From a z statistic to its p-value
Z-tests (one-sample with a known σ, proportion tests, large-sample tests) end in a z statistic read on the standard normal curve. Enter z and read the tail you need: left (alternative: less), right (alternative: greater), or the two-tailed default.
z = 1.96: two-tailed p = 0.0500, right-tailed p = 0.0250. Again, this is not a coincidence: 1.96 is the famous critical value that fences off the middle 95% of the normal curve. Our bakery z-test gives z = 2.1213, two-tailed p = 0.0339.
When is z the right curve?
When the standard error in your statistic's denominator is built from a known σ, from a proportion's null value (p₀(1−p₀) is known once p₀ is set), or from a sample large enough that the estimated SD behaves like a known one. With a small sample and an estimated SD, the honest curve is t, not z.
Frequently asked questions
What is the p-value for z = 1.645?
Right-tailed: 0.05. Two-tailed: 0.10. It is the one-sided 5% critical value.
Why is my two-tailed p double the one-tailed?
Because the two-tailed test counts extremes in both directions. By construction it is twice the one-sided area for a symmetric curve, when the statistic falls on the alternative's side.