Bell Curve / z-Score and Percentile
Where a value sits on a normal curve, as standard deviations from the mean and as a percentile.
Example
You enter
- Raw score 85
- Sample mean 75
- Sample standard deviation 10
You get
- z-score 1
- Percentile 84.13 %
- Curve letter A
Details, formula, and sources
z = (x - mu) / sigma; percentile from the standard normal CDF (Abramowitz + Stegun 26.2.17); curve letter band per the 68-95-99.7 rule. Teacher governs whether a normative curve is appropriate.
z = (raw - mean) / sd. percentile = standard-normal-CDF(z) * 100, via the Abramowitz + Stegun 26.2.17 approximation (~7.5e-8 absolute error). Curve bands: z >= 2 A+; 1 to 2 A; 0 to 1 B; -1 to 0 C; -2 to -1 D; below -2 F.
Abramowitz + Stegun, Handbook of Mathematical Functions, formula 26.2.17 (1965; National Bureau of Standards Applied Mathematics Series 55). Empirical 68-95-99.7 rule for the curve bands; pre-CCSS norm-referenced grading convention.
AMS 55 is public domain; free at dlmf.nist.gov and nist.gov/pml.
Estimate only. Readability formulas and similar metrics are derived from a representative population and have known edge-case noise. The classroom teacher governs final text selection, grade placement, and assessment decisions.
Field names used by the API: raw_score, mean, sd, z_score, percentile, curve_letter
- Population vs sample SD the tile takes a single SD; the user picks whether it is the population or sample valueconvention
- Approximation Abramowitz + Stegun 26.2.17 is accurate to ~7.5e-8 absolute error in the standard-normal CDFAMS 55 §26.2