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FIGURE 3-4: Smaller effects need larger samples.

This inverse relationship between sample size and effect size takes on a very simple mathematical

form (at least to a good approximation): The required sample size is inversely proportional to the

square of the effect size that can be detected. Or, equivalently, the detectable effect size is

inversely proportional to the square root of the sample size. So, quadrupling your sample size

allows you to detect effect sizes only one-half as large.

How to do power calculations

Power calculations can be an important step in the design of a research study because they

estimate how many individuals you will need in your sample to achieve the objectives of your

study. You don’t want your study to be underpowered, because then it will have a high risk of

missing real effects. You also don’t want your study to be overpowered, because then it’s larger,

costlier, and more time-consuming than necessary. You need to include a power/sample-size

calculation for research proposals submitted for funding and for any protocol you submit to a

human research ethical review board for approval. You can perform power calculations using

several different methods:

Computer software: The larger statistics packages such as SPSS, SAS, and R enable you to

perform a wide range of power calculations. Chapter 4 describes these different packages. There

are also programs specially designed for conducting power calculations, such as PS and G*Power,

which are described in Chapter 4.

Web pages: Many of the more common power calculations can be performed online using web-

based calculators. An example of one of these is here: