Welch's t-test Tool
Means tool
Compare means of two independent groups with fan charts and narrative insights.
QUICK START: Choose Your Path
MARKETING SCENARIOS
You're leading a marketing analytics project for an e-commerce brand that is A/B testing email campaigns. Group 1 might represent subscribers who received an optimized subject line, while Group 2 saw the current control subject line. After the campaign, you collected key performance indicators such as click-through rate, average order value, or downstream revenue per recipient.
Use the tool below to plug in the sample statistics for each variant, evaluate whether the observed lift is meaningful, and decide if the new creative warrants a broader rollout. Adjust the hypothesized difference to model the minimum effect size that would justify marketing spend.
INPUTS & SETTINGS
Select Data Entry / Upload Mode
Enter aggregated statistics for each group. Scenarios and uploads overwrite these fields automatically.
| Group | Name | Mean (μ) | Std. deviation (s) | Sample size (n) |
|---|---|---|---|---|
| Group 1 | ||||
| Group 2 |
Upload raw data
Upload a long-format file with columns like group,value. We’ll calculate the means, standard deviations, and sample sizes for the first two groups discovered.
Drag & Drop raw data file (.csv, .tsv, .txt)
Long format with headers like group,value; at least two unique groups. Up to 2,000 rows.
No raw file uploaded.
Analysis Settings
YOUR DECISION
Enter data to see your result
We'll calculate whether the group means differ significantly
VISUAL OUTPUT
Difference Fan Chart
Visual Output Settings
Means Fan Chart
Difference Fan Chart
TEST RESULTS
💡 How to Interpret These Results
Reading the t-statistic and p-value:
- t-statistic: Measures how many "standard errors" apart the two group means are. Larger |t| = stronger evidence of a difference.
- p < 0.05: Strong evidence that the means differ. Reject H₀ (means are equal).
- p ≥ 0.05: Insufficient evidence. Cannot conclude the group means differ significantly.
- Very small p (<0.001): Very strong evidence. The difference is highly unlikely due to chance alone.
Understanding the confidence interval:
- The CI shows the plausible range for the true difference between population means
- If the CI excludes zero, the difference is statistically significant at that confidence level
- Narrower intervals = more precision (often from larger samples or less variance)
- The fan chart visualizes this—see where zero falls relative to the bands
Effect size (Cohen's d):
- |d| < 0.2: Trivial effect—difference is negligible in practical terms
- 0.2 ≤ |d| < 0.5: Small effect—detectable but may not matter much operationally
- 0.5 ≤ |d| < 0.8: Medium effect—noticeable and often practically meaningful
- |d| ≥ 0.8: Large effect—substantial difference worth acting on
Statistical vs. practical significance:
A result can be statistically significant (p < 0.05) but have a trivial effect size. With large samples, even tiny differences become "significant." Always pair p-values with effect size (Cohen's d) and confidence intervals to make sound business decisions.
APA-Style Statistical Reporting
Managerial Interpretation
Summary of Estimates
| Measure | Estimate | Standard Error | Lower Bound | Upper Bound |
|---|
DIAGNOSTICS & ASSUMPTIONS
Diagnostics & Assumption Tests
Provide group summaries to review sample size balance, variance ratios, power, and interval diagnostics.
📚 STATISTICAL REFERENCE
Welch's t-Test Equations & Theory
The Welch t-test evaluates whether two independent group means differ when variances may be unequal.
Test statistic: $$t = \frac{\bar{x}_1 - \bar{x}_2 - \Delta_0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$
Confidence interval: $$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2,\,\nu} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$
Key concepts:
- Δ₀ (delta-zero): The hypothesized difference between means (usually 0)
- Welch's adjustment: Uses modified degrees of freedom when variances differ
- Two-tailed test: Detects differences in either direction
As degrees of freedom increase, Welch's t behaves like a z-test. When samples are small or variances highly imbalanced, keep the heavier tails in mind and report the estimated power.