When performing multiple statistical significance tests on the same data, the Bonferroni adjustment can be applied to make it more "difficult" for any one test to be statistically significant. For example, when reviewing multiple correlation coefficients from a correlation matrix, accepting and interpreting the correlations that are statistically significant at the conventional .05 level may be inappropriate, given that multiple tests are performed. Specifically, the alpha error probability of erroneously accepting the observed correlation coefficient as not-equal-to-zero when in fact (in the population) it is equal to zero may be much larger than .05 in this case.

The Bonferroni adjustment usually is accomplished by dividing the alpha level (usually set to .05, .01, etc.) by the number of tests being performing. For instance, suppose you performed multiple tests of individual correlations from the same correlation matrix. The Bonferroni adjusted level of significance for any one correlation would be:

Any test that results in a p-value of less than .01 would be considered statistically significant; correlations with a probability value greater than .01 (including those with p-values between .01 and .05) would be considered non-significant.