A Type I error occurs when a test rejects a null hypothesis that is actually true. A Type II error occurs when it fails to reject a null hypothesis that is actually false. The distinction comes down to the test’s decision versus reality; because reality is not known when you run a test, a result can’t tell you with certainty which kind of error, if any, occurred.
How to identify each error
A hypothesis test compares evidence with a null hypothesis, often written as H₀. The test leads to one of two decisions: reject H₀, or fail to reject H₀. Separately, H₀ is either true or false in reality. Pairing those two dimensions gives four possible outcomes:
| Actual state | Reject the null hypothesis | Fail to reject the null hypothesis |
|---|---|---|
| H₀ is true | Type I error | Correct decision |
| H₀ is false | Correct rejection | Type II error |
Use “fail to reject” rather than “accept” the null hypothesis. A test that does not find sufficient evidence against H₀ has not proved it true. Penn State’s STAT 500 course emphasizes this distinction.
Type I error: a false positive
A Type I error means rejecting a true null hypothesis. Its probability under the null is denoted by alpha (α), also called the significance level. For example, if H₀ says a system has no effect, a Type I error is concluding that it has an effect when it does not.
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Type II error: a false negative
A Type II error means failing to reject a false null hypothesis. Its probability is denoted by beta (β). In the same example, it is failing to detect an effect that really exists.
Alpha, beta, and statistical power
Alpha describes the Type I error probability under H₀. Beta describes the Type II error probability for a particular alternative hypothesis—the situation in which H₀ is false. The NIST Engineering Statistics Handbook defines a test’s power as the probability of rejecting the null when it is false, written as 1 − β.
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Beta is not one fixed property of a test regardless of what the alternative is. The chance of missing an effect depends on how large that effect is relative to variability, as well as on study design. A test may have greater power to detect a large effect than a small one under otherwise similar conditions.
How study design affects the trade-off
For a fixed test and sample size, lowering α generally makes it harder to reject H₀ and may increase β. Increasing the sample size can improve power; so can reducing standard error or studying an effect that is larger relative to variability. These relationships depend on the test’s assumptions and design, so they are not guarantees in every setting. NIST discusses these power relationships in its process-characterization guidance, and Penn State’s STAT 200 materials explain the role of sample size.
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When comparing plans, consider the chosen Type I error tolerance, power for a clearly specified effect or alternative, sample size and variability, and the practical costs of false positives versus missed effects. The appropriate balance depends on the application; neither error type is universally more serious.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.A courtroom example
The analogy works if the hypotheses are stated first. Suppose H₀ is “the defendant is not guilty.” Convicting an innocent person is a Type I error: rejecting a true null. Failing to convict a guilty person is a Type II error: failing to reject a false null. The analogy illustrates the decision table, but real statistical tests and legal standards are not interchangeable.
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