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A Type I error is rejecting a null hypothesis that is true; a Type II error is failing to reject a null hypothesis that is false. Their probabilities are called alpha (α) and beta (β), respectively. The distinction depends on both what a test concludes and what is actually true—something the test decision alone cannot tell you.
How the two errors differ
A hypothesis test compares evidence with a null hypothesis, usually written H₀. Its decision is either to reject H₀ or to fail to reject it. The null hypothesis may in reality be true or false, giving four possible combinations:
| What is true | Test rejects H₀ | Test fails to reject H₀ |
|---|---|---|
| H₀ is true | Type I error (probability associated with α) | Correct decision |
| H₀ is false | Correct rejection | Type II error (probability associated with β) |
NIST’s Engineering Statistics Handbook and Penn State’s statistics materials use these standard definitions. In practice, the truth is unknown, so you generally cannot tell from one test result whether an error occurred.
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1Clear out junk files and repair common Windows errors2Fix the driver behind crashes, sound loss and screen glitches3Repair Windows errors before they cause bigger problemsType I error: a false positive
A Type I error occurs when the test rejects a true null hypothesis. In plain language, the test reports evidence against H₀ even though H₀ is true. The significance level, α, is the test’s specified Type I error probability under the null hypothesis.
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Type II error: a missed effect
A Type II error occurs when the test fails to reject a false null hypothesis. The test does not find sufficient evidence against H₀ even though an alternative is true. Its probability is denoted β.
Why “fail to reject” is not “accept”
Failing to reject H₀ means the test did not find enough evidence, under its chosen decision rule, to reject it. It does not prove H₀ is true. A study may miss a real difference because the effect is small, measurements are variable, or the sample provides limited information. Penn State emphasizes this distinction in its hypothesis-testing materials.
Alpha, beta, and statistical power
Alpha and beta describe different error risks. Alpha is tied to the null hypothesis; beta is tied to a particular alternative hypothesis. NIST notes that beta can be computed only with a specific alternative in view, because the chance of missing an effect changes with its size and the test’s conditions.
Statistical power is the probability of rejecting H₀ when the specified alternative is true:
Power = 1 − β
Thus, a test with lower β has greater power for that alternative. Power is not a single universal property of a test: it depends on which effect or alternative you want to detect, as well as the study design.
How study design changes the trade-off
For a fixed test and sample size, choosing a lower α generally makes rejection more difficult and can increase β. That may reduce the chance of a false positive while increasing the chance of missing a real effect. The relationship is not independent of the test’s assumptions and design.
Increasing sample size can improve power. So can reducing standard error or studying an effect that is larger relative to the variability in the data. These are design relationships, not guarantees; a power calculation must specify the alternative and relevant assumptions.
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- Set the false-positive tolerance: identify the α level appropriate to the decision and its consequences.
- Name the effect you need to detect: state the alternative or effect size for which you want to assess β or power.
- Account for sample size and variability: both affect how clearly the data can distinguish the alternative from H₀.
- Weigh the consequences: decide what a false positive and a missed effect would cost in the specific application.
A courtroom analogy, with the hypotheses stated
Suppose the null hypothesis is “the defendant is not guilty.” Convicting an innocent person is analogous to a Type I error: rejecting a true null. Failing to convict a guilty person is analogous to a Type II error: failing to reject a false null. The analogy illustrates the definitions, but it does not establish that one error is always more serious. Which risk matters more depends on the application and how the hypotheses are framed.
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