# Power function

In statistics, the power function is a function that links the true value of a parameter to the probability of rejecting a null hypothesis about the value of that parameter.

## Definition

Here is a more formal definition.

Definition In a test of hypothesis about a parameter , let the null hypothesis beThe power function is a function that gives, for any , the probability of rejecting the null hypothesis when the true parameter is equal to .

Note that the power function depends on the null hypothesis: if we change , also the power function changes.

## Example

Suppose that we are testing the null hypothesis that the true parameter is equal to zero:

Suppose that the value of the power function at is

What does this mean? It means that if the true parameter is equal to , then there is a 50% probability that the test will reject the (false) null hypothesis that the parameter is equal to .

## Terminology

The parameter is often called alternative hypothesis and is called power against the alternative .

## Power and size

The size of a test is the probability of rejecting the null hypothesis when it is true.

Therefore, whenthe power function evaluated at gives the size of the test:

## Graph of the power function

We plot below the graph of a typical power function.

It plots the probability of rejecting an alternative in a z-test for the mean of a normal distribution, in which:

• is the unknown mean of the distribution;

• the variance of the distribution is known: ;

• the null is ;

• the size of the test is equal to 5%;

• the sample is made of 100 independent draws from the distribution.

Note that the minimum of the graph corresponds to the null and it is equal to the size of the test.

The power function, known in closed form, iswhere is the cumulative distribution function of the normal distribution, is the critical value corresponding to a 5% size, and is the number of draws.

## How to derive the power function

For examples of how to derive the power function, see the lectures:

## Dependence on sample size

Usually, the power of a test is an increasing function of sample size: the more observations we have, the more powerful the test.

## More details

You can find a more exhaustive explanation of the concept of power function in the lecture entitled Hypothesis testing.

Some related concepts are found in the following glossary entries:

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