A p-value is the probability of observing test results at least as extreme as what you got, assuming the null hypothesis is true. The Toolenza calculator converts a z-statistic (or t-statistic with degrees of freedom) into a one- or two-tailed p-value via the normal / Student-t distribution.

What p-value actually means (and doesn't)

It does not mean "the probability the null is true" — that's the famous misinterpretation. It means: if the null were true, how often would we see this result or one more extreme by chance.

A p-value of 0.03 means: if there's truly no effect, we'd still see a result this strong (or stronger) 3% of the time by random sampling.

The 0.05 threshold

Fisher proposed 0.05 in 1925 as a conventional cutoff, with the caveat that it shouldn't be treated as a rule. A century later, "p < 0.05 = significant" has hardened into automatic-reasoning across the social sciences — and the replication crisis is the consequence. Modern best practice:

• Pre-register your hypothesis before collecting data. P-hacking (re-running analyses until p < 0.05 appears) is the #1 source of irreproducible findings.
• Report effect size, not just p. A tiny p-value with a tiny effect is statistical significance without practical significance.
• Use confidence intervals to communicate the uncertainty range, not p-values to communicate a binary verdict.
• For exploratory analyses, use 0.01 or stricter; reserve 0.05 for confirmatory work.

Pitfalls

Multiple comparisons — if you test 20 outcomes at p < 0.05, expect 1 false positive by chance. Adjust with Bonferroni or false-discovery-rate methods. Small samples — p-values from n < 30 are unreliable for non-normal data; use bootstrapping or exact tests instead.