Plus-Four Confidence Interval for Difference Between Two Proportions
Use this calculator for confidence intervals comparing two proportions when sample sizes are small or proportions are extreme (works better than standard method).
When to Use
- Small sample sizes (n < 40 for either group)
- Extreme proportions near 0% or 100%
- Comparing two groups with binary outcomes
- Better coverage than standard method for these cases
Formula basis: Add 1 success and 1 failure to each group before calculating.
How to Use
Step 1: Enter group 1 sample size and successes
Step 2: Enter group 2 sample size and successes
Step 3: Select confidence level (typically 95%)
Step 4: Click “Calculate”
| Plus Four Confidence interval for Difference Between proportions | ||
|---|---|---|
| Sample 1 | Sample 2 | |
| Sample size | ||
| No. of successes | ||
| Confidence Level ($1-\alpha$) | ||
| Results | ||
| Sample proportions: | ||
| Standard Error of Diff. of prop.: | ||
| Z-critical value: | ||
| Margin of Error: | ||
| Lower Confidence Limits: | ||
| Upper Confidence Limits: | ||
Plus-Four Adjustment
For each group:
- Add 1 success
- Add 1 failure
- New sample size: n’ = n + 2
- New proportion: p̂’ = (k + 1)/(n + 2)
Then calculate standard difference CI with adjusted values.
When to Use
PLUS-FOUR:
- Either n < 40, OR
- Either p̂ < 0.1 or p̂ > 0.9
STANDARD method:
- Both n ≥ 40 AND
- 0.1 ≤ p̂ ≤ 0.9 (both groups)
Worked Example
Scenario: Testing treatment effectiveness. Control (n=15): 1 improved. Treatment (n=15): 5 improved.
Standard method fails: np̂ < 5 for control group
Plus-Four adjustment:
- Control: p̂₁ = (1+1)/(15+2) = 0.118
- Treatment: p̂₂ = (5+1)/(15+2) = 0.353
- Difference: 0.353 - 0.118 = 0.235
- SE = √(0.118×0.882/17 + 0.353×0.647/17) = 0.131
- E = 1.96 × 0.131 = 0.257
- CI = 0.235 ± 0.257 = [-0.022, 0.492]
Interpretation: Improvement rate in treatment group is 2.2% lower to 49.2% higher than control (95% confidence).
Standard method: CI for Two Proportions Related: Tutorial