One pattern under four tests
Every test statistic on this page is the same sentence: (what you saw − what the null expects) ÷ standard error. The families differ only in the ingredients. A z for a mean divides by σ/√n with σ known; a t divides by s/√n and pays for estimating σ with heavier tails and n − 1 degrees of freedom. A one-proportion z uses the null's own p0 inside its standard error, because the calculation lives inside the null's world. The two-proportion z pools both groups into one proportion for the same reason: under the null there is only one proportion to estimate.
Mean, σ known: x̄ = 52 against μ0 = 50, σ = 10, n = 64: z = 2 ÷ 1.25 = 1.600, two-tailed p = 0.1096. One proportion: 64 successes in 120 against p0 = 0.5: z = 0.730, p = 0.465. Same skeleton, different flesh. Watching the standard error swap ingredients across the four panels is the fastest way to see why each test is built the way it is.
When this page is the right tool
Textbooks, papers and reports hand you summary statistics, not raw data: a mean and SD from a published table, counts from a press release. This page rebuilds the statistic and p-value from those summaries so you can check a claim or finish an exam-style problem. When you hold the raw data, the dedicated test pages on this site are the better route: they run the same core maths and add the assumption checks, effect sizes and confidence intervals that a bare statistic leaves out.
Feeding a sample SD into the z family because the problem "feels" z-like. If σ is not genuinely known from outside the sample, the t family is the honest choice, especially at small n, where the difference in tails is largest.
Frequently asked questions
What is a test statistic, in one sentence?
A standardised distance between your data and the null hypothesis, measured in standard errors, whose distribution under the null is known.
Why does the one-proportion z use p0 in its standard error?
Because the p-value is computed inside the null hypothesis's world, where p equals p0. The CI for a proportion, a different task, uses the sample proportion instead.
Which tail should I choose?
The one your hypothesis named before seeing data: greater, less, or different in either direction (two-tailed, the default). Switching tails after peeking invalidates the p-value.