How this calculator works
Mean z tests use population standard deviations as known constants. Replacing them with estimated sample SDs changes the reference distribution and generally calls for t inference. The proportion test estimates the observed proportion but computes its standard error using the null probability. The normal approximation is questionable when either null expected binomial count is small; the result includes a count check. Independent random observations are required.
Formula
One mean: z = (x̄ - μ0)/(σ/√n). Two means: z = (x̄1 - x̄2 - Δ0)/√(σ1²/n1 + σ2²/n2). Proportion: z = (p̂ - p0)/√(p0(1 - p0)/n).
Worked example
With mean 102, null mean 100, population SD 10 and n = 100, z = 2 and two-sided p ≈ 0.0455003. For 60 successes in 100 against p0 = 0.5, the same statistic results.
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Frequently asked questions
Can I use the sample SD?
Use a t test when the population SD is estimated from the sample, especially for small samples.
Does a large sample fix a biased sample?
No. A large convenience or biased sample can give a very precise estimate of the wrong target.
What is the proportion count check?
Both n × p0 and n × (1 - p0) should be reasonably large. Ten is a common practical screening threshold.