Worked example
Two-sided positive z
- Input
- z = 1.96; two-sided
- Output
- p ≈ 0.049996
The two symmetric tails each contribute about 0.025.
P-Value Calculator
Convert a supplied z statistic into a tail probability while keeping the alternative hypothesis explicit. Displayed p-values are limited to six decimal places, matching the tested accuracy of the current numerical approximation. This page does not derive a z statistic from raw data or treat a small p-value as the probability that a hypothesis is true.
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How to use this calculator
Enter a z statistic and whether the alternative is two-sided, lower-tail, or upper-tail.
Review how the z statistic was derived, whether normal-model assumptions hold, and whether the tail was selected before seeing the data.
Read p-value under the standard normal model and the entered z statistic and compare the displayed assumptions before reusing the answer.
Formula
For standard normal CDF Φ, lower-tail p = Φ(z), upper-tail p = 1 − Φ(z), and two-sided p = 2 × [1 − Φ(|z|)]. Results are clamped to the probability interval from 0 through 1.
Worked example
A z statistic of 1.96 gives a two-sided p-value of about 0.049996 at the supported display precision; z = 0 gives 0.5 in either single tail and 1.0 for a two-sided calculation.
Reference guide
Formula details and result guidance
Follow the variables, derivation, precision rule, and worked cases below to reproduce the calculation independently.
Variables
| Symbol | Meaning | Unit | Role |
|---|---|---|---|
| z | Standard normal test statistic | standard deviations | Input |
| Φ(z) | Standard normal cumulative probability | probability | Intermediate |
| p | p-value for the selected tail | probability | Output |
Derivation
- The left-tail probability is p = Φ(z), while the right-tail probability is p = 1 − Φ(z).
- For a symmetric two-sided z test, p = 2 × min[Φ(z), 1 − Φ(z)].
- The probability is bounded to [0, 1] after numerical evaluation of the standard normal CDF.
Rounding and precision
The normal CDF is evaluated numerically and p-values display no more than 6 decimal places, matching the tested accuracy of the current approximation. Values shown as 0 are below display resolution, not proof of an impossible null result.
Worked and boundary examples
Worked example
Right-tail test
- Input
- z = 2.3263; right-tail
- Output
- p ≈ 0.01
The selected alternative determines which tail is counted.
Boundary case
Center of distribution
- Input
- z = 0; two-sided
- Output
- p = 1
A centered statistic is not extreme in either direction.
Common mistakes
- Selecting two-sided output after forming a directional hypothesis from the observed result.
- Interpreting p as the probability that the null hypothesis is true.
- Using a z distribution when the sampling model or test statistic requires another reference distribution.
How to interpret the result
The p-value is the probability, under the stated null model, of a test statistic at least as extreme in the selected tail direction. It is not an effect size or a posterior probability of a hypothesis.
Scope
Appropriate for
- Reproducing a standard-normal tail probability from a known z statistic.
- Checking one- and two-sided z-test arithmetic.
Do not use for
- Choosing the test or tail after looking at the result.
- Making a substantive decision without assumptions, effect size, uncertainty, and context.
Tools for the same task
Next tools in the workflow
Evidence and sources
- A p-value measures tail extremeness under the null model, not the probability that the null hypothesis is true.National Institute of Standards and Technology: What are confidence intervals and p-values?Supports: P-values as tail probabilities under a specified null model and preselected significance levels. Reviewed 2026-08-22.
- The standard normal probability distribution is symmetric, and its CDF provides tail probabilities for z.National Institute of Standards and Technology: Normal DistributionSupports: The standard normal density, cumulative probability, and quantile relationship. Reviewed 2026-08-22.
- The lower, upper, or two-sided tail must match the alternative hypothesis selected before interpretation.OpenStax: Introductory Statistics 2e — Full Hypothesis-Test ExamplesSupports: Null-hypothesis tests, p-values, confidence intervals, and distribution selection. Reviewed 2026-08-22.
Assumptions and limits
- Use this only when a standard normal reference distribution is justified. It is not a substitute for study design, effect-size analysis, multiple-testing correction, or domain-specific interpretation.
- P-Value Calculator uses only the values entered in the form and does not infer an institution, policy, contract, population, or project condition.
- Working calculations retain full numeric precision; display rounding is separate and should not be copied into a later calculation when the unrounded value is available.
Sources
- National Institute of Standards and Technology: What are confidence intervals and p-values?P-values as tail probabilities under a specified null model. Reviewed 2026-08-22.
- National Institute of Standards and Technology: Normal DistributionThe standard normal distribution, cumulative probability, and quantiles. Reviewed 2026-08-22.
- OpenStax: Introductory Statistics 2e — Full Hypothesis-Test ExamplesNormal tails, p-values, confidence intervals, and statistical interpretation. Reviewed 2026-08-22.
Related calculators
Verification
How this calculator is checked
Trust comes from reproducible evidence, not model confidence. You can inspect the formula, assumptions, worked example, and cited sources on this page.
- Versioned calculationFormula logic is kept separate from the interface and covered by automated registry and behavior checks.
- Cited evidenceSources linked for independent checking: 3.
- Transparent AI useAI may assist drafting or adversarial review. Agreement between models is not proof, and no human expert review is claimed unless a named reviewer is shown.
Frequently asked questions
How does the P-Value Calculator work?
For standard normal CDF Φ, lower-tail p = Φ(z), upper-tail p = 1 − Φ(z), and two-sided p = 2 × [1 − Φ(|z|)]. Results are clamped to the probability interval from 0 through 1.
What should I enter in the P-Value Calculator?
Enter a z statistic and whether the alternative is two-sided, lower-tail, or upper-tail, then verify how the z statistic was derived, whether normal-model assumptions hold, and whether the tail was selected before seeing the data.
When should I avoid relying only on the P-Value Calculator?
Use this only when a standard normal reference distribution is justified. It is not a substitute for study design, effect-size analysis, multiple-testing correction, or domain-specific interpretation.
Last updated 2026-08-25 · standard-normal-tail-six-decimal-v2 · content-2026-08-22 · Published by YunFanLabs