Proportions such as approval ratings or defect rates always carry sampling uncertainty.
How to use the Confidence Interval for a Proportion Calculator
- Successes
- Sample size
- Confidence level
Results update as you type, and every field accepts the units shown in its menu.
How it works
The Wilson interval adjusts the simple estimate so the interval stays within 0–100% and has good coverage.
Inputs explained
| Field | What to enter |
|---|---|
| Successes | A number |
| Sample size | A number |
| Confidence level | Choose one: 90%, 95%, 99% |
What each result means
- Sample proportion = [Sample proportion]×100
- Lower bound = ([Centre] – [Half width])×100
- Upper bound = ([Centre] + [Half width])×100
Example values
How the main result changes when Successes changes and everything else stays at the example values above:
| Successes | Lower bound (%) |
|---|---|
| 21 | 11.74 |
| 31.5 | 19.2 |
| 42 | 27.05 |
| 52.5 | 35.2 |
| 63 | 43.63 |
| 84 | 61.28 |
Formula
Wilson: (p̂ + z²/2n ± z√(p̂(1 − p̂)/n + z²/4n²)) ÷ (1 + z²/n)Worked example
Inputs — Successes: 42; Sample size: 120; Confidence level: 95%.
Results — Sample proportion = 35 %; Lower bound = 27.05 %; Upper bound = 43.88 %.
Frequently asked questions
Why Wilson instead of the simple formula?
The simple (Wald) interval performs poorly for small samples or extreme proportions.
How do I narrow the interval?
Increase the sample size.
What formula does the Confidence Interval for a Proportion Calculator use?
Wilson: (p̂ + z²/2n ± z√(p̂(1 − p̂)/n + z²/4n²)) ÷ (1 + z²/n).