Advanced Type II Error Calculator

Calculator Inputs

Test design

Tail direction

Alpha level

Null value (θ₀)

For one-sample tests, this is the null mean. For two-sample tests, use the null mean difference.

Alternative value (θ₁)

Enter the practical effect you want the test to detect.

Sigma 1

Use the planning-stage standard deviation for sample 1.

Sample size n1

Example Data Table

Scenario	Alpha	Tail	θ₀	θ₁	σ₁	σ₂	n₁	n₂	Beta	Power
One-sample launch metric	0.05	Right	50	55	12	—	49	—	0.1017	0.8983
Two-sample model uplift	0.05	Two	0	5	9	11	64	64	0.1964	0.8036
Two-sample loss reduction	0.01	Left	0	-4.5	8	9	80	75	0.1696	0.8304

These rows illustrate planning-stage scenarios for mean testing with z-based approximations and known or assumed standard deviations.

Formula Used

Type II error is the probability of failing to reject the null hypothesis when the alternative value is actually true.

One-sample standard error: SE = σ / √n

Two-sample standard error: SE = √[(σ₁² / n₁) + (σ₂² / n₂)]

Right-tailed critical boundary: c = θ₀ + z(1 − α) × SE

Left-tailed critical boundary: c = θ₀ − z(1 − α) × SE

Two-tailed boundaries: cL = θ₀ − z(1 − α/2) × SE and cU = θ₀ + z(1 − α/2) × SE

Beta for a right-tailed test: β = Φ[(c − θ₁) / SE]

Beta for a left-tailed test: β = 1 − Φ[(c − θ₁) / SE]

Beta for a two-tailed test: β = Φ[(cU − θ₁) / SE] − Φ[(cL − θ₁) / SE]

Power: 1 − β

This page uses normal-distribution planning formulas. They work best when standard deviations are known or carefully estimated before testing.

How to Use This Calculator

Choose one-sample mode for a single mean, or two-sample mode for a difference between independent means.
Set the tail direction to match your research claim. Use two-tailed when changes in both directions matter.
Enter alpha, the null value, and the alternative value you consider practically important.
Provide planning standard deviations and sample sizes. For two-sample mode, fill both groups.
Press the calculate button. The result appears below the header and above the form.
Review beta, power, critical rule, and effect measures. Export the output to CSV or PDF for documentation.

FAQs

1. What does Type II error mean?

Type II error is the chance that a test misses a real effect. It happens when the null is not rejected even though the alternative value is true.

2. How is beta related to power?

Power equals 1 minus beta. Lower beta means the design is more likely to detect the effect you care about during testing.

3. Why does larger sample size reduce beta?

Larger samples shrink standard error. That moves the alternative distribution farther from the rejection boundary in standardized units and improves detection sensitivity.

4. When should I use a two-tailed option?

Use a two-tailed test when effects in either direction matter. It splits alpha across both tails, usually increasing beta compared with a one-tailed design.

5. Does lower alpha always help?

Lower alpha reduces false positives but also makes rejection harder. Unless you increase sample size or effect size, beta usually rises.

6. What values should I enter for sigma?

Use planning-stage standard deviations from reliable history, pilot studies, or validated assumptions. Weak sigma estimates can distort both beta and power projections.

7. Is this calculator suitable for A/B testing plans?

Yes, when your metric can be approximated with mean-based normal formulas. It is useful for planning uplift detection, sensitivity checks, and sample sizing discussions.

8. Why is the result only an approximation?

The page uses z-based formulas with assumed standard deviations. Real experiments may deviate because of non-normal data, unequal variance behavior, or estimation error.