What is a one-proportion z-test?

A one-proportion z-test compares a sample proportion (p̂) to a known or hypothesized population proportion (p₀) to decide whether the observed proportion differs significantly from p₀ under the assumption that the sampling distribution of the proportion is approximately normal.

When should I use a one-proportion z-test?

Use it when you have one categorical outcome with two levels (success/failure), a single random sample, independent observations, and the normal approximation is valid (both np₀ ≥ 10 and n(1−p₀) ≥ 10).

What is the formula for the one-proportion z-test?

z = (p̂ − p₀) / √(p₀(1−p₀)/n), where p̂ is the sample proportion, p₀ is the hypothesized proportion, and n is the sample size.

How do I interpret the p-value?

The p-value is the probability of observing a sample proportion as extreme as p̂ if the true population proportion equals p₀. If p < α (commonly 0.05), reject the null hypothesis.

Cohen's h is a standardized effect size for proportions: h = 2·arcsin(√p̂) − 2·arcsin(√p₀). Benchmarks: 0.2 small, 0.5 medium, 0.8 large.

What assumptions does the test require?

Independent observations, a single random sample, a binary outcome, and the normal approximation conditions np₀ ≥ 10 and n(1−p₀) ≥ 10.

What if the sample size is small?

If np₀ < 10 or n(1−p₀) < 10, use the exact binomial test instead of the normal-approximation z-test.

What is the difference between one-tailed and two-tailed?

A two-tailed test detects any difference from p₀. A one-tailed test detects a difference only in a pre-specified direction (greater than or less than).

How do I report results in APA style?

A one-proportion z-test indicated that the observed proportion (p̂ = ___) differed significantly from the hypothesized value (p₀ = ___), z = ___, p = ___, 95% CI [___, ___].

Can I use this tool for my research?

Yes, for exploratory and educational use. For formal research, verify results with peer-reviewed software (R, Python, SPSS, SAS) and cite the tool: STATS UNLOCK (2026). One-Proportion z-Test Calculator. https://statsunlock.com

One-Proportion z-Test Calculator (Free Online Tool) | StatsUnlock

📊 Data Input

Sample Dataset:

Each cluster (group) is one sample. Enter comma-separated binary values (1 = success, 0 = failure) or numeric measurements (a threshold below converts them to binary). Add more clusters to compare multiple samples side-by-side; each runs its own one-proportion z-test against the hypothesized p₀.

Upload CSV or Excel file:

Supports .csv, .txt, .xlsx, .xls — headers auto-detected. ✨ Multi-column mode: click any number of columns to load each one as its own cluster (group). Use the Select All shortcut to load every numeric column at once, or Clear to reset.

Enter summary counts directly. Suitable when you already have successes and sample size from a published study or summary table.

Successes (x)

Sample Size (n)

Group Name

⚙️ Test Configuration

Hypothesized Proportion (p₀)

Null hypothesis value (0–1).

Alternative Hypothesis

Significance Level (α)

Numeric threshold (only if data are not 0/1)

If your data are already 0/1, leave blank. Otherwise, values strictly greater than the threshold count as a "success".

Continuity Correction

📈 Results

Full Statistical Results

Per-Cluster Summary

📊 Visualizations

Four publication-ready, colorful plots auto-generated from your data — hover any chart for tooltips with full statistics.

🔔 Standard Normal Curve with Rejection Region

⚡ Power Curve — Effect of Sample Size

🎯 Effect Size (Cohen's h) Gauge

🌈 Binomial Sampling Distribution under H₀

Assumption Checks

💡 Plain-Language Interpretation

✍️ How to Write Your Results in Research

Five ready-to-use reporting templates — click 📋 Copy on any card to paste it directly into your manuscript, thesis, report, abstract, or pre-registration document.

🏁 Detailed Conclusion

📐 Technical Notes & Formulas

A. Formulas Used

z = (p̂ − p₀) / √( p₀(1 − p₀) / n ) Where: z = standardized test statistic (standard normal under H₀) p̂ = sample proportion = x / n p₀ = hypothesized population proportion x = number of successes in the sample n = sample size

Sample proportion: p̂ = x / n Standard error under H₀: SE₀ = √( p₀(1 − p₀) / n ) Standard error of p̂ (for CI): SE_p̂ = √( p̂(1 − p̂) / n ) 95% Confidence interval (Wald): p̂ ± z_(α/2) · SE_p̂

P-value: Two-tailed: p = 2 · (1 − Φ(|z|)) Right-tailed: p = 1 − Φ(z) Left-tailed: p = Φ(z) Where Φ(·) is the standard normal CDF.

Effect size — Cohen's h (for proportions): h = 2·arcsin(√p̂) − 2·arcsin(√p₀) Benchmarks (Cohen 1988): |h| ≈ 0.20 small effect |h| ≈ 0.50 medium effect |h| ≈ 0.80 large effect

Continuity correction (Yates): z_cc = (|p̂ − p₀| − 1/(2n)) / SE₀ · sign(p̂ − p₀)

B. Technical Notes

Assumptions: independent observations, single random sample, binary outcome, normal-approximation conditions (np₀ ≥ 10 and n(1−p₀) ≥ 10).
Small samples: if np₀ < 10 or n(1−p₀) < 10, use the exact binomial test instead of the normal-approximation z-test.
Confidence intervals: the Wald interval is reported; for proportions near 0 or 1 the Wilson score CI is more accurate (Newcombe, 1998).
Two- vs one-tailed: use two-tailed by default; only switch to one-tailed when direction is pre-specified before data collection.
Multi-cluster runs: each cluster is tested independently against the same p₀; this is not a pooled or comparative test (use a two-proportion z-test for pairwise comparison of two samples).

✅ When to Use the One-Proportion z-Test

The one-proportion z-test answers a single question: "Does the proportion of successes in my sample differ significantly from a known or hypothesized population proportion?"

Decision Checklist

✅ You have one sample with a binary outcome (success/failure, yes/no, 0/1).
✅ You have a known or hypothesized population proportion (p₀) to compare against.
✅ Observations are independent (no clustering, no repeated measures).
✅ Sample size satisfies np₀ ≥ 10 and n(1−p₀) ≥ 10.
❌ Do not use if comparing two independent proportions → use two-proportion z-test.
❌ Do not use if comparing paired binary outcomes → use McNemar's test.
❌ Do not use if expected counts are small → use exact binomial test.

Real-World Examples

Medical research: Test whether the recovery rate in a single-arm trial (62 of 100) exceeds the 50% historical standard.
Quality control: Test whether a factory's defect rate (14 of 200) differs from the contractual maximum of 5%.
Education: Test whether the pass rate on a new exam (59 of 80) differs from the established 70% benchmark.
Marketing: Test whether a campaign's click-through rate (90 of 500) differs from the industry benchmark of 20%.
Ecology: Test whether the species presence rate at 120 survey sites differs from a previously published 30% occupancy estimate.

Sample Size Guidance

Minimum: both np₀ ≥ 10 and n(1−p₀) ≥ 10. For p₀ near 0.5, n = 40 is usually adequate; for p₀ near 0.1 or 0.9, n ≥ 100 is needed. For 80% power at α = 0.05 to detect a 10-percentage-point difference from p₀ = 0.5, n ≈ 200.

📖 How to Use This Tool — Step by Step

Enter your data — paste comma-separated values (e.g., 52, 48, 55, 61, 47) or 0/1 binary outcomes. Each cluster in the list is one sample. Edit the cluster name to label it (e.g., "Site A").
Choose a sample dataset — pick one of the five built-in datasets from the dropdown to see a fully worked example.
Configure the test — set the hypothesized proportion p₀, choose two- or one-tailed, and pick α (commonly 0.05). If your data are not already 0/1, set a threshold (values above it count as a "success").
Click ▶ Run One-Proportion z-Test — results appear instantly below.
Read the summary cards — z, p, p̂, p₀, and Cohen's h. Green = significant; amber = borderline; rose = non-significant.
Inspect the full results table — every value with a short description of what it means.
Examine the four colorful plots — observed vs hypothesized proportion bar with CI; standard normal curve with rejection regions; success/failure pie; and multi-cluster comparison.
Check the assumption badges — green = pass; amber = caution; red = use the exact binomial test instead.
Read the interpretation, conclusion, and reporting examples — auto-filled with your computed values. Click 📋 Copy on any card to paste directly into your write-up.
Export — Download Doc (.txt) for editing, or Download PDF for printing/sharing.

❓ Frequently Asked Questions

Q1. What is the one-proportion z-test and when should I use it?

The one-proportion z-test compares a single sample proportion (p̂) to a hypothesized or known population proportion (p₀). Use it when you have one sample, a binary outcome (success/failure), and the sample is large enough that the normal approximation is valid — typically np₀ ≥ 10 and n(1−p₀) ≥ 10. Example: testing whether a factory's observed defect rate of 7% differs significantly from the contractual maximum of 5%.

Q2. What is the p-value and how do I interpret it for this test?

The p-value is the probability of observing a sample proportion as extreme as p̂ (or more extreme) if the true population proportion really equals p₀. A p-value of 0.03 means: if the null hypothesis were true, there is a 3% chance of seeing data this extreme by random sampling alone. It is not the probability that H₀ is true.

Q3. Does statistical significance mean practical importance?

No. With very large n, even tiny differences from p₀ can be statistically significant (p < 0.05) yet practically meaningless. Always interpret the effect size (Cohen's h) alongside the p-value. A small h (≈ 0.2) means the observed proportion only barely shifts from p₀; a large h (≥ 0.8) means a real-world large shift.

Q4. What is Cohen's h and how do I interpret it?

Cohen's h is the standardized effect size for proportions: h = 2·arcsin(√p̂) − 2·arcsin(√p₀). Cohen (1988) benchmarks: |h| ≈ 0.2 is small (proportions differ by roughly a percentage-point or two near 0.5), 0.5 is medium, and 0.8 is large. h is preferred over the raw difference p̂ − p₀ because it has more stable statistical properties near the 0 and 1 boundaries.

Q5. What assumptions does the one-proportion z-test require?

Four assumptions: (1) independent observations, (2) a single random sample from the target population, (3) a binary outcome, and (4) the normal approximation conditions np₀ ≥ 10 and n(1−p₀) ≥ 10. If the normal approximation fails, switch to the exact binomial test; if observations are clustered, use a mixed-effects GLM with a binomial response.

Q6. How large a sample do I need?

The normal-approximation rule requires both np₀ ≥ 10 and n(1−p₀) ≥ 10. For p₀ ≈ 0.5, that means n ≥ 20; for p₀ ≈ 0.1, n ≥ 100. For 80% power at α = 0.05 to detect a 10-point shift from p₀ = 0.50, n ≈ 200; for the same shift from p₀ = 0.10, n ≈ 350. Use the dedicated power-analysis tool on StatsUnlock for exact figures.

Q7. One-tailed or two-tailed — which should I choose?

Use two-tailed by default. A two-tailed test detects deviations in either direction (p̂ > p₀ or p̂ < p₀). Use a one-tailed test only when the direction is pre-specified before data collection on theoretical grounds — for example, when testing whether a new manufacturing process produces fewer defects than the current standard.

Q8. How do I report one-proportion z-test results in APA 7th edition?

Example: "A one-proportion z-test indicated that the observed proportion (p̂ = 0.62) differed significantly from the hypothesized value (p₀ = 0.50), z = 2.94, p = .003, h = 0.24, 95% CI [0.54, 0.70]." Always report z, p (three decimals or < .001), the effect size, and the confidence interval. See Section "How to Write Your Results" above for five filled-in style templates.

Q9. Can I use this calculator for my published research or thesis?

Yes, for exploratory analysis, teaching, and most coursework. For formal peer-reviewed publication, verify the numbers in R (prop.test()), Python (statsmodels.stats.proportion.proportions_ztest), SPSS, or SAS, and cite the tool: STATS UNLOCK. (2026). One-Proportion z-Test Calculator. https://statsunlock.com.

Q10. What does a non-significant result mean — is my hypothesis wrong?

A non-significant result (p > α) does not prove H₀ is true. It means the data do not provide sufficient evidence to reject it. The cause may be a genuinely small effect, low statistical power, or insufficient sample size. Always inspect the effect size (Cohen's h) and the confidence interval; if the CI includes p₀ but is also very wide, the study was underpowered. Consider a Bayes Factor to quantify evidence for H₀.

📚 References

The following references support the statistical methods used in this one-proportion z-test calculator, covering p-value interpretation, effect size, and best practices in hypothesis testing for proportions.

Agresti, A., & Coull, B. A. (1998). Approximate is better than "exact" for interval estimation of binomial proportions. The American Statistician, 52(2), 119–126. https://doi.org/10.1080/00031305.1998.10480550
Brown, L. D., Cai, T. T., & DasGupta, A. (2001). Interval estimation for a binomial proportion. Statistical Science, 16(2), 101–133. https://doi.org/10.1214/ss/1009213286
Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine, 17(8), 857–872. https://doi.org/10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E
Wilson, E. B. (1927). Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association, 22(158), 209–212. https://doi.org/10.1080/01621459.1927.10502953
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
Fleiss, J. L., Levin, B., & Paik, M. C. (2003). Statistical methods for rates and proportions (3rd ed.). John Wiley & Sons. https://doi.org/10.1002/0471445428
Agresti, A. (2018). An introduction to categorical data analysis (3rd ed.). John Wiley & Sons.
Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the behavioral sciences (10th ed.). Cengage Learning.
American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). APA. https://doi.org/10.1037/0000165-000
Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129–133. https://doi.org/10.1080/00031305.2016.1154108
Greenland, S., Senn, S. J., Rothman, K. J., Carlin, J. B., Poole, C., Goodman, S. N., & Altman, D. G. (2016). Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337–350. https://doi.org/10.1007/s10654-016-0149-3
R Core Team. (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/
NIST/SEMATECH. (2013). e-Handbook of statistical methods: 1-Sample Test for proportions. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm
STATS UNLOCK. (2026). One-Proportion z-Test Calculator. https://statsunlock.com/one-proportion-z-test-calculator/

📝 Cite This Tool

If you used the One-Proportion z-Test Calculator in your research, thesis, dissertation, manuscript, or report, please cite it using the format below. Click 📋 Copy to copy the citation directly to your clipboard.

APA 7th Edition

STATS UNLOCK. (2026). One-Proportion z-Test Calculator. Retrieved from https://statsunlock.com/one-proportion-z-test-calculator/