P-Value Calculator

Compute p-values from z-test statistics for left-tailed, right-tailed, and two-tailed hypothesis tests. Compare against alpha and get a clear significance decision.

P-Value

0.04999565

Moderate evidence against the null hypothesis

Tail Type

Two-Tailed

Alpha

0.05

Decision

Significant

Formula

p = 2 × (1 − Φ(|z|))

Inference

Reject the null hypothesis because p (0.049996) < α (0.05).

What Is a P-Value?

A p-value quantifies how surprising your observed statistic is under the null hypothesis. If the null hypothesis is true, the p-value is the probability of seeing a result as extreme or more extreme than the one observed. Small p-values suggest the data are less compatible with the null model and may justify rejection at a chosen significance level (alpha).

Tail Direction and Formula

For z-tests, p-values are derived from the standard normal cumulative distribution function Φ(z).

  • Left-tailed: p = Φ(z)
  • Right-tailed: p = 1 − Φ(z)
  • Two-tailed: p = 2 × (1 − Φ(|z|))

Tail choice must match your hypothesis statement. If your alternative is directional, use one-tailed. If your alternative is simply “different,” use two-tailed.

Worked Example

Suppose your test yields z = 2.10 with a two-tailed hypothesis.

  • Φ(2.10) ≈ 0.9821
  • Upper tail = 1 − 0.9821 = 0.0179
  • Two-tailed p = 2 × 0.0179 = 0.0358

At alpha = 0.05, p = 0.0358 < 0.05, so you reject the null hypothesis.

How to Interpret P-Values Responsibly

  • P-value is not effect size: significance does not imply practical impact.
  • P-value is not certainty: it does not give the probability that the null is true.
  • Context matters: sample design, assumptions, and multiple testing can shift interpretation.
  • Use companion metrics: confidence intervals and effect sizes should be reported together.

Common Alpha Levels

Alpha (α)Confidence EquivalentTypical Use
0.1090%Exploratory analysis
0.0595%General scientific reporting
0.0199%High-stakes or strict inference

Sources and References

  • Fisher, R. A. Statistical Methods for Research Workers.
  • Wasserstein, R. L., & Lazar, N. A. (2016). The ASA Statement on p-values.
  • NIST/SEMATECH e-Handbook of Statistical Methods — Hypothesis Testing.

Frequently Asked Questions

What is a p-value?
A p-value is the probability of observing a test statistic at least as extreme as the one you obtained, assuming the null hypothesis is true. Smaller p-values indicate stronger evidence against the null hypothesis. A p-value does not measure effect size, practical importance, or the probability that the null hypothesis is true. It only measures compatibility of data with the null model.
What is the difference between one-tailed and two-tailed tests?
A one-tailed test checks for an effect in a specific direction (greater than or less than), while a two-tailed test checks for any difference regardless of direction. Two-tailed tests split probability across both extremes and are generally more conservative. You should choose tail type before seeing results, based on research design and hypothesis wording.
What does p < 0.05 actually mean?
It means that if the null hypothesis were true, data at least this extreme would occur less than 5% of the time under the assumed model. This threshold is a convention, not a universal law. In high-risk fields, stricter thresholds such as 0.01 may be preferred. In exploratory work, p-values should be interpreted alongside effect sizes and confidence intervals.
Can a p-value prove my hypothesis is true?
No. A p-value can indicate whether data are inconsistent with the null hypothesis, but it cannot prove your alternative hypothesis. Statistical significance is not the same as truth, causality, or practical value. Good conclusions combine p-values with effect magnitude, confidence intervals, study quality, assumptions, and external evidence.
Why can tiny effects become significant with large samples?
As sample size grows, standard errors shrink, making it easier to detect even very small differences. This can produce very small p-values for effects that are statistically significant but practically trivial. That is why reporting effect size and confidence intervals is critical, especially in large datasets and A/B testing environments.
Should I use this calculator for t-tests too?
This tool computes p-values from z-statistics (normal distribution). For large samples this is often close to t-test results, but for small samples you should use t-distribution methods with degrees of freedom. If your workflow requires exact t-test p-values, use a dedicated t-test calculator or statistical package for full precision.

Related Tools