Calculator
Formula Used
General interval: Estimate ± Critical value × Standard error
Welch mean interval: (x̄1 - x̄2) ± t* × √(s1²/n1 + s2²/n2)
Pooled mean interval: (x̄1 - x̄2) ± t* × sp × √(1/n1 + 1/n2)
Pooled variance: sp² = [((n1 - 1)s1²) + ((n2 - 1)s2²)] / (n1 + n2 - 2)
Known sigma interval: (x̄1 - x̄2) ± z* × √(σ1²/n1 + σ2²/n2)
Two proportion interval: (p̂1 - p̂2) ± z* × √[p̂1(1 - p̂1)/n1 + p̂2(1 - p̂2)/n2]
How to Use This Calculator
- Select two means or two proportions.
- Choose summary statistics or raw values for means.
- Enter the sample data in the matching fields.
- Keep 95 for a standard 95% confidence interval.
- Choose Welch unless equal variance is justified.
- Select the subtraction order used in your report.
- Press Submit to show results above the form.
- Use CSV or PDF download for saved output.
Example Data Table
| Example | Sample 1 | Sample 2 | Suggested method |
|---|---|---|---|
| Two means | n = 40, mean = 82, s = 10 | n = 35, mean = 76, s = 12 | Welch t interval |
| Two proportions | 54 successes from 120 | 42 successes from 110 | Two proportion z interval |
| Equal variance means | n = 28, mean = 15.2, s = 3.1 | n = 30, mean = 13.9, s = 3.0 | Pooled t interval |
Understanding the Two Sample Interval
A two sample confidence interval estimates the true difference between two population values. It may compare two averages. It may also compare two proportions. The calculator uses sample information to build a range around the observed difference. A 95% interval means the method captures the true difference in about 95 out of 100 repeated studies.
Why This Calculator Helps
Manual work can be slow. Two sample problems need careful choices. You must decide whether the data uses means or proportions. You must also choose the variance assumption. Independent means often use Welch's method. Equal spread data may use a pooled method. Known population standard deviations use a z method. Proportions also use a z method.
What The Results Mean
The estimate is the observed difference between sample one and sample two. The standard error measures sampling noise. The critical value sets the width of the interval. The margin of error is the critical value times the standard error. The lower and upper limits form the confidence interval.
Using Mean Data
For means, enter each sample size, mean, and standard deviation. You can also paste raw lists. The tool then calculates the mean and sample deviation. Use Welch's option when the groups are independent and spreads may differ. Use pooled only when equal variance is reasonable. Use the known sigma option only for population deviations.
Using Proportion Data
For proportions, enter successes and sample sizes. The calculator converts them into sample proportions. It then finds the difference and standard error. This works for survey rates, pass rates, defect rates, and similar outcomes.
Good Practice
Check that both samples are random or representative. Avoid using this method for strongly biased samples. For tiny samples, inspect the data first. Outliers can change mean intervals. For proportions, very small counts may need exact methods. Report the order of subtraction. This page can calculate sample one minus sample two, or the reverse.
Exporting Results
Use the CSV export for spreadsheets. Use the PDF export for a quick report. Keep the selected method with your answer. It explains why the interval width was chosen. Always include units when values have units. This keeps the interpretation clear and prevents report confusion later.
FAQs
What does a 95% confidence interval mean?
It means the method would capture the true population difference in about 95% of many repeated samples. It does not mean there is a 95% chance that one computed interval contains the fixed true value.
Should I use Welch or pooled t?
Use Welch for most independent mean comparisons. It handles unequal standard deviations better. Use pooled t only when equal population variances are a reasonable assumption.
Can I calculate a two proportion interval?
Yes. Select two independent proportions. Then enter successes and sample sizes for both groups. The calculator uses the usual large sample z interval for the difference.
Can I paste raw sample data?
Yes. Select raw sample lists. Paste numbers separated by commas, spaces, semicolons, or new lines. The tool calculates each mean, sample size, and sample standard deviation.
What is the margin of error?
The margin of error is the critical value multiplied by the standard error. It shows how far the confidence limit sits from the observed sample difference.
Which difference order should I choose?
Choose the order used in your report. Sample 1 minus Sample 2 gives a positive estimate when sample one is larger. The reverse changes the sign.
Why is my interval very wide?
A wide interval usually comes from small sample sizes, high variation, or a higher confidence level. For proportions, rates near the middle can also increase standard error.
Can I export the calculation?
Yes. Use the CSV button for spreadsheet use. Use the PDF button for a simple report containing the method, estimate, limits, and interpretation.