- Altman, D. G. (1991). Practical Statistics for Medical Research. Chapman & Hall.
- For assessment-specific guidance, consult the manual of the test you are using: it reports its own norms, standard errors and confidence intervals.