Z Distribution Calculator
Calculate a standard normal percentile, left or right tail, two-tailed probability, or area between two z-scores. You can also standardize a raw score or work backward from a percentile.
Written and checked by TheToolNet Editorial Team · Last reviewed: September 12, 2026 · How we check statistics tools
The percentile below z is 100 × Φ(z). The right tail is 1 − Φ(z).
Calculate z-score probability or percentile
Choose what you know. Results include the selected z-score, all relevant tails, a normal-curve marker, calculation steps, and a probability check.
What the z distribution calculator can solve
| Starting information | Calculation | Result |
|---|---|---|
| Z-score | Φ(z) | Percentile and tail probabilities |
| Percentile p | Φ⁻¹(p ÷ 100) | Z-score |
| Raw score x | (x − μ) ÷ σ | Z-score and probabilities |
| Lower and upper z | Φ(upper) − Φ(lower) | Probability between them |
Z-score probability example
For z = 1.96, Φ(1.96) ≈ 0.9750. That places the value at about the 97.5th percentile. The right-tail probability is about 0.0250, and the symmetric two-tailed probability is about 0.0500.
To verify a rounded answer manually, use the z-score table for statistics.
Left-tail, right-tail, and two-tailed probability
- Left tail: P(Z ≤ z) = Φ(z).
- Right tail: P(Z ≥ z) = 1 − Φ(z).
- Symmetric two tails: 2 × min[Φ(z), 1 − Φ(z)].
- Between two values: Φ(upper) − Φ(lower).
A two-tailed probability here measures outcomes at least as far from zero as |z| under a standard normal model. A complete hypothesis test may require other choices, including the correct null model and whether its assumptions hold.
When should you use the z distribution?
Use standard normal probabilities after a value has been appropriately standardized and the analysis justifies a normal or large-sample approximation. For a raw score, the standard deviation must be positive. When the population SD is unknown and the sample is small, a t distribution may be more appropriate.
For spreadsheet work, see how to calculate a z-score in Excel.
Frequently asked questions
Is a z-score the same as a percentile?
No. A z-score measures signed distance from the mean in standard deviations. A percentile is the percentage of the reference distribution below that score.
Can a z-score be negative?
Yes. A negative z-score is below the mean; zero is at the mean; and a positive score is above it.
Why are extreme results rounded to 0 or 1?
Very large positive or negative z-scores have probabilities extremely close to 1 or 0. Display precision can hide the tiny remaining tail.