Confidence Interval for Difference Between Proportions Calculator
Calculate Two-Proportion difference confidence intervals ($(\hat{p}_1 - \hat{p}_2) \pm z^* \cdot \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$) for A/B testing.
Calculate Difference Between Two Proportions CI
Enter success count and sample size for Group 1 and Group 2.
Calculation Results
Calculated using difference between proportions confidence interval equations: \text{CI} = (\hat{p}_1 - \hat{p}_2) \pm z^* \cdot \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}
Quick Summary
The Confidence Interval for Difference Between Proportions Calculator evaluates proportion difference bounds ($(\hat{p}_1 - \hat{p}_2) \pm z^* \cdot \text{SE}$), percentage margin of error, and unpooled standard error.
Formula Explanation
\text{CI} = (\hat{p}_1 - \hat{p}_2) \pm z^* \cdot \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}
\text{SE}_{\text{unpooled}} = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}How It Works
This calculator evaluates the difference between two success rates (e.g. Variant A vs Variant B conversion rate). The Confidence Interval for Difference Between Proportions Calculator evaluates point estimate $\hat{p}_1 - \hat{p}_2$ and unpooled standard error.
Step-by-Step Worked Example
Practical Problem: Variant A ($x_1 = 45, n_1 = 100 \rightarrow \hat{p}_1 = 0.45$) vs Variant B ($x_2 = 30, n_2 = 100 \rightarrow \hat{p}_2 = 0.30$). Calculate 95% confidence interval for proportion difference.
- Step 1: Calculate sample proportions ($\hat{p}_1, \hat{p}_2$): $\hat{p}_1 = 0.45$, $\hat{p}_2 = 0.30$.
- Step 2: Calculate proportion difference ($\hat{p}_1 - \hat{p}_2$): $0.45 - 0.30 = \mathbf{+0.1500\text{ (15.00\%)}}.$
- Step 3: Calculate unpooled standard error ($\text{SE}_{\text{unpooled}}$): $\sqrt{\frac{0.45 \times 0.55}{100} + \frac{0.30 \times 0.70}{100}} = \sqrt{0.002475 + 0.002100} = \sqrt{0.004575} = \mathbf{0.067639}$.
- Step 4: Calculate Margin of Error ($\text{ME}$): $1.960 \times 0.067639 = \mathbf{0.13257\text{ (13.26\%)}}.$
- Step 5: Compute lower and upper bounds: Lower $= 15.0\% - 13.26\% = \mathbf{+1.74\%}$; Upper $= 15.0\% + 13.26\% = \mathbf{+28.26\%\text{ (95\% CI: [1.74\%, 28.26\%])}}.$ Excludes zero!
Real-World Calculation Examples
Scenario 1: Website A/B Test (45/100 vs 30/100)
Parameters: Diff = +15.0%
Result: [+1.74%, +28.26%] (Variant A is significantly better).
Scenario 2: Clinical Medical Vaccine Trial Efficacy
Parameters: Vaccine vs Placebo infection rate
Result: 95% CI for risk reduction percentage.
Scenario 3: Male vs Female Support for Ballot Measure
Parameters: Survey voting preference
Result: Confidence interval for gender preference gap.
Scenario 4: Factory Line A vs Line B Defect Rate
Parameters: Manufacturing quality audit
Result: 99% CI comparison of defect rates.
Key Benefits of Using This Calculator
A/B Testing Decision Utility
Essential tool for conversion optimization and digital marketing experiments.
Unpooled Standard Error Model
Uses unpooled standard error recommended for confidence interval construction.
Multi-Confidence Level Options
Supports 90%, 95%, and 99% confidence levels.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What does the confidence interval for difference between proportions measure?
It estimates the plausible percentage point difference between two population proportions $p_1 - p_2$.
Why is unpooled standard error used for confidence intervals?
Confidence intervals do NOT assume a null hypothesis $p_1 = p_2$, so unpooled sample proportions $\hat{p}_1, \hat{p}_2$ provide the correct standard error.
What does it mean if the confidence interval includes zero (0%)?
If the interval contains 0% (e.g. [-2.1%, +5.4%]), the proportion difference is not statistically significant at $\alpha = 0.05$.
How do I calculate two-proportion CI in Excel?
Calculate `diff = p1 - p2` and `ME = 1.96 * SQRT((p1*(1-p1)/n1) + (p2*(1-p2)/n2))`.
What are normal approximation requirements?
$n_1 \hat{p}_1 \ge 10, n_1(1-\hat{p}_1) \ge 10$ AND $n_2 \hat{p}_2 \ge 10, n_2(1-\hat{p}_2) \ge 10$.
What is Relative Risk (RR)?
$\text{Relative Risk} = \frac{\hat{p}_1}{\hat{p}_2}$.
What is Number Needed to Treat (NNT)?
$\text{NNT} = \frac{1}{\hat{p}_1 - \hat{p}_2}$ (reciprocal of absolute risk reduction).
Can sample sizes $n_1$ and $n_2$ be different?
Yes — the formula accommodates unequal sample sizes.
How does confidence interval relate to Two-Proportion Z-test?
If a 95% CI excludes 0%, a 2-tailed Two-Proportion Z-test will yield $p < 0.05$.
What non-parametric test replaces 2-proportion CI for small samples?
Fisher's Exact Test.