Every hypothesis test can make one of two formal mistakes: reject a null hypothesis that is actually true (a Type I error, or false positive) or fail to reject a null hypothesis that is actually false (a Type II error, or false negative). The complete logic is easiest to see by crossing the test’s decision with the null hypothesis’s actual state.
The four possible outcomes
The table shows outcomes for a specified hypothesis-testing procedure. “Reject” and “fail to reject” are decisions made by the test; they are not direct proof that the alternative hypothesis is true or that the null hypothesis is true.
| Actual state | Reject the null hypothesis | Fail to reject the null hypothesis |
|---|---|---|
| Null hypothesis is true | Type I error False positive Probability: α |
Correct non-rejection |
| Null hypothesis is false | Correct detection Contributes to statistical power |
Type II error False negative Probability: β |
These standard definitions are summarized by the Journal of Pharmacology & Pharmacotherapeutics review and OpenStax’s outcomes table.
What each error means
Type I error: a false alarm
A Type I error occurs when the null hypothesis is true but the test rejects it. It is commonly called a false positive. The symbol α (alpha) denotes the probability of this error under the null hypothesis for the chosen testing procedure. For example, if the null says there is no effect, a Type I error is concluding that an effect exists when it does not.
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Type II error: a missed effect
A Type II error occurs when the null hypothesis is false but the test fails to reject it. It is commonly called a false negative. The symbol β (beta) denotes this probability for a specified alternative or effect. If the null says there is no effect, a Type II error is missing a real effect.
“False positive” and “false negative” are useful memory aids, but everyday uses of those terms vary by application. Always identify the null hypothesis and the test decision first; that is what determines the formal error label.
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How alpha, beta and power relate
Alpha (α)
Alpha is the probability of rejecting a true null hypothesis: α = P(Type I error). It is a property of the testing rule under the null, not the probability that the null hypothesis is true after seeing your data.
Beta (β)
Beta is the probability of failing to reject a false null hypothesis for a specified alternative: β = P(Type II error). Because the value depends on which alternative effect is being considered, a study does not have one context-free beta for every possible effect.
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Power
Statistical power is the probability of correctly rejecting the null when the specified alternative is true: power = 1 − β. In other words, power is the test’s chance of detecting the effect you planned for. The PMC review and StatPearls overview describe this conditional interpretation.
What changes the risk of a Type II error?
Power, and therefore β, depends on the entire design rather than on a single universal percentage. Important factors include:
- Sample size: larger samples generally provide more information and increase power when other design features stay fixed.
- Effect size: larger departures from the null are generally easier to detect, increasing power.
- Population variance: greater variability makes a given effect harder to distinguish from noise.
- Significance level: changing α changes the rejection threshold and therefore affects power.
At fixed sample size, effect size and variability, lowering α to reduce the chance of a Type I error can also lower power and increase β. The direction and size of that change depend on the test and alternative; there is no universal numerical trade-off independent of assumptions. See the CDC statistical considerations and StatPearls for the design dependence.
Why “fail to reject” does not prove the null
A non-significant result means the observed data did not cross the procedure’s rejection threshold. It does not establish that the null hypothesis is true. If the study has low power, a real effect may remain undetected, producing a Type II error or an inconclusive result. The National Academies’ statistical reference guide warns that a non-significant finding in a low-powered study can be inconclusive rather than a reliable negative.
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A quick way to apply the picture
- State the null hypothesis. Write exactly what “no effect,” “no difference” or another baseline claim means in your analysis.
- Record the test decision. The procedure either rejects the null or fails to reject it at the selected significance level.
- Compare the decision with the actual state. If the null is true, rejection is Type I; if the null is false, non-rejection is Type II.
- Interpret probabilities conditionally. α describes the long-run Type I error rate under a true null; β describes Type II risk for a specified false-null alternative; power is 1 − β.
Illustrative analogy
OpenStax uses a tomato-plant example to show the logic: let the null hypothesis be “the plant is alive.” Calling the plant alive when it is actually dead corresponds to failing to reject a false null—a Type II error. The example is an analogy for matching a decision to the underlying state; the same four-cell structure applies to scientific, medical and engineering tests.
Type I versus Type II at a glance
| Comparison | Type I | Type II |
|---|---|---|
| Null-hypothesis state | True | False |
| Test decision | Reject | Fail to reject |
| Common everyday label | False positive; false alarm | False negative; missed effect |
| Notation | α | β |
| Design question | How costly is an unwarranted signal? | How costly is missing a real effect? |
Bias can also produce misleading positive or negative findings, but bias is not itself the formal definition of a Type I or Type II error. Those labels describe the relationship between a hypothesis-testing decision and the null hypothesis’s actual state.
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