Plus-Four Confidence Interval for Single Proportion
Use this calculator for confidence intervals when sample size is small or proportion is near 0 or 1 (performs better than standard method in these cases).
When to Use This Calculator
- Small samples (n < 40)
- Extreme proportions near 0% or 100% (p̂ < 0.1 or p̂ > 0.9)
- Proportions fail requirements for standard method
- Better coverage than standard method for these scenarios
Formula basis: Add 2 successes and 2 failures (4 total) before calculating.
How to Use
Step 1: Enter sample size (n)
Step 2: Enter number of successes (k)
Step 3: Select confidence level (typically 95%)
Step 4: Click “Calculate”
| Confidence Interval Calculator for proportion | |
|---|---|
| Sample Size ($n$) | |
| Number of successes ($k$) | |
| Confidence Level ($1-\alpha$) | |
| Results | |
| Estimate of proportion: ($\hat{p}$) | |
| Standard Error of proportion: ($SE$) | |
| Z-critical value: ($Z_{\alpha/2}$) | |
| Margin of Error: ($E$) | |
| Lower Confidence Limits: | |
| Upper Confidence Limits: | |
Plus-Four Formula
Adjust sample:
- Add 2 successes and 2 failures
- New n = n + 4
- New p̂ = (k + 2)/(n + 4)
Then apply standard formula: $$CI = \hat{p}{adj} \pm z{\alpha/2} \times \sqrt{\frac{\hat{p}{adj}(1-\hat{p}{adj})}{n+4}}$$
When to Use Plus-Four
Use PLUS-FOUR when:
- n < 40, OR
- p̂ < 0.1 or p̂ > 0.9
Use STANDARD when:
- n ≥ 40 AND
- 0.1 ≤ p̂ ≤ 0.9
Worked Example
Scenario: Testing rare event. Out of 20 units, 1 failed. Estimate failure proportion.
Standard method fails: np̂ = 1 < 5
Plus-Four method:
- New count: k’ = 1 + 2 = 3
- New total: n’ = 20 + 4 = 24
- p̂’ = 3/24 = 0.125
- SE = √(0.125 × 0.875 / 24) = 0.0670
- E = 1.96 × 0.0670 = 0.131
- CI = 0.125 ± 0.131 = [-0.006, 0.256] → [0, 0.256] (bounded by 0-1)
Interpretation: We’re 95% confident the failure rate is between 0% and 25.6%.
Standard method: CI for Single Proportion Related: Tutorial