Z-score Calculator
Convert between raw scores, Z-scores and normal probabilities. Three tools on one page: compute Z from a raw score, convert any single probability back to a Z, and find the area between two Z-scores — all with live standard-normal CDF, no Z-table required.
Tool 1 — Z-score Calculator
Enter a raw score and the distribution's mean and standard deviation.
Tool 2 — Z-score and Probability Converter
Provide any ONE value below to convert between Z-score and probability. This replaces the classic Z-table lookup.
Tool 3 — Probability Between Two Z-scores
Enter a left and right Z-score; the tool shades the area between them.
Z-scores explained, step by step
Each concept below has a plain-English definition, its formula, a small diagram and a worked example — all in one card so you never have to jump between sections to piece it together.
1. What a z-score really measures
A z-score converts any raw value into the number of standard deviations it sits from its distribution's mean. That makes scores from different tests or scales directly comparable.
2. Positive, negative and zero z
The sign of z tells you which side of the mean the value falls on. Positive z is above the mean, negative is below, and z = 0 is exactly at the mean. The magnitude tells you how far away in standard-deviation units.
3. Converting z to a probability
The standard-normal CDF Φ(z) gives the fraction of a normal distribution that lies below any z-score. That is what a Z-table looks up — this calculator computes it live from the error function so any z or probability works exactly.
4. Landmarks worth memorising
A handful of z-values show up everywhere: ±1 covers about 68% of the distribution, ±1.96 covers 95% (the classic 95% confidence cutoff), and ±2.576 covers 99%. Anything in between smoothly interpolates.
Common Z-score landmarks
| Z | P(x < Z) | Percentile | Note |
|---|---|---|---|
| -3 | 0.00135 | 0.13% | ≈ 0.1st percentile |
| -2 | 0.02275 | 2.28% | ≈ 2.3rd percentile |
| -1.96 | 0.02500 | 2.50% | lower 95% cutoff |
| -1 | 0.15866 | 15.87% | ≈ 1 SD below |
| -0.5 | 0.30854 | 30.85% | |
| 0 | 0.50000 | 50.00% | mean |
| 0.5 | 0.69146 | 69.15% | |
| 1 | 0.84134 | 84.13% | ≈ 1 SD above |
| 1.645 | 0.95002 | 95.00% | 90% one-tail cutoff |
| 1.96 | 0.97500 | 97.50% | 95% two-tail cutoff |
| 2 | 0.97725 | 97.72% | ≈ 97.7th percentile |
| 2.576 | 0.99500 | 99.50% | 99% two-tail cutoff |
| 3 | 0.99865 | 99.87% | ≈ 99.9th percentile |
Features of this calculator
- Three tools on one page — raw-score Z, Z↔probability converter, and area between two Z-scores
- Standard-normal CDF computed live via the error function (5-decimal precision) — no static Z-table lookup
- Small bell-curve icons next to each probability line so the shaded region is always visible
- Full-size bell curve with dashed Z1/Z2 markers for the two-bound tool
- Step-by-step working with the formula substituted for every result
- Inverse normal (Acklam) for the converter — enter any probability, get the exact Z