Two means, two known sigmas
The two-sample z-test mirrors the one-sample version's bargain: the normal curve is earned only by genuinely known population standard deviations, both of them, one per group. Given that, the standard error of the difference combines the two known spreads, and the gap between sample means is read on the normal curve.
Two production lines, five measured parts each: means 84.6 and 75.6, with long-documented σ = 5 and σ = 4. The standard error of the difference is √(25/5 + 16/5) = 2.864; z = 9 ÷ 2.864 = 3.143, two-tailed p = 0.0017. The lines' true means differ. The interval for the gap (about 3.4 to 14.6 at 95%) excludes zero by a wide margin.
The same trap as the one-sample z: pasting sample SDs into the σ boxes of a five-per-group study. With samples that small the t-test's wider fence is not pedantry, it is the correct price of estimating spread.
Frequently asked questions
When would both sigmas ever be known?
Calibrated instruments, standardised manufacturing gauges, and large-scale standardised tests, processes whose spread has been documented over thousands of observations.
Why does this page ask for raw data if the sigmas are given?
It still needs each group's mean and n from the data. If you only have summary statistics, the same arithmetic runs from the two-sample t page's framework, or compute means on the mean calculator first.