Calculate Type II Error
Use this calculator for a one-sample z test when the population standard deviation is known.
Example Data Table
| Scenario | μ₀ | μ₁ | σ | n | α | Direction |
|---|---|---|---|---|---|---|
| Quality improvement | 100 | 105 | 15 | 36 | 0.05 | Right-tailed |
| Cost reduction | 250 | 238 | 30 | 64 | 0.05 | Left-tailed |
| Process change | 50 | 54 | 10 | 49 | 0.05 | Two-sided |
Formula Used
Standard error: SE = σ ÷ √n
Right-tailed critical mean: c = μ₀ + z1−α × SE
Left-tailed critical mean: c = μ₀ − z1−α × SE
Two-sided boundaries: cL = μ₀ − z1−α/2 × SE and cU = μ₀ + z1−α/2 × SE
Beta: The calculator finds the chance that the sample mean stays in the non-rejection region when μ₁ is true.
Φ represents the standard normal cumulative distribution. Power equals 1 − β.
How to Use This Calculator
- Enter the mean stated by the null hypothesis.
- Enter the meaningful true mean you want to detect.
- Add the known population standard deviation and planned sample size.
- Choose alpha and select the test direction before viewing results.
- Submit the form. Review beta, power, critical values, and the planning note.
Understanding Type II Error
A Type II error happens when a test misses a real effect. The null hypothesis remains unchallenged, even though the alternative condition is true. Its probability is called beta, written as beta. A lower beta means the test is more likely to detect the difference you care about.
Beta and Statistical Power
Statistical power is the opposite side of beta. Power equals one minus beta. A test with beta of 0.20 has power of 0.80. This means it has an eighty percent chance of detecting the specified true difference. Researchers often target power near 0.80. The right target depends on cost, risk, and the importance of missing an effect.
What This Tool Calculates
This calculator uses a one-sample z test for a population mean. It assumes the population standard deviation is known. You enter a null mean, a true mean, standard deviation, sample size, alpha level, and test direction. The tool creates critical decision boundaries. It then finds the probability a sample mean falls inside the non-rejection region under the true mean.
Why Effect Size Matters
The distance between the true mean and null mean drives detection. A larger distance is easier to detect. A small distance needs more precise data or a bigger sample. Standard deviation also matters. Greater spread makes sample means less predictable. This raises beta when every other setting stays fixed. Consider a realistic difference, not only a statistically convenient one.
The Role of Sample Size
A larger sample makes the standard error smaller. The critical boundary moves closer to the null mean in standard-error units. That usually lowers beta and increases power. However, sample size cannot correct a weak research question. Select enough observations to detect a meaningful difference. Also consider data quality, missing values, and design limits before collecting records.
One-Sided and Two-Sided Choices
A one-sided test focuses on one direction. It can have more power for that direction at the same alpha level. A two-sided test checks for meaningful change in either direction. Its non-rejection region sits between two critical boundaries. Choose the direction before seeing results. Changing it after analysis can distort the stated error rate.
Reading Your Results
The result card reports beta, power, standard error, and decision thresholds. Beta is the chance of missing the entered true mean difference. Power is the chance of detecting it. The critical values show where rejection begins. Review them with the assumptions shown on the page. A result is only as useful as its chosen effect, deviation estimate, and study design.
Good Planning Habits
Run several scenarios before finalizing a plan. Change the true mean to examine small, moderate, and large effects. Compare different sample sizes. Keep alpha consistent unless your decision rules truly change. Use expert knowledge to select practical inputs. This approach turns beta from a technical number into a planning tool for better studies.
Frequently Asked Questions
What is a Type II error?
It occurs when a real difference exists, but the test does not reject the null hypothesis. Its probability is beta.
What does beta mean?
Beta is the estimated probability of failing to detect the specified true mean difference under the selected assumptions.
How is power related to beta?
Power equals one minus beta. Lower beta produces higher power when the same test setup is used.
Can I use this for a t test?
No. This page uses a z-test model with a known population standard deviation. Use a dedicated t-test power tool when the deviation is estimated from the sample.
Why does a larger sample lower beta?
A larger sample reduces standard error. Sample means become more precise, so a meaningful difference is easier to detect.
Should I choose a one-sided test for more power?
Only when a single direction was justified before data collection. Do not choose a one-sided test merely because it gives a more favorable result.
What alpha level should I enter?
Use the alpha level specified by your study plan or decision rule. A common value is 0.05.
What true mean should I use?
Enter the smallest practical difference worth detecting. This makes the calculation useful for realistic planning.
Can beta be zero?
It can become extremely small, but exact zero is unusual in continuous statistical models. The displayed value may round to zero.
Does a low beta prove the alternative hypothesis?
No. It describes planned detection ability under chosen assumptions. It does not prove any hypothesis after observing data.
What should I do after viewing beta?
Compare realistic scenarios, then plan confident, transparent statistical decisions.