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# Statistical power

- Concept

Power is the probability that a study detects an effect that truly exists. An underpowered study can miss real effects.

Updated Nov 5, 2026

## Key points

- Power = 1 − β.
- It rises with a larger sample, a larger effect, a higher α and less variability (SD) in the measurement.
- A two-sided test has slightly less power than a one-sided test at the same α.
- Paired designs and more precise measurements raise power by reducing variability.
- Calculating power after the study from the observed effect adds nothing; the CI shows better which effects were ruled out.

## What it is

Statistical power (1 − β) is the probability of rejecting the null hypothesis when it is false, that is, of finding a real difference of the expected size. It is set when the study is designed, usually at 80% or 90%. An underpowered study can give a false negative, and when it does come out significant by chance it tends to exaggerate the size of the effect.

## Worked example

- Data: a trial compares two diets on systolic pressure; SD = 12 mmHg, two-sided α = 0.05, 40 people per group.
- SE of the difference = 12 × √(2 / 40) = 12 × 0.224 = 2.68 mmHg.
- If the true effect is 5 mmHg: 5 / 2.68 = 1.86; power ≈ normal probability of (1.86 − 1.96) ≈ 46%.
- With 91 people per group, the SE falls to 1.78 and power rises to about 80%.
- With 40 per group but a true effect of 8 mmHg: 8 / 2.68 = 2.98; power ≈ probability of (2.98 − 1.96) ≈ 85%.
- Reading: with 40 per group, a 5 mmHg effect has less than a 50% chance of being detected; more people or a bigger effect raises power.

## What raises and what lowers power

| Factor | To raise power | Comment |
| --- | --- | --- |
| Sample size | More participants | The most used lever |
| Effect size | Larger true effect | Not under control; choose the smallest relevant effect |
| α level | Higher α (0.10 instead of 0.05) | At the cost of more false positives |
| Variability | More precise measurements, paired design | Smaller SD, smaller SE |
| Type of test | One-sided (if justified in advance) | Two-sided is the standard |

## Markers

- 1 − β
- 80%
- 90%

## Common pitfalls

- Reading a non-significant result from a small study as proof of no effect.
- Planning the study around an unrealistically large expected effect so that a small sample comes out.

## Memory aid

‘More people, more effect, less noise’: the three ways to give a study power.

## Test yourself

Think of the answer, then tap the question to see it.

## What is statistical power?

The probability of rejecting the null hypothesis when it is false: detecting a real effect. It equals 1 − β.

## Which four factors increase power?

A larger sample size, a larger effect, a higher α and less variability in the measurement.

## Why does a paired design usually have more power?

Because each person is their own control, removing between-person variability and so reducing the standard error.

## A 30-patient trial finds no difference. What should be checked before concluding?

Its power and the confidence interval of the effect: with few patients, important effects may still be compatible with the data.

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## Connections

## Related 4

- [Type I and type II errors](https://durgical.com/en/epidemiology/type-i-and-type-ii-errors/)
- [Sample size](https://durgical.com/en/epidemiology/sample-size/)
- [Hypothesis testing](https://durgical.com/en/epidemiology/hypothesis-testing/)
- [Standard error](https://durgical.com/en/epidemiology/standard-error/)

[← All of epidemiology](https://durgical.com/en/epidemiology/)

Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.

Sources: [Sample sizes required](https://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); [What are statistical tests?](https://www.itl.nist.gov/div898/handbook/prc/section1/prc13.htm) (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012).

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Representative model, for educational purposes only.

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