Confidence Interval for Difference of Two Means (Z-Distribution)
Use this calculator to estimate the confidence interval for the difference between two population means when both population standard deviations are known (rare but theoretically important).
When to Use This Calculator
- Two independent samples from two different populations
- Both population standard deviations (σ) are known (rarely true in practice)
- Large sample sizes (typically n₁, n₂ ≥ 30, or normal populations assumed)
- Comparing average values between two groups (e.g., product A vs product B)
- Want to know if one mean is likely higher/lower than the other
Note: In practice, use the t-distribution version instead (when σ is unknown), which is more common.
How to Use This Calculator
Step 1: Enter Sample 1 mean and sample size
Step 2: Enter Sample 2 mean and sample size
Step 3: Enter both population standard deviations (σ₁ and σ₂)
Step 4: Select the confidence level (typically 95%)
Step 5: Click “Calculate”
Step 6: Results show:
- Standard Error of difference
- Z-critical value
- Margin of Error
- Lower and Upper Confidence Limits for the difference
| Confidence Interval Calculator for Difference of means | ||
|---|---|---|
| Sample 1 | Sample 2 | |
| Sample Mean | ||
| Sample Size | ||
| Population Standard Deviation | ||
| Confidence Level ($1-\alpha$) | ||
| Results | ||
| Standard Error of Diff. of Means: | ||
| Z-critical value: ($Z_{\alpha/2}$) | ||
| Margin of Error: ($E$) | ||
| Lower Confidence Limits: | ||
| Upper Confidence Limits: | ||
Theory & Formula
The $100(1-\alpha)%$ confidence interval for difference of two means (σ₁, σ₂ known) is:
$$CI = (\overline{x}_1 - \overline{x}2) \pm z{\alpha/2} \times \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$$
Where:
- $\overline{x}_1, \overline{x}_2$ = sample means
- $\sigma_1, \sigma_2$ = known population standard deviations
- $n_1, n_2$ = sample sizes
- $z_{\alpha/2}$ = z-critical value
Standard Error: $SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$
Assumptions
- Two independent samples - no relationship between observations
- Population SDs are known - σ₁ and σ₂ are known (rare)
- Large samples OR normal populations - n₁, n₂ ≥ 30 or populations approximately normal
- Random samples - data collected without bias
Worked Example
Scenario: Comparing average salary between two departments. Department 1: n₁=50, mean=$52,000, σ₁=$5,000. Department 2: n₂=45, mean=$48,000, σ₂=$4,500.
Solution:
- $SE = \sqrt{\frac{5000^2}{50} + \frac{4500^2}{45}} = \sqrt{500000 + 450000} = 976.24$
- For 95% CI: $z_{0.025} = 1.96$
- $E = 1.96 \times 976.24 = 1,913$
- $CI = (52000 - 48000) \pm 1913 = 4000 \pm 1913 = [2,087, 5,913]$
Interpretation: We’re 95% confident that Department 1’s mean salary is between $2,087 and $5,913 higher than Department 2.
Key Differences: Z vs T for Two Means
| Feature | Z-Distribution (σ known) | T-Distribution (σ unknown) |
|---|---|---|
| When to Use | Both σ known (rare) | Both σ unknown (typical) |
| Critical Values | Smaller | Larger (wider CI) |
| Intervals | Narrower | Wider |
| Real-world | Almost never used | Standard choice |
Recommendation: Use t-distribution version for real data.
Tutorial: CI for Two Means T-Distribution Version: CI for Two Means (T-Distribution)