# Confidence Interval for a Proportion Calculator

URL: https://speedycalculator.com/calculators/statistics/sample-proportion/
Last updated: 2026-10-25

> This calculator finds a confidence interval for a population proportion from the number of successes and the sample size, using the Wilson score interval, which works well even for small samples.

## Inputs

- Successes
- Sample size
- Confidence level

## Results

- Sample proportion
- Lower bound
- Upper bound

## Formula

```
Wilson: (p̂ + z²/2n ± z√(p̂(1 − p̂)/n + z²/4n²)) ÷ (1 + z²/n)
```

## Explanation

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

FieldWhat to enterSuccessesA numberSample sizeA numberConfidence levelChoose 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:

SuccessesLower bound (%)2111.7431.519.24227.0552.535.26343.638461.28

## Worked example

Inputs — Successes: 42; Sample size: 120; Confidence level: 95%.

Results — Sample proportion = 35 %; **Lower bound = 27.05 %**; Upper bound = 43.88 %.

## FAQ

**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).


## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)