### Computing the minimum effect size that can be detected with a specified level of statistical power

Rearranging the equation (Formula 1):

to solve *d*, we obtain:

Suppose we have only 25 cases available for study. What is the minimum effect size that we could expect to detect with power = 80% using one-tailed alpha = .05?

In this scenario, *n* = 25*,* and using the WISE *p*–*z* converter, we determine. Z_{α} = 1.645 and *Z _{β}* = -0.842.

Applying the formula we find *d* = (2.487 / 5) = .497

This tells us that if the actual effect is smaller than about .50, then power for the contemplated study is less than 80%. We may decide that the study is not worth conducting because it is important to design a study that is likely to detect an effect that is less than .50, say .40.

If we can specify a minimum effect size that is important to detect, then we can use that value along with *n* and alpha error to compute power. In this example, if d = .40, alpha = .05, and *n* = 25, we can use the WISE Power Applet to find Power = .638. We may decide that it is not worth conducting the study with such low power.

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