What is a p-value in statistics?

A p-value is the probability of obtaining test results at least as extreme as observed, assuming the null hypothesis is true. A small p-value (typically < 0.05) suggests the data is unlikely under the null hypothesis and provides evidence to reject it. P-values range from 0 to 1.

What does p < 0.05 mean?

p < 0.05 means there is less than a 5% probability of obtaining your results (or more extreme) if the null hypothesis were true. By convention, p < 0.05 is considered statistically significant. However, statistical significance does not equal practical significance or prove causation.

What is the difference between one-tailed and two-tailed tests?

A one-tailed test checks for an effect in one direction (e.g., only whether treatment increases scores). A two-tailed test checks for an effect in either direction. Two-tailed tests are more conservative (require stronger evidence) and are preferred unless you have a strong directional hypothesis before collecting data.

Can a p-value tell me if my hypothesis is true?

No. A p-value only tells you the probability of observing your data if the null hypothesis were true - it does not give the probability that the null hypothesis is true or false. A low p-value means the data is surprising under the null hypothesis, not that the alternative hypothesis is confirmed. Always consider effect size and confidence intervals alongside p-values.

P-Value Calculator

Select the distribution type, enter your test statistic and degrees of freedom (if needed), and choose the tail.

P-Value — Complete Guide

What is a p-value?

The p-value is the probability of obtaining a test result at least as extreme as the observed result, assuming the null hypothesis H₀ is true. It does not tell you the probability that H₀ is true; it measures how compatible your data are with H₀.

How to interpret a p-value

p-value	Interpretation
p < 0.001	Very strong evidence against H₀
0.001 ≤ p < 0.01	Strong evidence against H₀
0.01 ≤ p < 0.05	Moderate evidence against H₀ (significant at α = 0.05)
0.05 ≤ p < 0.10	Weak evidence; marginal significance
p ≥ 0.10	Little to no evidence against H₀

Z-distribution (standard normal)

Used when the test statistic follows a standard normal distribution (large samples, known σ).

Two-tailed: p = 2 × (1 − Φ(|z|)), where Φ is the standard normal CDF
Left-tailed: p = Φ(z)
Right-tailed: p = 1 − Φ(z)

Example: z = 1.96 (two-tailed) ⇒ p ≈ 0.05

χ²-distribution (chi-square)

Used in goodness-of-fit and independence tests. Always right-tailed.

p = 1 − Fχ²(χ², df), where Fχ² is the chi-square CDF with df degrees of freedom.

Example: χ² = 5.99, df = 2 (two-tailed) ⇒ p ≈ 0.05

Common misconceptions

A p-value is not the probability that H₀ is true.
Statistical significance does not imply practical significance.
p ≥ 0.05 does not prove H₀ is true; it only means insufficient evidence to reject it.
The threshold α = 0.05 is a convention, not a law — choose based on context.

Step-by-step example (Z-test, two-tailed)

Compute the test statistic: z = (x̄ − μ₀) / (σ / √n)
Find the area beyond |z| in the standard normal distribution.
Multiply by 2 for a two-tailed test: p = 2 × P(Z > |z|)
Compare p to α: if p < α, reject H₀.

References

Wasserstein, R.L. & Lazar, N.A. (2016). The ASA Statement on p-values: Context, Process, and Purpose. The American Statistician, 70(2), 129–133.
Fisher, R.A. (1925). Statistical Methods for Research Workers. Oliver & Boyd.
Neyman, J. & Pearson, E.S. (1933). On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society A, 231, 289–337.
Casella, G. & Berger, R.L. (2002). Statistical Inference (2nd ed.). Duxbury.

Related Calculators

How to Interpret a P-Value

The p-value is the probability of observing a test result at least as extreme as the one obtained, assuming the null hypothesis is true. A p-value of 0.03 means: if the null hypothesis were true, there is only a 3% chance of seeing data this extreme by chance. Conventionally, p < 0.05 is considered statistically significant — meaning the result is unlikely due to chance alone. p < 0.01 is highly significant; p < 0.001 is very highly significant.

Common misinterpretations: A p-value is NOT the probability that the null hypothesis is true. It does NOT measure effect size or practical importance. A statistically significant result with p = 0.04 might describe a trivially small real-world difference with a huge sample size. Always report effect size (Cohen's d, correlation coefficient, etc.) alongside p-values. With very large samples, almost any trivial difference becomes statistically significant.

P-Value Significance Thresholds

P-Value Range	Interpretation	Common Notation
< 0.001	Very highly significant	***
0.001–0.01	Highly significant	**
0.01–0.05	Significant	*
0.05–0.10	Marginal / trend	†
> 0.10	Not significant	ns