# Power of a test

Power is the probability of rejecting the null when a real effect exists: power = 1 - beta. Power rises with a larger sample, a bigger true effect and a larger alpha.

Statistics (university) · Part 06: Hypothesis testing · Card 5 of 9 · Last updated 2026-10-19

## Analogy

How good a metal detector is at finding a coin that really is buried.

## Example

**Power by hand**

H0: mean 100, true mean 104
sigma 15, n 100, alpha 0.05 one-sided
Reject if mean > 100 + 1.645 x 1.5 = 102.47
Power = P(Z > (102.47 - 104)/1.5)
= P(Z > -1.02) = 0.85

## In practice

Studies are planned for about 80 percent power so a real effect is likely to show up.

## Practice

### Flashcards

- **What is power?** 1 minus beta: the chance of detecting a real effect.

### Interview questions

**Explain Type I and Type II errors using a medical screening example, and say how sample size affects them.**

- Type I is a false alarm
- Type II is a missed case
- Alpha is the Type I rate
- More data raises power
- Lowering alpha raises beta

## Questions people ask

### How can I increase the power of a study?

Collect more data, since this shrinks the standard error. Reduce measurement noise, aim for a larger effect, or accept a higher alpha. Of these, sample size is usually the practical lever, which is why power is calculated before data collection.

### What is the link between power and Type II error?

Power equals 1 minus the Type II error probability beta. If beta is 0.15, power is 0.85, meaning an 85 percent chance of detecting the effect when it exists. Low power means real effects are often missed, so a study with 30 percent power wastes time and money.

## Related concepts

- [Type I and Type II errors](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-6/statistics-university-part-6-concept-4)
- [Null and alternative hypotheses](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-6/statistics-university-part-6-concept-1)
- [p-value](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-6/statistics-university-part-6-concept-2)
- [Significance level and decision](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-6/statistics-university-part-6-concept-3)

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