The Minimally Important Difference Significant Criterion for Sample Size

For a wide range of tests of single-df hypotheses, the sample size needed to achieve 50% power is readily approximated by setting N such that a significance test conducted on data that fit one's assumptions perfectly just barely achieves statistical significance at one's chosen alpha level. If the effect size assumed in establishing one's N is the minimally important effect size (i.e., that effect size such that population differences or correlations smaller than that are not of any practical or theoretical significance, whether statistically significant or not), then 50% power is optimal, because the probability of rejecting the null hypothesis should be greater than .5 when the population difference is of practical or theoretical significance but lower than .5 when it is not. Moreover, the actual power of the test in this case will be considerably higher than .5, exceeding .95 for a population difference two or more times as large as the minimally important difference (MID). This minimally important difference significant (MIDS) criterion extends naturally to specific comparisons following (or substituting for) overall tests such as the ANOVA