Calculator inputs
This page estimates β for a planned z-based mean test using a known or planning standard deviation.
Example data table
All examples below use an upper-tailed test with μ₀ = 50, μ₁ = 55, σ = 10, and α = 0.05.
| Sample size (n) | Standard error | Upper critical mean | Type II error (β) | Power |
|---|---|---|---|---|
| 16 | 2.5000 | 54.1121 | 0.3610 | 0.6390 |
| 25 | 2.0000 | 53.2897 | 0.1961 | 0.8039 |
| 64 | 1.2500 | 52.0560 | 0.0093 | 0.9907 |
Formula used
Standard error:
SE = σ / √n
Upper-tailed test:
Reject H₀ if x̄ ≥ μ₀ + z(1−α) × SE
β = Φ((critical − μ₁) / SE)
Lower-tailed test:
Reject H₀ if x̄ ≤ μ₀ + z(α) × SE
β = 1 − Φ((critical − μ₁) / SE)
Two-tailed test:
Reject H₀ if x̄ ≤ μ₀ − z(1−α/2) × SE or x̄ ≥ μ₀ + z(1−α/2) × SE
β = Φ((upper critical − μ₁)/SE) − Φ((lower critical − μ₁)/SE)
Power:
Power = 1 − β
How to use this calculator
- Enter the null mean you want to test.
- Enter the true mean you want the study to detect.
- Provide a population or planning standard deviation.
- Enter the sample size and significance level.
- Select upper-tailed, lower-tailed, or two-tailed testing.
- Click the calculate button to show β, power, and critical cutoffs.
- Review the chart to see how β changes across possible true means.
- Use the CSV or PDF buttons to export the current result set.
FAQs
1. What is a type two error?
A type two error happens when a false null hypothesis is not rejected. It measures missed-detection risk. The symbol β represents that probability.
2. How is power related to β?
Power equals 1 minus β. A higher power means the test is better at detecting the selected alternative mean.
3. Why does sample size matter so much?
Larger samples reduce the standard error. That usually moves the alternative distribution farther from the acceptance region, which lowers β and raises power.
4. What does the true mean input represent?
It is the effect you want to detect under the alternative hypothesis. β is always conditional on that chosen value.
5. Does lowering alpha always help?
Lower alpha makes rejection harder. That reduces type one error, but it often increases β unless you also increase sample size or effect size.
6. When should I use a two-tailed test?
Use a two-tailed test when changes in either direction matter. Because alpha is split across both tails, β is often larger than in a one-tailed design.
7. What standard deviation should I enter?
Use a reliable population value or a planning estimate from prior studies, pilot data, or domain knowledge. Poor SD estimates can distort β and power.
8. Is this calculator suitable for planning studies?
Yes. It is useful for quick planning, sensitivity checks, and sample-size intuition when you are evaluating a z-based mean test framework.