Example Data Table
| Case | n | p0 | p1 | Alpha | Alternative | Use |
|---|---|---|---|---|---|---|
| Quality pass rate | 200 | 0.50 | 0.60 | 0.05 | Two-sided | Detect a changed pass rate. |
| Conversion uplift | 350 | 0.12 | 0.16 | 0.05 | Greater | Check marketing improvement. |
| Defect reduction | 500 | 0.08 | 0.05 | 0.01 | Less | Plan stricter quality evidence. |
Formula Used
The calculator uses a one-sample proportion framework. The null value is p0. The expected true value is p1.
Standard error under the null: SE0 = sqrt[p0 × (1 - p0) / n]
Standard error under the alternative: SE1 = sqrt[p1 × (1 - p1) / n]
Test statistic: z = (p hat - p0) / SE0
For a greater test, power = P(p hat exceeds the upper rejection boundary when p = p1).
For a less test, power = P(p hat falls below the lower rejection boundary when p = p1).
For a two-sided test, power is the sum of both rejection tail probabilities.
Exact mode sums binomial probabilities in the rejection region. Cohen h = 2asin(sqrt(p1)) - 2asin(sqrt(p0)).
How to Use This Calculator
- Enter the planned sample size.
- Enter the null proportion used in the hypothesis.
- Enter the alternative proportion you want to detect.
- Select alpha and the alternative direction.
- Choose normal approximation or exact binomial mode.
- Add observed successes when you also want p hat and z.
- Press Calculate to view power above the form.
- Use CSV or PDF buttons for saved reports.
Understanding One-Sample Proportion Power
Power planning checks whether a one-sample proportion test can detect a real difference. The calculator compares a null proportion with an expected true proportion. It then estimates the chance of rejecting the null hypothesis when that expected value is correct. This chance is the statistical power. Its complement is beta, which is the risk of missing the effect.
Why Power Matters
A study with low power can waste time and money. It may fail even when the practical effect is meaningful. A study with very high power may require a larger sample than needed. Good planning balances risk, cost, and precision. This tool helps you review that balance before collecting data.
Inputs That Drive Results
The null proportion is the benchmark. The alternative proportion is the value you hope to detect. The sample size controls precision. The alpha level sets the false positive risk. The alternative type decides whether the test looks upward, downward, or in both directions. Each choice changes the rejection region and the final power.
Advanced Options
The calculator includes normal approximation and exact binomial methods. The normal method is quick and works well for large samples. The exact method uses binomial probabilities and is useful for smaller samples. Continuity correction can make normal results more conservative. The tool also reports Cohen's h, critical boundaries, expected successes, and a sample size estimate.
Interpreting Output
Power near 0.80 is often used as a planning target. This is not a universal rule. Regulatory, medical, business, and academic settings may need stricter values. Review beta beside power. A beta of 0.20 means a twenty percent miss risk under the chosen alternative. Also inspect the critical count. It shows how many successes would trigger rejection.
Practical Use
Start with realistic assumptions. Use prior studies, pilot data, or domain knowledge. Try several alternative proportions. Small changes can require many more observations. Export the results for records. Share the table with reviewers, clients, or team members. The calculation is a planning guide, not a replacement for study design judgment. Sensitivity checks are helpful. Run optimistic and conservative cases. Compare one-sided and two-sided choices. Document every assumption so later readers know the planning basis and the decision context.
FAQs
What is power in a one-sample proportion test?
Power is the probability of rejecting the null hypothesis when the chosen alternative proportion is true. Higher power means lower miss risk.
What does beta mean?
Beta is the chance of failing to reject the null when the alternative is true. It equals one minus power.
When should I use exact binomial mode?
Use exact binomial mode for smaller samples or proportions near zero or one. It uses binomial probabilities instead of a normal approximation.
Why is continuity correction included?
Continuity correction adjusts a continuous normal approximation for discrete count data. It often gives a more conservative power estimate.
What is Cohen h?
Cohen h is an effect size for two proportions. It helps compare the practical size of the difference between p0 and p1.
Can this calculator find required sample size?
Yes. It gives an approximate required sample size for the selected target power, alpha, and expected alternative proportion.
Does a two-sided test reduce power?
Often yes. A two-sided test splits alpha across two tails. This can require more evidence than a one-sided test.
Can I export the result?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a compact report summary.