Two-Sample z-Test Sample Size (Known Variance)

Methodology

This calculator implements the two-sample z-test for two independent means when the population variances are known. The null hypothesis is H₀: μ₁ − μ₂ = 0, with either a two-sided alternative or a one-sided alternative.

Test Statistic

When the population variances are known, the standard error of the difference between two independent sample means is

For the usual superiority calculation, the null difference δ₀ is zero. The sample-size calculation uses the anticipated difference δ = |μ₁ − μ₂| as the effect to be detected.

Power Calculation

Under the anticipated alternative, the standardized effect is

For a two-sided test with significance level α, the exact normal approximation used here is

For a one-sided test, the corresponding expression is

where Φ is the standard normal cumulative distribution function. Rather than relying on a rounded closed-form approximation, the calculator searches over integer sample sizes and returns the first allocation whose calculated power is at least the requested target.

Group Allocation

With equal allocation, N₁ = N₂. With ratio allocation, the calculator uses N₂ = R × N₁ and searches over integer N₁, rounding N₂ upward to the next whole participant when necessary.

For equal allocation, the familiar approximate closed-form expression follows directly from the standardized effect:

The actual implementation performs the power calculation directly and searches for the smallest integer sample size, which also accounts for the small contribution of the opposite tail in a two-sided test.

Worked Validation Example

this method provides a validation example using a two-sided two-sample z-test with 90% target power, α = 0.05, equal allocation, a mean difference of 2, and known standard deviations of 4.1 and 5.3. Equivalently, the known variances entered here are σ₁² = 16.81 and σ₂² = 28.09.

The JavaScript implementation below reproduces this validation result: 117 participants per group gives power below 0.90, whereas 118 per group gives power approximately 0.90013. Thus 118 per group is the smallest integer allocation satisfying the target.

Assumptions

• The two groups are independent.
• The outcome is modeled using normal distributions.
• The population variances are known rather than estimated from the sample.
• The specified mean difference represents the effect the study is designed to detect.
• The significance level and target power are specified before the sample-size calculation.

References

the software, LLC. (2023). Two-Sample Z-Tests Allowing Unequal Variance. this method Sample Size Software, Chapter 428. the relevant methodological literature describes sample-size and power calculations for one- and two-sided two-sample z-tests with known unequal variances and provides the worked validation example used above.
this method procedure documentation

the software, LLC. (2023). Two-Sample Z-Tests Assuming Equal Variance. this method Sample Size Software, Chapter 426. This documentation gives the corresponding known-common-variance two-sample z-test and cites Julious (2010) and Chow, Shao, and Wang (2008) as references for the power and sample-size methodology.
this method equal-variance procedure documentation

Julious, S. A. (2010). Sample Sizes for Clinical Trials. Boca Raton: CRC Press/Taylor & Francis. Chapter 3 covers sample-size calculations for parallel-group superiority trials with normal data.

Julious, S. A., Tan, S. B., & Machin, D. (2010). Sample Size Calculations for Clinical Trials. In An Introduction to Statistics in Early Phase Trials, pp. 37–53. Wiley. DOI: 10.1002/9780470686164.ch3 .

Statsols /. Advanced User Manual, Section 7.2.5.1, “Two Means (GST1).” The documentation expresses the maximum information for an inequality two-sample z-test in terms of σ₁²/N₁ + σ₂²/N₂ and relates the information to the standardized mean difference.
Advanced User Manual