The new multiple range test proposed by Duncan makes use of special protection levels based upon degrees of freedom. Let formula_59 be the protection level for testing the significance of a difference between two means; that is, the probability that a significant difference between two means will not be found if the population means are equal. Duncan reasons that one has p-1 degrees of freedom for testing p ranked mean, and hence one may conduct p-1 independent tests, each with protection level formula_59 . Hence, the joint protection level is: