Z-Test for Difference Between Two Population Means
Use this calculator to test if two population means differ when both population standard deviations are known (rare scenario).
When to Use
- Two independent samples from different populations
- Both population SDs known (σ₁ and σ₂)
- Large samples or normal populations
- Testing if means differ between two groups
- Note: Use t-test instead in almost all practical situations
How to Use
Step 1: Enter sample 1 mean and sample 2 mean
Step 2: Enter both population standard deviations
Step 3: Enter both sample sizes
Step 4: Enter level of significance (α, typically 0.05)
Step 5: Select tail type (left, right, or two-tailed)
Step 6: Click “Calculate”
Interpret: If p-value < α, means differ significantly
| Z test Calculator for two means | |||
|---|---|---|---|
| Sample 1 | Sample 2 | ||
| Sample Mean | |||
| Standard Deviation | |||
| Sample Size | |||
| Level of Significance ($\alpha$) | |||
| Tail | Left tailed Right tailed Two tailed | ||
| Results | |||
| Standard Error of Diff. of Means: | |||
| Test Statistics Z: | |||
| Z-critical value(s): | |||
| p-value: | |||
Test Formula
$$Z = \frac{(\overline{x}_1 - \overline{x}_2) - 0}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$
Where:
- $\overline{x}_1, \overline{x}_2$ = sample means
- $\sigma_1, \sigma_2$ = known population standard deviations
- $n_1, n_2$ = sample sizes
Worked Example
Scenario: Comparing average salary between two departments. Dept A (n₁=50, x̄₁=$55K, σ₁=$8K) vs Dept B (n₂=60, x̄₂=$52K, σ₂=$7K).
Hypotheses:
- H₀: μ₁ = μ₂ (salaries are equal)
- H₁: μ₁ ≠ μ₂ (salaries differ, two-tailed)
Calculation:
- $SE = \sqrt{\frac{64M}{50} + \frac{49M}{60}} = 1.37$
- $Z = (55-52)/1.37 = 2.19$
- For α=0.05, critical Z = ±1.96
- p-value = 0.029
Decision: p-value (0.029) < α (0.05), reject H₀. Significant evidence that departments have different average salaries.
⚠️ Use t-test instead - Population SDs are rarely known in practice
Related: T-test for Two Means, Tutorial