Confidence Interval for Difference of Two Means (Unequal Variances - Welch’s Method)
Use this calculator for confidence intervals when comparing two independent means where sample standard deviations are unknown and unequal (Welch’s method is more robust).
When to Use This Calculator
- Two independent samples with different variances
- Unequal population variances - don’t assume σ₁ = σ₂
- Sample SDs unknown - estimated from samples
- Most robust choice - use this unless you’ve verified equal variances with Levene’s test
- Default recommendation when unsure whether variances are equal
How to Use
Step 1: Enter both sample means
Step 2: Enter both sample sizes
Step 3: Enter both sample standard deviations
Step 4: Select confidence level (typically 95%)
Step 5: Click “Calculate”
| Confidence Interval Calculator for Difference between means | ||
|---|---|---|
| Sample 1 | Sample 2 | |
| Sample Mean | ||
| Sample Size | ||
| Standard Deviation | ||
| Confidence Level ($1-\alpha$) | ||
| Results | ||
| Standard Error of Diff. of Means: | ||
| Degrees of Freedom: | ||
| t-critical value: ($t_{\alpha/2,df}$) | ||
| Margin of Error: ($E$) | ||
| Lower Confidence Limits: | ||
| Upper Confidence Limits: | ||
Welch’s Method (Unequal Variances)
$$CI = (\overline{x}_1 - \overline{x}2) \pm t{\alpha/2, df} \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$
Welch-Satterthwaite degrees of freedom: $$df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}}$$
Advantages of Welch’s Method
- Doesn’t assume equal variances - more conservative
- Better Type I error control - maintains correct rejection rate
- Recommended by statisticians - robust choice when unsure
- Use by default - when you haven’t tested for equal variances
Worked Example
Scenario: Testing if Brand A (s₁=15, n₁=30, x̄₁=85) differs from Brand B (s₂=25, n₂=25, x̄₂=75).
Solution:
- $SE = \sqrt{\frac{225}{30} + \frac{625}{25}} = 4.15$
- Welch df ≈ 40
- $t_{0.025,40} = 2.021$
- E = 2.021 × 4.15 = 8.39
- CI = 10 ± 8.39 = [1.61, 18.39]
Interpretation: We’re 95% confident Brand A differs from Brand B by 1.61 to 18.39 points.
When variances are equal: Use equal variances method
Related: Tutorial