Example Data Table
| Scenario |
Successes |
Sample size |
Claimed proportion |
Alternative |
Alpha |
| Email clicks | 65 | 100 | 0.55 | Two sided | 0.05 |
| Quality pass rate | 184 | 220 | 0.80 | Greater | 0.01 |
| Survey approval | 92 | 150 | 0.70 | Less | 0.05 |
| Defect-free items | 470 | 500 | 0.95 | Two sided | 0.10 |
Formula Used
The one sample z test for a proportion starts with the sample proportion.
p̂ = x / n
The null standard error is based on the claimed proportion.
SE0 = sqrt[p0(1 - p0) / n]
With finite population correction, multiply the standard error by this factor.
FPC = sqrt[(N - n) / (N - 1)]
The test statistic is:
z = (p̂ - p0) / SE0
For a two sided test, the p value is:
p value = 2 × [1 - Φ(|z|)]
For a right tailed test, it is 1 - Φ(z). For a left tailed test, it is Φ(z).
The Wilson score interval is used by default. The Wald interval is also available.
How to Use This Calculator
- Enter the number of successes from your sample.
- Enter the full sample size.
- Enter the hypothesized population proportion as a decimal.
- Select the alternative hypothesis for your study question.
- Choose the significance level and confidence level.
- Select the interval method.
- Add population size only when the population is known.
- Use continuity correction when you want a count-adjusted normal approximation.
- Press Calculate and review the result above the form.
- Download CSV or PDF when you need a saved report.
Understanding One Sample Proportion Testing
What the Test Measures
A one sample z test for proportions checks one population rate. It compares a sample proportion with a claimed value. The method works best when the sample is large. It also needs expected successes and expected failures to be high enough.
Where It Helps
This calculator is useful for surveys, audits, product testing, and conversion tracking. You enter the number of successes, total trials, and hypothesized proportion. The tool then returns the sample proportion, standard error, z score, p value, confidence interval, and decision.
Hypotheses and Direction
The null hypothesis says the population proportion equals the target value. The alternative hypothesis says the proportion is different, greater, or smaller. A two tailed test looks for any difference. A right tailed test checks whether the rate is higher. A left tailed test checks whether the rate is lower.
Z Score and P Value
The z statistic measures distance from the target rate. It uses the null proportion in the standard error. A large absolute z value shows stronger evidence against the null claim. The p value converts that distance into probability. Smaller p values give stronger evidence.
Advanced Options
The calculator also supports optional continuity correction. That correction can improve the normal approximation for count data. It is often helpful when the sample is not very large. You can also add a finite population size. This adjusts the standard error when sampling without replacement from a known population.
Confidence Intervals
Confidence intervals describe a likely range for the true proportion. A Wald interval is simple and common. A Wilson interval is often more stable near zero, near one, or with smaller samples.
Interpreting Results
Use the result with context. Statistical significance does not always mean practical importance. A tiny change can be significant with a huge sample. A large change may be uncertain with a small sample. Review sample design, random selection, and data quality before making decisions.
Reporting Your Work
This page is built for transparent work. It shows formulas, assumptions, and export options. Download the CSV for spreadsheets. Download the PDF for reports. Use the example table to understand typical inputs. Then replace the values with your own study data.
FAQs
What is a one sample z test for proportions?
It tests whether one population proportion differs from a claimed value. It uses sample successes, sample size, and a hypothesized proportion to calculate a z statistic and p value.
When should I use this calculator?
Use it when your data has success or failure outcomes. Examples include pass rates, approval rates, conversion rates, click rates, or defect-free rates.
What does p0 mean?
p0 is the hypothesized population proportion. It is the claimed target rate under the null hypothesis. Enter it as a decimal, such as 0.60.
What does the p value mean?
The p value estimates how unusual your sample result is if the null claim is true. A smaller p value gives stronger evidence against the null hypothesis.
What is a two sided test?
A two sided test checks whether the true proportion is either higher or lower than the claimed proportion. It is used when direction is not specified.
Should I use continuity correction?
Continuity correction adjusts the count before using the normal curve. It can be helpful with smaller samples, but it may also make the test more conservative.
What is the Wilson interval?
The Wilson interval is a confidence interval for a proportion. It often behaves better than the Wald interval, especially near zero, near one, or with smaller samples.
Can this replace an exact binomial test?
No. If expected successes or failures are small, an exact binomial test may be better. This calculator warns when the normal approximation may be weak.