Example Data Table
| Scenario |
Group 1 Successes |
Group 1 Total |
Group 2 Successes |
Group 2 Total |
Confidence |
| Email campaign response |
42 |
200 |
30 |
180 |
95% |
| Defect comparison |
18 |
250 |
31 |
260 |
95% |
| Pass rate study |
86 |
120 |
74 |
115 |
99% |
Formula Used
Sample proportions are calculated as p1 = x1 / n1 and p2 = x2 / n2.
The observed difference is d = p1 - p2.
For a zero null difference, the pooled proportion is pc = (x1 + x2) / (n1 + n2).
The pooled standard error is SE = sqrt(pc(1 - pc)(1 / n1 + 1 / n2)).
The unpooled standard error is SE = sqrt(p1(1 - p1) / n1 + p2(1 - p2) / n2).
The test statistic is z = ((p1 - p2) - null difference) / SE.
The confidence interval is d ± z critical × unpooled SE.
How To Use This Calculator
Enter the success count and total sample size for each group.
Choose a confidence level. Use 95 for common reporting.
Enter the null difference. Use 0 when testing equal proportions.
Select the alternative hypothesis that matches your claim.
Apply continuity correction when you want a conservative normal approximation.
Press Calculate. The result appears above the form.
Use CSV or PDF buttons to save the current result.
Testing Two Proportions
A two proportion test compares rates from two independent groups. It is common in surveys, quality checks, medical studies, and marketing tests. Each group has a count of successes and a total sample size. The calculator turns those counts into sample proportions. It then tests whether the difference is large enough to matter statistically.
What The Test Measures
The main value is the difference between two proportions. A positive value means group one has the higher rate. A negative value means group two has the higher rate. The z statistic compares the observed difference with the null difference. The p value shows how unusual that result is, if the null claim is true.
Advanced Result Details
This tool also reports a confidence interval. The interval gives a likely range for the true difference. It uses the unpooled standard error for the estimate. The test uses a pooled error when the null difference is zero. That approach matches the standard two proportion z test. When the null difference is not zero, the calculator uses an unpooled error for the test.
Useful Effect Measures
Statistical significance is not the whole story. The risk ratio compares the two observed rates. The odds ratio compares odds of success. Cohen's h gives a scale-free effect size for proportions. These measures help judge practical importance. A tiny p value may still describe a small business effect.
Interpreting The Output
Check assumptions before trusting the answer. The samples should be independent. Counts should come from random or representative data. Expected successes and failures should usually be large enough. Small samples may need an exact method. Use the selected alternative hypothesis to match your research question. Choose two sided when any difference matters. Choose greater when group one is expected to be higher. Choose less when group one is expected to be lower.
Why It Helps
Manual testing can be slow and error prone. This calculator keeps every important value together. It shows formulas, interval estimates, p values, and export options. You can test campaign rates, pass rates, defect rates, or response rates. The example table gives clear sample data. CSV and PDF buttons help save results for reports. Use exported files during review.
FAQs
What is a two proportion test?
It is a z test that compares two independent sample proportions. It checks whether the observed difference is likely under a selected null hypothesis.
When should I use this calculator?
Use it when both groups have success counts and total counts. Examples include pass rates, response rates, defect rates, and conversion rates.
What does the p value mean?
The p value estimates how unusual your observed result is under the null hypothesis. Smaller values give stronger evidence against the null claim.
What is the null difference?
The null difference is the claimed difference between proportions. Most tests use zero, which means the two population proportions are assumed equal.
Why is a confidence interval included?
The confidence interval gives a likely range for the true difference. It helps show both direction and practical size of the effect.
What is continuity correction?
Continuity correction adjusts the normal approximation for count data. It can make the test more conservative, especially with smaller samples.
What is Cohen's h?
Cohen's h is an effect size for two proportions. It helps compare the size of differences without depending only on sample size.
Can I use this for paired data?
No. This calculator is for independent groups. Paired binary data usually needs McNemar's test or another matched design method.