Statistics · step 13 of 13

Type I / Type II errors & power

A Type I error rejects a true H₀ — a false positive, occurring with probability α. A Type II error fails to reject a false H₀ — a false negative, with probability β. Power is 1 − β, the chance of detecting a real effect. The two error rates trade off: lowering α raises β unless you increase the sample size, which is the only way to improve both.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Fix the definitionsType I: reject a true null. Type II: fail to reject a false null. Write them out — the labels are easy to swap under pressure.
Recognise α as your choiceThe significance level IS the Type I error rate, chosen before the test.
Identify what drives powerPower increases with sample size, with effect size, with larger α, and with smaller variance.
Note the trade-offFor fixed n, reducing α increases β. Only a larger sample reduces both.

Worked example

A medical test for a serious disease is being designed. Which error type is worse, and how should α be set?

Type I error: declaring disease when the patient is healthy — unnecessary treatment and anxiety.
Type II error: missing a real disease — potentially fatal delay.
Here the Type II error is far more costly.
So use a larger α (more willing to flag) and increase the sample or test sensitivity to raise power.

Answer. The Type II error dominates, so the design should favour power: a higher α, a more sensitive test, or a larger sample — accepting more false positives to avoid missed cases.

Where marks get dropped

These are the specific errors that cost credit on type i / type ii errors & power questions — QED's rubric penalises each of them separately.

Swapping the two error types. Type I goes with α and with rejecting; Type II goes with β and with failing to reject.
Treating power as a property of the test alone. It depends on the effect size you want to detect, so power must always be stated "to detect an effect of size d".
Reducing α to 0.01 without increasing n and expecting no cost. β rises and real effects get missed.

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Type I / Type II errors & power — frequently asked questions

What is a typical target power?

80% is the conventional minimum, with 90% preferred in clinical work. Power below 50% means the study is more likely to miss a real effect than find it.

How do I increase power?

Increase n (the main lever), increase α, reduce measurement noise, or use a more efficient design such as pairing.

What is an underpowered study?

One unlikely to detect the effect it seeks. Worse, its significant results are more likely to be exaggerated or false — the winner’s curse.

The rest of Statistics

Describing data, distributions, estimation and hypothesis tests. Each subtopic below has its own method, worked example and mark-losing traps.

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