Two Sample Z Test Calculator

Select null and alternative hypotheses:

Sample 1

Sample Mean

Population Standard Deviation

Sample Size

Sample 2

Sample Mean

Population Standard Deviation

Sample Size

Significance Level

About This Test

Two-Sample Z-Test

This tests whether the means of two independent populations differ, when the standard deviation of both populations is already known — not estimated from the samples. This is fairly specialized; most real comparisons only know sample standard deviations, in which case one of the Two-Sample T-Test calculators is what you actually want.

When to Use It

Comparing two manufacturing processes with long-established, known standard deviations
Comparing two standardized measurement systems with published variability
Any comparison where both population standard deviations are genuinely known, not estimated

The Formula

\[ Z = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}} \]

Worked Example

Factory A has a known population standard deviation of 0.5mm; a sample of 50 parts has a mean of 25.3mm. Factory B has a known standard deviation of 0.4mm; a sample of 60 parts has a mean of 25.1mm. At α = 0.05, do the factories' true averages differ?

\( \bar{X}_1 = 25.3, \ \bar{X}_2 = 25.1, \ \sigma_1 = 0.5, \ \sigma_2 = 0.4, \ n_1 = 50, \ n_2 = 60 \)
\( Z \approx 2.28 \), two-tailed critical value ≈ ±1.96
p-value ≈ 0.0224
Since 0.0224 < 0.05, we reject H₀ — the factories' true averages likely differ.

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