Two-Proportion Z-Test Calculator
The two-proportion z-test compares success proportions from two independent groups using a pooled standard error and a two-sided normal approximation.
How to use this calculator
- Enter integer success counts and total counts. Each success count must be between zero and its group total.
- Check the units and definitions before comparing results with another source.
- Change one input at a time when comparing scenarios, and keep the reporting period and assumptions consistent.
Formula used by the two proportion z test calculator
p̂ = (x₁+x₂)/(n₁+n₂); z = (p̂₁−p̂₂)/√(p̂(1−p̂)(1/n₁+1/n₂))
- p̂ = (x₁+x₂)/(n₁+n₂); z = (p̂₁−p̂₂)/√(p̂(1−p̂)(1/n₁+1/n₂))
- The normal approximation may be poor with sparse or small samples. Observations should be independent and the outcome definitions consistent between groups.
- Inputs are checked for required values and valid ranges; displayed results are rounded for readability.
Worked examples
Worked example
Group 1 has 45 successes from 100 observations; group 2 has 30 from 100.
The observed proportions are 45% and 30%; the pooled z statistic is approximately 2.191 and the two-sided p-value is about 0.0285.
The observed proportions are 45% and 30%; the pooled z statistic is approximately 2.191 and the two-sided p-value is about 0.0285.
Understanding the result
The two-proportion z-test compares success proportions from two independent groups using a pooled standard error and a two-sided normal approximation.
The normal approximation may be poor with sparse or small samples. Observations should be independent and the outcome definitions consistent between groups.
Important notes
The normal approximation may be poor with sparse or small samples. Observations should be independent and the outcome definitions consistent between groups.
The result applies to the numbers and assumptions entered here; verify specifications for safety-critical or regulated uses.
Frequently asked questions
Why are the groups pooled?
Under the null hypothesis that proportions are equal, the combined sample estimates the common proportion used in the standard error.
Is a low p-value proof that the change caused the difference?
No. The test addresses a statistical hypothesis under assumptions and does not establish causality or practical importance.
How do I use this calculator?
Enter integer success counts and total counts. Each success count must be between zero and its group total.
Affiliate recommendations
Tools related to Two-Proportion Z-Test Calculator
These tools are selected for the calculator topic. The Calc Studio may earn a commission from purchases made through affiliate links.
- Growth Wise SAT & ACT prep Visit partner ↗
SAT and ACT study tools, flashcards, and preparation resources for students.
See the Affiliate Disclosure for details.
Related calculators
- Proportion Calculator — Solve a simple proportion with three known values.
- Test Grade Calculator — Calculate the score needed on a final assessment to reach a target overall grade.
- Pooled Standard Deviation Calculator — Combine two sample standard deviations into a pooled estimate when the equal-variance assumption is appropriate.
- Sample Size for a Proportion Calculator — Estimate sample size for a proportion at a chosen confidence z-score and margin of error.
- 2×2 Matrix Calculator — Add, subtract and multiply 2×2 matrices and calculate each determinant with transparent row and column inputs.
- Circle Calculator — Calculate circle area, circumference, and diameter from radius.
Related explanations
Learn the method behind the number or compare the terminology before using the calculator.
- How to Calculate Percent Error — Learn the percent error formula and how to compare an experimental value with a reference or accepted value.
- CAGR Explained: Compound Annual Growth Rate — Learn the CAGR formula and how to use it to compare growth over multiple years.
This page provides a calculation tool plus explanatory information so you can understand what the result means and when the underlying method may not fit your situation.