Type I and Type II Errors Explained
Every hypothesis test can go wrong in exactly two ways. Learn what they are, how to visualize the trade-off, and why you cannot eliminate both simultaneously.
Key takeaways
- A Type I error (α) occurs when you reject the null hypothesis even though it is true. You conclude there is an effect when there is not.
- A Type II error (β) occurs when you fail to reject the null hypothesis even though it is false. You miss a real effect.
- These two errors exist in tension: tightening your criterion to avoid one makes the other more likely.
- You control α directly (e.g., set it to 0.05). You control β indirectly by choosing your sample size.
§1What is a Type I Error?
A Type I error is a false positive: you reject the null hypothesis (H₀) when it is actually true. You conclude there is an effect, a difference, or a relationship when in reality there is none. This is also called the significance level (α).
Imagine you are running a drug trial. The null hypothesis is that the drug has no benefit over placebo. A Type I error would mean you conclude the drug works when it actually does not. Your study, by random chance, generated results that looked like evidence of benefit—but the drug is useless. You might approve an ineffective treatment, waste patients' money, or expose them to side effects for no gain.
The probability of a Type I error is α. By convention, researchers often set α = 0.05, meaning they accept a 5% risk of falsely rejecting H₀. This does not mean "the probability that your result is due to chance" (a common misconception). Instead, it means: if H₀ were true and you repeated your study many times, you would see a result this extreme or more extreme about 5% of the time by random variation alone.
§2What is a Type II Error?
A Type II error is a false negative: you fail to reject the null hypothesis even though it is false. You conclude there is no effect when in reality an effect exists. This error rate is called β (beta).
Using the drug trial again: suppose the drug actually does work. A Type II error would mean your study fails to detect this benefit. By bad luck, your sample did not capture the effect strongly enough, or your sample was too small. You might discontinue a promising treatment, deny patients access to a life-saving drug, or abandon a real scientific finding.
The probability of a Type II error is β. The complement, 1 − β, is called statistical power—the probability that your test will correctly detect an effect if it truly exists. Researchers typically aim for power around 0.80 (an 80% chance of detecting a real effect if it is there), which means accepting a 20% risk of a Type II error.
§3The α and β Trade-Off: Interactive Visualization
Here is the fundamental tension: as you make it harder to reject H₀ (lowering α to avoid false positives), you make it easier to miss a real effect (increasing β, lowering power). Drag the slider below to see this trade-off in action. The blue curve is the distribution if H₀ is true (mean = 0). The gray curve is the distribution if a real effect exists (mean = 2.5, a moderate effect size). The vertical line is your decision cutoff. To the right of the line, you reject H₀. α (the red shaded area under the blue curve) is the Type I error rate; β (the faded red area under the gray curve) is the Type II error rate.
α (Type I error)
0.0505
β (Type II error)
0.2119
§4How Sample Size Affects the Trade-Off
Here is the key insight: increasing your sample size does not move the α-β frontier—it shrinks it. When you collect more data, both distributions (null and alternative) narrow around their means. This means you can detect smaller effects without raising your false positive rate.
Specifically, as n increases, the standard error decreases. A test with n = 30 is more sensitive than one with n = 10, so you are more likely to spot a real effect. Meanwhile, α (Type I error) stays fixed—you control it by setting your significance threshold, not by collecting more data. This is why sample size planning is essential: you decide in advance how much power you want (i.e., how small β should be), then calculate the n you need to achieve it.
§5Comparison: Type I vs. Type II
| Aspect | Type I Error | Type II Error |
|---|---|---|
| Definition | Rejecting H₀ when it is true | Failing to reject H₀ when it is false |
| Also called | False positive, significance level | False negative, miss |
| Symbol | α | β |
| Controlled by | Your significance threshold (e.g., α = 0.05) | Your sample size (via the desired power, 1 − β) |
| Real-world example | Approving an ineffective drug, releasing an innocent person with DNA evidence contaminated by error, publishing a false finding that wastes future research effort | Rejecting an effective drug, convicting the guilty person when weak evidence is present, failing to notice a real problem in your data |
§6FAQ
Can you eliminate both errors at once?
No. Decreasing α and β simultaneously would require an infinitely large sample size. In practice, you set α (e.g., to 0.05) according to how much you tolerate false positives, then you choose a sample size large enough to achieve your desired power (1 − β).
How does statistical power relate to Type II error?
Power is exactly 1 − β. A test with 80% power has a 20% chance of a Type II error (missing a real effect). Researchers usually target 80% power as a convention, which means accepting a 20% risk of a false negative.
Why is α conventionally set to 0.05?
This is an arbitrary convention dating to Sir Ronald Fisher's work in the 1920s. He suggested 0.05 as a reasonable threshold when no better benchmark existed. It is not sacred: your field, the stakes of a false positive, and your research question should guide your choice of α.
Which error is worse, Type I or Type II?
It depends on the context. A Type I error might waste money on an ineffective drug; a Type II error might withhold a life-saving treatment. Criminal law treats Type I errors (convicting the innocent) as graver than Type II errors (freeing the guilty). Always consider your field and stakes.
How can I calculate the sample size needed to control both α and β?
Use our sample size calculator. You specify your desired α (e.g., 0.05), your desired power (e.g., 0.80), the effect size you want to detect, and the test type. The calculator will give you the minimum sample size needed.
What happens to Type II error as sample size increases?
β decreases (power increases). A larger sample gives you more evidence, so you are more likely to detect a real effect if one exists. α (Type I error rate) does not change with sample size—it is set by your significance threshold (e.g., 0.05).
§7Cross-References
Dive deeper into related concepts:
- Sample size calculator — Plan your study to achieve your desired power and control both α and β.
- T-test calculator — Perform a t-test and interpret the p-value in light of Type I and Type II error.
- P-value calculator — Understand what a p-value actually means (the probability of seeing data this extreme if H₀ is true).
- Which statistical test should I use? — Choose the right test for your data and learn how the test controls α.
§8Sources
- OpenStax Introductory Statistics 2e — Hypothesis testing chapter; canonical coverage of Type I and Type II errors with worked examples.
- NIST/SEMATECH e-Handbook of Statistical Methods — Authoritative reference on statistical power, Type I and Type II errors, and sample size planning.
- Wikipedia: Type I and type II errors — Overview of definitions, applications, and historical context.
- Statistics How To: Hypothesis Testing — Clear explanations with examples for students.