Calculator
Example Data Table
| Example | Group 1 successes | Group 1 size | Group 2 successes | Group 2 size | Difference | Approx p value |
|---|---|---|---|---|---|---|
| Website conversion test | 64 | 200 | 45 | 180 | 0.070 | 0.132 |
| Training pass rate | 82 | 150 | 61 | 145 | 0.126 | 0.030 |
| Survey preference | 210 | 500 | 185 | 480 | 0.035 | 0.269 |
Formula Used
The sample proportions are p1 = x1 / n1 and p2 = x2 / n2.
The observed difference is d = p1 - p2.
For the common pooled test, p pooled = (x1 + x2) / (n1 + n2).
The pooled standard error is SE = sqrt(p pooled × (1 - p pooled) × (1 / n1 + 1 / n2)).
The unpooled standard error is SE = sqrt(p1 × (1 - p1) / n1 + p2 × (1 - p2) / n2).
The z statistic is z = ((p1 - p2) - d0) / SE. A continuity correction subtracts a small adjustment from the tested difference.
The confidence interval is (p1 - p2) ± z critical × unpooled SE.
Risk ratio, odds ratio, Cohen h, power, and estimated sample size are added as practical reporting measures.
How to Use This Calculator
- Enter successes for the first independent group.
- Enter the total sample size for the first group.
- Enter successes and sample size for the second group.
- Set the null difference. Use zero for equal proportions.
- Choose the alternative hypothesis before calculating.
- Enter alpha, confidence level, and desired power.
- Select pooled testing or continuity correction when needed.
- Press Calculate. The result appears above the form.
- Use CSV or PDF buttons to export the current calculation.
Two Sample Proportion Test Guide
What This Test Compares
A two sample proportion test compares two independent rates. It is useful when each observation has two possible outcomes. Examples include pass or fail, buy or not buy, click or no click, and recover or not recover. The calculator uses successes and sample sizes. It then estimates each sample proportion and the difference between them.
How the Test Works
The main test asks whether the observed difference is larger than random sampling variation. The default null value is zero. That means both population proportions are treated as equal. You may also test a planned margin by entering a different hypothesized difference. The tool supports two sided, left tailed, and right tailed tests. It also offers an optional continuity correction for conservative reporting.
Reading the Output
The output includes the z statistic, p value, confidence interval, pooled estimate, unpooled standard error, odds ratio, risk ratio, and Cohen's h. These measures help you report both significance and size. A small p value can show evidence against the null hypothesis. Effect sizes show whether the practical difference is meaningful. Both ideas matter in clear reporting.
Approximation Rules
Use the normal approximation only when sample counts are large enough. A common check is at least five successes and five failures in each group. When counts are very small, an exact method may be better. The calculator warns you when this condition is weak.
Planning and Exporting
This page also estimates observed power and an equal group sample size. These values are approximate. They help when planning a future study or reviewing a completed one. CSV and PDF exports make it easier to save the final numbers.
Practical Uses
The calculator is helpful for surveys, quality checks, marketing tests, classroom examples, and clinical summaries. It keeps all core values on one page. This makes review faster. It also reduces manual rounding mistakes. You can compare the raw proportions, the adjusted test result, and the interval estimate together. This helps readers see both uncertainty and direction without searching across separate notes during audits and reviews.
Reporting Tips
Good inputs create useful conclusions. Enter counts carefully. Choose the alternative before looking at the p value. Report the confidence level and alpha with the result. Mention whether a pooled test or continuity correction was used. Small checks make final statistical choices easier to explain.
FAQs
1. What is a two sample proportion test?
It is a z test that compares two independent proportions. Each group must have successes and total observations. The test checks whether the population proportions are likely equal or meaningfully different.
2. When should I use this calculator?
Use it when both groups have binary outcomes. Common cases include conversion rates, pass rates, response rates, defect rates, and survey choices.
3. What is the null hypothesis?
The default null hypothesis says the two population proportions are equal. In difference form, it says p1 - p2 equals zero.
4. What does the p value mean?
The p value shows how unusual the observed difference is under the null hypothesis. A smaller value gives stronger evidence against the null.
5. Should I use pooled standard error?
Use pooled standard error for the common equality test when the hypothesized difference is zero. Use unpooled standard error for confidence intervals and nonzero null differences.
6. What is continuity correction?
Continuity correction adjusts the z statistic for discrete count data. It can make the test more conservative, especially for smaller samples.
7. What is Cohen's h?
Cohen's h is an effect size for two proportions. It uses an arcsine transformation and helps describe practical difference beyond the p value.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet use. Use the PDF button for a simple saved report with inputs, statistics, and decisions.