Calculator
Example Data Table
| Case | Sample 1 | Sample 2 | Suggested method | Question |
|---|---|---|---|---|
| Exam scores | n = 36, mean = 82.4, sd = 10.1 | n = 40, mean = 77.8, sd = 11.4 | Welch means | Do average scores differ? |
| Before and after | Before raw list | After raw list | Paired means | Did paired values change? |
| Conversion rate | 64 successes from 120 | 50 successes from 115 | Two proportions | Are proportions different? |
Formula Used
Welch means: t = ((x̄1 - x̄2) - d0) / sqrt(s1^2/n1 + s2^2/n2). Degrees of freedom use the Welch-Satterthwaite formula.
Pooled means: sp = sqrt(((n1 - 1)s1^2 + (n2 - 1)s2^2) / (n1 + n2 - 2)). Then SE = sp sqrt(1/n1 + 1/n2).
Paired means: t = (d̄ - d0) / (sd / sqrt(n)). The confidence interval is d̄ plus or minus t critical times SE.
Two proportions: z = ((p1 - p2) - d0) / SE. The test uses pooled SE when d0 equals zero. The interval uses unpooled SE.
How to Use This Calculator
- Choose the inference method that matches your study design.
- Select the alternative hypothesis and confidence level.
- Enter summary statistics, raw lists, or proportion counts.
- Use the null difference field for the tested population difference.
- Press Calculate to view the result above the form.
- Download the table as a CSV or PDF file.
Understanding Two Sample Inference
Two sample inference helps compare two independent or paired groups. It asks whether an observed difference is large enough to show a real population difference. The calculator supports mean differences, paired differences, and proportion differences. It also reports confidence intervals, test decisions, and effect sizes. These outputs help explain both significance and practical size.
When To Use It
Use the mean option when each group has a numerical outcome. Use Welch inference when standard deviations differ, or when equal variance is doubtful. Use the pooled option only when equal spread is reasonable. Use the paired option when observations belong together, such as before and after values. Use the proportion option when each group has counts of successes and totals.
Interpreting Results
The test statistic measures distance from the null hypothesis in standard error units. A small p value suggests the observed difference would be unusual if the null statement were true. The confidence interval gives a range of likely values for the population difference. If a two sided interval excludes the null value, the matching test usually rejects at that confidence level.
Effect Size Matters
Statistical significance depends on sample size. Large studies can detect tiny differences. Small studies may miss useful effects. Effect sizes add context. Cohen's d or Hedges' g describes standardized mean separation. Cohen's h describes standardized proportion separation. Report effect size with the confidence interval for a clearer conclusion.
Good Data Practice
Check that samples were collected fairly. Look for extreme outliers, entry mistakes, and mismatched paired values. Make sure units match across samples. For proportions, ensure successes are not larger than totals. For small samples, normal methods may be rough. A larger sample usually improves approximation quality.
Reporting Tip
A useful report states the method, null difference, alternative direction, statistic, degrees of freedom, p value, interval, and decision. Add the sample summaries too. This calculator gives those parts together, so results can be copied into homework, lab notes, dashboards, or quality reports.
Limitations To Remember
This tool uses large sample approximations and classical formulas. It does not replace study design, randomization, or subject knowledge. Treat results as evidence, not proof. Confirm assumptions before making important decisions. Use judgment with every conclusion.
FAQs
1. What is two sample inference?
Two sample inference compares two groups. It estimates a population difference and tests whether that difference is likely to be zero or another chosen value.
2. When should I use Welch instead of pooled testing?
Use Welch testing when sample standard deviations differ, sample sizes differ, or equal variance is not trusted. It is often the safer default.
3. What does the null difference mean?
The null difference is the value tested by the hypothesis test. Most studies use zero, meaning no population difference between groups.
4. Can I enter raw data?
Yes. Paste comma, space, or line separated values. For paired testing, both raw lists must have the same number of values.
5. Why is the p value important?
The p value measures how unusual the sample result is under the null hypothesis. Smaller values give stronger evidence against the null.
6. What does the confidence interval show?
The confidence interval gives a plausible range for the true population difference. Wider intervals show more uncertainty in the estimate.
7. What is an effect size?
An effect size describes the practical size of a difference. It helps judge importance beyond the p value alone.
8. Can this calculator prove causation?
No. It supports statistical comparison. Causation depends on study design, controls, random assignment, and subject knowledge.