Turn any z score into clear probability values. Switch tails, two-sides, or percentiles instantly here. Download results as CSV and PDF in seconds easily.
Sample values using Phi(z) = P(Z ≤ z).
| z | Phi(z) | Right-tail (1 - Phi) | Interpretation |
|---|---|---|---|
| 0.00 | 0.500000 | 0.500000 | Half the mass lies below zero. |
| 1.00 | 0.841345 | 0.158655 | About 84.13% is below 1. |
| 1.64 | 0.949497 | 0.050503 | Common one-sided 5% threshold. |
| 1.96 | 0.975002 | 0.024998 | Classic two-sided 95% critical value. |
| -1.96 | 0.024998 | 0.975002 | Symmetric lower tail at 2.5%. |
The Z table is based on the standard normal distribution, where Z ~ N(0,1). Its probability density function is:
The cumulative distribution function gives the left-tail probability:
This page computes Phi(z) using a stable polynomial approximation and computes inverse values using a high-accuracy rational approximation.
The standard normal model has mean 0 and standard deviation 1. Any normal measurement x can be converted into a z score using z = (x - mu) / sigma, which makes units comparable. This calculator evaluates Phi(z), the cumulative probability from negative infinity to z, and also provides complementary tail areas for decisions.
A table entry is interpreted as Phi(z) = P(Z ≤ z). For z = 0.00, Phi(z) = 0.500000, which splits the curve in half. For z = 1.00, Phi(z) is about 0.841345 and the right tail is 0.158655. For z = -1.96, Phi(z) is about 0.024998, aligning with the lower 2.5% region.
In testing, a two-sided p-value doubles the smaller tail probability. The calculator uses p = 2 * min(Phi(z), 1 - Phi(z)). With z = 1.96, p is about 0.0500, matching the familiar 5% threshold. With z = 2.33, Phi(z) is about 0.9901, so p is roughly 0.0198.
Range probabilities rely on subtraction of cumulative values. For z1 = -1.00 and z2 = 1.00, Phi(1.00) - Phi(-1.00) is about 0.682689, the classic one-sigma coverage. A central area between -z and +z is computed as 2 * Phi(z) - 1, and the outside area is 1 minus that central result.
Inverse lookup turns a probability into a cutoff z. If p = 0.975, the left-tail percentile is z about 1.96, often used for 95% intervals. If a right-tail probability is 0.05, the cutoff is z about 1.64485. For two-tail alpha = 0.01, the critical value is z about 2.57583, leaving 0.5% in each tail.
Professional reporting benefits from consistent rounding and clear labels. Using six probability decimals and four z decimals keeps tables readable while preserving accuracy for most analyses. The mini table shows nearby hundredth cells so you can cross-check outputs. Exporting to CSV supports spreadsheet auditing, while the PDF option creates an attachment for reviews and records. Use chart to confirm shaded region visually quickly.
Phi(z) is the cumulative probability that a standard normal variable is less than or equal to z. It is the left-tail area under the curve up to that z score.
Use right-tail when your question is about values above a cutoff. Use two-tail when extreme values on both sides matter, such as testing whether a mean differs from a target in either direction.
Select an inverse mode and enter the desired probability. For example, a left-tail probability of 0.975 returns the z cutoff near 1.96, which is commonly used for 95% confidence levels.
Negative z scores lie left of the mean, so less area accumulates before reaching them. By symmetry, Phi(-z) equals 1 minus Phi(z), which helps validate results and check rounding.
The calculator uses standard numerical approximations that closely match published tables for typical z ranges. Small differences can occur in the sixth decimal place due to rounding and interpolation methods.
Most applications fall between -3 and 3, where almost all probability mass lies. Values beyond about +/-4 are rare, but the tool still computes them and the chart expands to fit.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.