P-value Calculator

P-value calculator: z-score to tail probability

This page converts z-scores to p-values (or p-values to z) on the standard normal curve, compares p to α, and shades the matching tail. Calculations run locally; Center mode is for z-table area only.

What this calculator does

Computes tail probabilities on the standard normal distribution from a z-score or inverts a p-value back to z. Inputs under Test parameters: input mode, z or p, significance level α (0.01–0.10), and tail type (left, right, two-tailed, or center). Outputs: p-value to six decimals, optional z, Reject or Fail to reject null hypothesis when the tail mode supports hypothesis testing, interactive bell curve with shaded area, and Logic trace showing Φ(z) substitution. Uses the error function for Φ(z); nothing is sent to a server. Does not run t-tests, chi-square tests, or ANOVA from raw data.

Hypothesis-test modes:

Left-, right-, and two-tailed shading with p ≤ α decision.
Center (0 to Z):

Area between 0 and z for z-table practice; decision badge N/A.

How the Math Works

Let Φ(z) be the standard normal cumulative distribution function. This page evaluates Φ with the error function, then applies the active tail rule. Left-tailed:

P = Φ(z)

Right-tailed:

P = 1 − Φ(z)

Two-tailed:

P = 2 × (1 − Φ(|z|))

Center:

P = |Φ(z) − 0.5|

Worked example: z = 1.96, two-tailed → Φ(1.96) ≈ 0.975 → P ≈ 2 × 0.025 = 0.05. At α = 0.05 that sits on the rejection boundary. Inverse mode solves z from p using Newton–Raphson on Φ(z) − p = 0.

Significance level α and the decision badge

The α slider sets the largest p-value you will label statistically significant on this page. Common classroom default: 0.05. When p ≤ α in a tail test mode, the result panel shows Reject null hypothesis; otherwise Fail to reject null hypothesis. That wording follows introductory hypothesis-testing convention: failing to reject is not the same as proving the null true. Raising α makes rejection easier and increases Type I error risk; lowering α is stricter. This widget does not adjust for multiple comparisons.

Choosing a tail on this page

Pick the tail before you read p. A z = 2.0 yields P ≈ 0.023 in a right-tailed test but P ≈ 0.046 two-tailed because both tails count. The curve shading and Logic trace always reflect the tail button you pressed. This calculator does not standardize raw scores from x, μ, and σ, and it does not compute confidence intervals from sample data.

P-value Calculator FAQ

How does this page turn a z-score into a p-value?

Choose Calculate P-value from Z-score, enter your test statistic, and pick a tail type. The tool evaluates Φ(z) with the error function, then applies the active tail rule. The shaded area on the normal curve and the Logic trace line both show the same probability with your numbers substituted.

When does the decision badge say reject the null hypothesis?

For left-, right-, and two-tailed modes, the Decision row compares your p-value to the significance level α on the slider (1%–10%, default 5%). If p ≤ α, the badge reads Reject null hypothesis. If p > α, it reads Fail to reject null hypothesis. Center (0 to Z) mode shows table-style area only; the decision badge is N/A because that area is not a hypothesis-test p-value.

What are the tail formulas on this calculator?

Left-tailed:

P = Φ(z)

area below z. Right-tailed:

P = 1 − Φ(z)

area above z. Two-tailed:

P = 2 × (1 − Φ(|z|))

combined area in both tails beyond ±|z|. Center (0 to Z):

P = |Φ(z) − 0.5|

area between 0 and z, like many textbook z-tables.

How do I pick left-tailed, right-tailed, or two-tailed?

Match the buttons to your alternative hypothesis. Right-tailed when you test whether a statistic is greater than a reference (μ > μ₀). Left-tailed for less than. Two-tailed for different from in either direction. Picking the wrong tail changes the shaded region and can flip a significance call at the margin.

Can I start from a p-value and get the z-score?

Yes. Switch input mode to Calculate Z-score from P-value and enter a probability between 0 and 1 (exclusive). The page inverts Φ(z) with a Newton–Raphson search on the standard normal density. In two-tailed mode, entering p-value 0.05 returns z ≈ 1.96; entering 0.01 returns z ≈ 2.58.

What does hover on the normal curve show?

Move the pointer under the bell curve to read a live z coordinate. In tail modes the tooltip shows Φ(z), the cumulative probability up to that z. In Center (0 to Z) mode it shows the area from 0 to that z. Red dashed lines mark your entered statistic; green dashed lines mark α critical values in two-tailed view.

How is this different from the Z-score Calculator?

This page assumes you already have a standardized z (or a p-value) for a hypothesis test. The Z-score Calculator converts raw scores with z = (x − μ) / σ, finds percentiles, or computes probability between two raw values.

Does a small p-value prove the alternative hypothesis?

No. A p-value measures how surprising your statistic would be if the null were true. It does not measure effect size, practical importance, or whether the null is literally true. This tool reports the probability and an α-based decision label; interpretation still belongs to your study design and field standards.

Test parameters

Reading your p-value and decision

Example: z = 1.96, two-tailed, α = 5%

Inverse mode: p-value to z

Center mode and the Decision badge