Pooled Proportion Calculator

Calculator Form

Enter two binomial samples to compute the pooled proportion, z test, p-value, confidence interval, and supporting statistics.

Sample 1 label

Sample 1 successes

Sample 1 size

Sample 2 label

Sample 2 successes

Sample 2 size

Hypothesis tail

Confidence level

Decimal places

Example Data Table

This sample shows how the calculator behaves with a typical two-sample proportion comparison.

Example item	Value
Sample 1 successes	54
Sample 1 size	120
Sample 2 successes	39
Sample 2 size	110
Sample 1 proportion	0.4500
Sample 2 proportion	0.3545
Pooled proportion	0.4043
Z statistic	1.4735
Two-tailed p-value	0.1406
95% confidence interval for difference	-0.0307 to 0.2216

Formula Used

Sample proportions
p1 = x1 / n1
p2 = x2 / n2

Pooled proportion
p̂ = (x1 + x2) / (n1 + n2)

Pooled standard error for H0: p1 = p2
SEpooled = √[ p̂(1 - p̂) × (1/n1 + 1/n2) ]

Z test statistic
z = (p1 - p2) / SEpooled

Confidence interval for the difference
(p1 - p2) ± z* × √[ p1(1-p1)/n1 + p2(1-p2)/n2 ]

The pooled estimate is mainly used for hypothesis testing when the null assumes equal population proportions. The interval estimate for the difference usually uses the unpooled standard error.

How to Use This Calculator

Enter a label for each sample to personalize the report.
Type the number of successes for both samples.
Enter the total sample size for each group.
Select a two-tailed, left-tailed, or right-tailed test.
Choose the confidence level you want for interval estimation.
Pick the number of decimal places for the output.
Click Calculate Now to display results above the form.
Use the download buttons to save the report as CSV or PDF.

Use this tool for A/B testing, survey comparisons, treatment-control studies, quality checks, and any problem involving two observed proportions from separate samples.

Frequently Asked Questions

1) What is a pooled proportion?

A pooled proportion combines successes from both samples and divides by the combined sample size. It creates one shared estimate under the null assumption that both population proportions are equal.

2) When should I use the pooled method?

Use it when testing whether two population proportions are equal. It is most common in two-proportion z tests, especially for experiments, surveys, and quality-control comparisons.

3) Is the pooled proportion just a simple average?

No. It is a weighted estimate based on sample sizes. Larger samples contribute more to the pooled result, so it usually differs from the plain average of the two sample proportions.

4) Why does the confidence interval use an unpooled error?

For estimation, the two population proportions are not forced to be equal. That is why interval estimation usually relies on the unpooled standard error, even when the test statistic uses the pooled version.

5) What does a large p-value mean here?

A large p-value means the observed difference is not unusual under the null model. It does not prove the proportions are equal. It only shows weak evidence against equality.

6) Can I run a one-tailed test?

Yes. Choose left-tailed or right-tailed when you have a directional hypothesis decided before seeing the data. Use a two-tailed test when any difference matters.

7) What happens if successes exceed the sample size?

That input is invalid. Successes must stay between zero and the total sample size. The calculator checks this and shows an error instead of producing misleading statistics.

8) Why does sample size matter so much?

Larger samples reduce standard error and make estimates more stable. Smaller samples create wider intervals and weaker test power, even when the observed difference looks large.