Calculator Inputs
Example Data Table
This example table shows how beta risk changes with different assumptions.
| Case | μ₀ | μ₁ | σ | n | α | Tail | β | Power |
|---|---|---|---|---|---|---|---|---|
| 1 | 50.00 | 54.00 | 10.00 | 16 | 0.05 | Right Tailed | 0.517888 | 0.482112 |
| 2 | 100.00 | 94.00 | 12.00 | 25 | 0.05 | Left Tailed | 0.196235 | 0.803765 |
| 3 | 200.00 | 206.00 | 15.00 | 36 | 0.01 | Two Tailed | 0.569786 | 0.430214 |
| 4 | 75.00 | 80.00 | 9.00 | 49 | 0.10 | Right Tailed | 0.004563 | 0.995437 |
Formula Used
SE = σ / √n
Critical Mean = μ₀ + z1-α × SE
β = Φ((Critical Mean - μ₁) / SE)
Critical Mean = μ₀ - z1-α × SE
β = 1 - Φ((Critical Mean - μ₁) / SE)
Lower Critical = μ₀ - z1-α/2 × SE
Upper Critical = μ₀ + z1-α/2 × SE
β = Φ((Upper Critical - μ₁) / SE) - Φ((Lower Critical - μ₁) / SE)
Here, Φ represents the standard normal cumulative distribution function. Type 2 error measures the chance of missing a real shift.
How to Use This Calculator
- Enter the null hypothesis mean in the first field.
- Enter the actual process mean you want to detect.
- Provide the known population standard deviation.
- Enter the sample size used in the test.
- Set the significance level, such as 0.05 or 0.01.
- Choose whether the test is left-tailed, right-tailed, or two-tailed.
- Click the calculate button to view beta, power, and thresholds.
- Use the graph to see how sample size changes beta risk.
- Download the summary as CSV or PDF when needed.
Frequently Asked Questions
1. What is a type 2 error?
A type 2 error happens when a test fails to reject the null hypothesis even though a real difference exists. It is commonly labeled beta and reflects missed detection risk.
2. What does beta tell me in engineering work?
Beta shows the chance of overlooking an actual process shift, performance drift, or quality problem. Lower beta usually means stronger detection capability for engineering decisions.
3. How is power related to beta?
Power equals 1 minus beta. It measures the chance of correctly detecting the specified effect. Higher power means your study is more likely to catch meaningful changes.
4. Why does sample size matter so much?
Larger samples reduce standard error, tighten the sampling distribution, and usually reduce beta. That makes real shifts easier to detect with the same significance level.
5. What happens when alpha changes?
Lower alpha makes rejection harder and often increases beta. Higher alpha makes rejection easier and can lower beta, but it also increases type 1 error risk.
6. When should I use a two-tailed test?
Use a two-tailed test when deviations in both directions matter. It splits alpha across both tails, which often raises beta compared with a one-tailed design.
7. Does this calculator assume a known standard deviation?
Yes. This page uses a z-based normal model with known population standard deviation. It works well for planning and instructional use when that assumption is appropriate.
8. Can I use this for quality control planning?
Yes. It is useful for inspection design, process monitoring, reliability studies, and acceptance testing where you want to understand missed-detection probability before collecting data.