Two Sample Z Test Calculator
Calculate independent Two-Sample Z-test statistic ($Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$), mean difference, $p$-value, and 95% CI.
Calculate Independent Two-Sample Z-Test
Enter sample mean, population SD (σ), and sample size for Group 1 and Group 2.
Calculation Results
Calculated using Two-Sample Z-test equations: Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}
Quick Summary
The Two Sample Z Test Calculator evaluates independent group mean differences ($\bar{x}_1 - \bar{x}_2$), two-sample $Z$-statistic, $p$-values, and 95% confidence intervals.
Formula Explanation
Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}
\text{95\% CI} = (\bar{x}_1 - \bar{x}_2) \pm 1.96 \cdot \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}How It Works
An independent 2-sample Z-test compares two distinct population means when population standard deviations $\sigma_1$ and $\sigma_2$ are known. The Two Sample Z Test Calculator evaluates standard error of difference $\text{SE}_{\text{diff}} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$ and $Z$-statistic.
Step-by-Step Worked Example
Practical Problem: Group 1 ($n_1 = 40, \bar{x}_1 = 82, \sigma_1 = 10$) vs Group 2 ($n_2 = 40, \bar{x}_2 = 75, \sigma_2 = 12$). Calculate $Z$-statistic and test for significant difference ($\alpha = 0.05$).
- Step 1: Calculate mean difference ($\bar{x}_1 - \bar{x}_2$): $82 - 75 = \mathbf{+7.0000}$.
- Step 2: Calculate variance ratios ($\sigma_1^2/n_1$ and $\sigma_2^2/n_2$): $100/40 = 2.50$; $144/40 = 3.60$.
- Step 3: Calculate standard error of difference ($\text{SE}_{\text{diff}}$): $\sqrt{2.50 + 3.60} = \sqrt{6.10} = \mathbf{2.4698}$.
- Step 4: Divide mean difference by $\text{SE}_{\text{diff}}$: $Z = 7.0 / 2.4698 = \mathbf{+2.8342}$.
- Step 5: Calculate two-tailed p-value: $p = 2 \times (1 - \Phi(2.8342)) = \mathbf{0.0046}$ (Reject $H_0$ at 5% significance level).
Real-World Calculation Examples
Scenario 1: Test Scores (x̄1 = 82, x̄2 = 75, n = 40)
Parameters: Group 1 vs Group 2
Result: Z = +2.8342, p = 0.0046 (Statistically significant).
Scenario 2: Factory A vs Factory B Output Weight
Parameters: Factory A (x̄1 = 500g, σ1 = 5g) vs Factory B (x̄2 = 498g, σ2 = 6g)
Result: Significant weight difference.
Scenario 3: National Exam Regional Scores
Parameters: Region 1 vs Region 2 large sample Z-test
Result: 2-sample Z-test evaluation.
Scenario 4: Large Sample Medical Trial (n1 = n2 = 500)
Parameters: Large sample Z-test comparison
Result: Standard normal Z-test.
Key Benefits of Using This Calculator
Known Population SDs ($\sigma_1, \sigma_2$)
Calculates exact two-sample Z-test when population standard deviations are known.
95% CI of Difference
Outputs 95% confidence interval for mean difference $(\bar{x}_1 - \bar{x}_2)$.
Large Sample Precision
Leverages Central Limit Theorem for large sample inferential testing.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a Two-Sample Z-test?
A Two-Sample Z-test checks whether the difference between two independent sample means $\bar{x}_1 - \bar{x}_2$ is statistically significant when population SDs are known.
What is null hypothesis ($H_0$) in 2-sample Z-test?
$H_0: \mu_1 - \mu_2 = 0$ (no difference between population means).
When should I use 2-sample Z-test instead of 2-sample t-test?
Use a Z-test when population standard deviations $\sigma_1, \sigma_2$ are known or when both sample sizes are large ($n_1, n_2 \ge 30$).
How do I calculate a 2-sample Z-test in Excel?
Use formula =(xBar1 - xBar2) / SQRT((sigma1^2/n1) + (sigma2^2/n2)).
What is standard error of difference?
$\text{SE}_{\text{diff}} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$.
Can sample sizes $n_1$ and $n_2$ be different?
Yes — Two-Sample Z-tests handle unequal sample sizes ($n_1 \neq n_2$) seamlessly.
What is critical Z-value for 95% confidence?
Two-tailed critical $Z = \pm 1.96$.
What is 95% confidence interval formula for mean difference?
$(\bar{x}_1 - \bar{x}_2) \pm 1.96 \cdot \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$.
What non-parametric test replaces 2-sample Z-test?
The Mann-Whitney U test (Wilcoxon rank-sum test).
What is Cohen's d for two independent samples?
$\text{Cohen's d} = \frac{\bar{x}_1 - \bar{x}_2}{\sigma_{\text{pooled}}}$.