Formula Used
Sample proportion: p̂ = x / n
Standard error: SE = √[p̂(1 − p̂) / n]
Confidence interval: p̂ ± z* × SE
Plus-four adjustment: p̃ = (x + 2) / (n + 4)
Finite population correction: FPC = √[(N − n) / (N − 1)]
The calculator uses the selected method, confidence level, optional z value,
finite population correction, and continuity setting to create the final interval.
How to Use This Calculator
- Enter the number of successes.
- Enter the total sample size.
- Select or type the confidence level.
- Add a custom z value only when required.
- Choose the standard or adjusted method.
- Add population size when the population is finite.
- Select output format and decimal places.
- Press the calculate button to view results above the form.
- Use CSV or PDF buttons to save the result.
Example Data Table
| Example |
Successes |
Sample Size |
Confidence |
Method |
Approximate Interval |
Meaning |
| Survey approval |
56 |
100 |
95% |
Standard |
46.27% to 65.73% |
The true approval share is estimated inside this range. |
| Defect rate |
18 |
250 |
90% |
Standard |
4.52% to 9.88% |
The process defect share is estimated from sampled items. |
| Small pilot |
7 |
20 |
95% |
Plus-four |
18.89% to 56.11% |
The adjusted method supports small sample reporting. |
Understanding a One Proportion Interval
A one proportion z interval estimates a population share. It uses one sample, not two groups. The sample gives the observed proportion. The interval then adds and subtracts a margin of error. This produces a likely range for the true population proportion.
Why This Calculator Helps
Manual work can be easy to misplace. A small rounding error changes the final limits. This calculator keeps each step visible. It reports the sample proportion, selected z value, standard error, margin, and bounded interval. It also checks the common success failure rule. That rule helps users decide whether the normal model is reasonable.
What the Inputs Mean
Successes are the observations that match the event of interest. Sample size is the total number observed. Confidence level controls interval width. A larger confidence level gives a wider interval. A smaller confidence level gives a narrower interval. Decimal places control displayed precision only. They do not change the internal calculation. Population size is optional. Use it when sampling without replacement from a known finite population.
Reading the Result
The lower limit is the smallest plausible proportion. The upper limit is the largest plausible proportion. The midpoint is the observed or adjusted proportion used by the selected method. The margin of error shows the distance from that midpoint to either bound. Percent output is often easier for reports. Decimal output is useful for formulas and spreadsheets.
Practical Reporting Tips
Always describe the data source. State the sample size and confidence level. Mention the method used. If assumptions fail, treat the result as a rough guide. Use adjusted methods for small samples. Never present the interval as a guaranteed range. It is an estimate based on repeated sampling logic.
Common Use Cases
This tool helps with surveys, quality checks, classroom statistics, marketing tests, and simple public polls. It can estimate a defect rate, approval share, preference share, conversion rate, or pass rate. The export buttons let users save results for reports. The example table shows typical inputs and expected interpretation. Review inputs before quoting the interval. Store the CSV for audits. Save the PDF when sharing summaries with teammates. Keep raw counts nearby for later checks. This improves review and repeat use.
FAQs
What is a one proportion z interval?
It is a confidence interval for one population proportion. It uses sample successes, sample size, and a z critical value to estimate a plausible range for the true share.
When should I use this calculator?
Use it when you have one sample and a yes-or-no outcome. Common uses include surveys, defect checks, pass rates, conversion rates, and preference studies.
What are successes?
Successes are observations that match the event being measured. For example, approved voters, defective items, converted users, or students who passed can be counted as successes.
What confidence level should I choose?
Many reports use 95%. A higher level gives a wider interval. A lower level gives a narrower interval, but it carries less confidence.
What does the margin of error mean?
The margin of error is the distance from the estimated proportion to each interval limit. A larger margin creates a wider confidence interval.
Why use the plus-four method?
The plus-four method adjusts successes and sample size. It can give more stable results when the sample is small or proportions are near zero or one.
What is finite population correction?
Finite population correction reduces standard error when sampling without replacement from a known population. It is useful when the sample is more than a small fraction.
Can I export the answer?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a simple report summary.