Math & Statistics
Two-Proportion Difference Calculator
Compare two independent observed proportions, calculate group-specific rates, their signed difference, a pooled z statistic, an unpooled critical-value interval, and the gap from an entered practical comparison. Every numerator, denominator, standard error, and percentage-point conversion remains visible.
Group 2 minus group 1 (decimal)-
Group 1 proportion (decimal)-
Group 2 proportion (decimal)-
Pooled proportion (decimal)-
Pooled standard error-
Pooled z statistic-
Unpooled standard error-
Unpooled critical-value margin-
Lower difference bound-
Upper difference bound-
Entered comparison difference (decimal)-
Difference minus entered comparison-
Decision view
Two observed proportions with a difference-risk interval
Two observed proportions with a difference-risk intervalGroup composition bars feed a signed difference scale that separates the observed effect, practical comparison, and unpooled interval.
Exact scenario comparisonGroup 2 successes changes while all other entered assumptions remain constant.
| Group 2 successes | Group 2 minus group 1 (decimal) | Group 1 proportion (decimal) | Group 2 proportion (decimal) | Pooled proportion (decimal) | Pooled standard error | Pooled z statistic | Unpooled standard error | Unpooled critical-value margin | Lower difference bound | Upper difference bound | Entered comparison difference (decimal) | Difference minus entered comparison |
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How to use Two-Proportion Difference Calculator
- Enter successes and sample size for each independent group.
- Confirm every success count belongs to its matching denominator.
- Choose a critical value and a practical comparison difference.
- Interpret the observed difference, pooled z statistic, and unpooled interval as separate outputs.
Calculator guide
Understanding Two-Proportion Difference Calculator
A difference between two observed proportions needs two separate calculations: the descriptive difference itself and the uncertainty around that difference. This calculator also keeps the pooled standard error used for a null test separate from the unpooled standard error used for the displayed interval.
Calculation method
How the calculation works
Compute each observed proportion, their difference, a pooled test statistic, and a separate unpooled critical-value interval without conflating the two standard errors. In the Two-Proportion Difference Calculator, the live scenario varies group 2 successes and tracks group 2 minus group 1 (decimal) while the remaining results preserve the reconciliation path. Divide each success count by its own sample size, subtract group 1 from group 2, use the combined success share for the pooled test standard error, and use the two group-specific variances for the unpooled interval standard error.
Detailed calculation process
Separate the observed effect from its two uncertainty models
The defaults compare 84 of 150 successes with 112 of 170 successes and use a critical value of 1.96.
General formula: p1 = x1/n1; p2 = x2/n2; d = p2 - p1; pp = (x1+x2)/(n1+n2); SEp = sqrt[pp(1-pp)(1/n1+1/n2)]; SEu = sqrt[p1(1-p1)/n1 + p2(1-p2)/n2]; CI = d +/- c*SEu The observed difference d is descriptive. Pooling assumes one common rate for the z statistic, while the interval preserves each group's own observed rate; therefore the two standard errors should not be interchanged.
What each symbol means
Worked substitution with the default inputs
The defaults produce group proportions of 56.000% and 65.882%, an observed increase of 9.882 points, z = 1.811, and an unpooled interval from about -0.790 to 20.555 percentage points.
Interpretation map
Keep descriptive, statistical, and practical comparisons apart
A complete comparison reports all three layers without treating them as interchangeable.
Worked situations
Practical examples
- Group 1 records 84 successes among 150 observations, or 56%.
- Group 2 records 112 among 170, or about 65.882%.
- A 9.882-point observed gap can still have an interval that crosses zero.
Better inputs
Useful tips
- Report the two raw counts with the percentages.
- Use percentage points, not percent change, when describing p2 - p1.
- Predefine the practical comparison before reviewing results when possible.
Before relying on the result
Limitations and common mistakes
- The formulas assume two appropriate independent binomial samples.
- Small expected counts, clustering, weights, repeated measures, or missing outcomes require different methods.
- The interval is a normal approximation and does not quantify bias or multiplicity.
Reference
Key terms
- Percentage-point difference
- Direct subtraction of two proportions after expressing both as percentages.
- Pooled standard error
- Common-rate uncertainty used for the displayed z statistic.
- Unpooled standard error
- Uncertainty retaining each group's observed proportion for the interval.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Why are there two standard errors?
The pooled version represents a common-rate null model for z, while the interval uses the two observed group variances.
Is a 0.10 difference the same as 10% growth?
No. It is a 10-percentage-point difference; relative growth uses a different denominator.
What does an interval crossing zero mean?
Under this approximation, differences in both directions remain compatible with the selected critical-value procedure.
Can I use paired before-and-after data?
Not with this independent-group formula; paired binary outcomes require methods that account for matching.