What a p-value answers
A p-value is a conditional probability: assuming the null hypothesis (no real effect) is true, how likely is it to see a test statistic at least as extreme as the one you obtained? A small p-value means your result would be surprising under the null hypothesis, which is treated as evidence against it. A large p-value means the data is comfortably consistent with no effect.
This tool starts from a z statistic and uses the standard normal distribution to find that tail probability.
Choosing the right tail
The tail must match the hypothesis you set before collecting data, because it changes how the extreme region is defined.
- Two-tailed: you care about a difference in either direction; the p-value counts both tails.
- Right-tailed: you predicted the value would be larger than expected; only the upper tail counts.
- Left-tailed: you predicted it would be smaller; only the lower tail counts.
- A two-tailed p-value is roughly double the matching one-tailed value, so switching tails after seeing the data is a misuse.
Common misreadings
A p-value is not the probability that the null hypothesis is true, nor the probability that your result happened by chance. It also says nothing about the size or importance of an effect — with a large enough sample, a trivial difference can produce a tiny p-value.
Treat the 0.05 line as a convention, not a law of nature, and fix your threshold before running the test. Statistical significance is not proof; pair the p-value with effect sizes and confidence intervals, and remember this tool is a learning aid rather than a substitute for a full analysis.
Formula
left: Φ(z); right: 1 − Φ(z); two-tailed: 2·(1 − Φ(|z|))Frequently asked questions
- What counts as significant?
- A common threshold is 0.05: if the p-value is below it, the result is usually called statistically significant. The threshold should be chosen before running the test.