P-Value Calculator
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What Your Result Means
- P-value: The probability of observing a test statistic at least as extreme as yours, assuming the null hypothesis is true. Lower p-values suggest stronger evidence against the null hypothesis.
- Significant at α = 0.05: If p-value < 0.05, the result is considered statistically significant at the 5% level — a common threshold in research. This means there is less than a 5% probability the result occurred by chance alone.
- Significant at α = 0.01: If p-value < 0.01, the result is statistically significant at the 1% level — a stricter threshold often used when false positives are costly. Only a 1% probability of occurring by chance.
- Confidence level: Equal to (1 − p-value) × 100%. For example, a p-value of 0.05 gives a confidence level of 95%, indicating you can be 95% confident the null hypothesis should be rejected.
How This Calculator Works
Enter your test statistic (z or t value) from your statistical test. If you're performing a t-test, supply the degrees of freedom; leave it blank for a z-test. Select one-tailed (testing direction in one direction) or two-tailed (testing if a difference exists in either direction). The calculator uses the standard normal cumulative distribution function (CDF) to compute the p-value. For t-tests, it applies a normal approximation, which is accurate for large degrees of freedom but may drift slightly for small samples.
Quick Questions
What does a p-value actually tell me?
A p-value is the probability of your observed data (or more extreme data) occurring if the null hypothesis is true. It does not tell you the probability that the null hypothesis is true, nor does it measure the size or importance of an effect.
One-tailed vs two-tailed?
Use one-tailed if you predicted the direction of the effect in advance (e.g., "mean A is greater than mean B"). Use two-tailed if you only care whether a difference exists, without predicting direction. Two-tailed tests are more conservative and are standard when direction is unknown.
When to use a z-test vs t-test?
Use z-tests when the population standard deviation is known or the sample size is very large (>30). Use t-tests for small samples where you estimate the standard deviation from the sample. t-tests account for additional uncertainty in small samples.
Does p < 0.05 mean the result is true?
No. A small p-value indicates strong evidence against the null hypothesis, but it does not prove your alternative hypothesis is true. It also does not account for prior probability, effect size, or the number of tests you've run. Always consider context and avoid p-hacking.
Why does this calculator use a normal approximation for t-tests?
The normal approximation simplifies calculation and is accurate for many practical cases, especially with larger degrees of freedom. For very small samples or critical decisions, use a dedicated statistics package (R, Python, SAS) that computes exact t distribution values.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (hypothesis testing methodology and p-value definition)
- Penn State STAT 500 — Hypothesis Testing (z-test and t-test procedures)
- Wikipedia — Statistical Hypothesis Testing (p-value interpretation and background)
Method & review
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.