03/07/2022
Degrees of Freedom-
is defined as the number of scores that can vary independently in a sample, that is, the number of scores in a sample that can take on any possible value, with one score being completely dependent on the other n – 1 scores. For example, say you have a sample with n = 5 scores and you know the mean is M = 10. Four (n – 1) of the five scores could be any value, but for the mean to be M = 10 the value of that last (fifth) score depends on the other n – 1 scores. Thus, four scores (n – 1) can vary independently from each other, but one completely score depends on the others.
Say four (n – 1) of five scores are 7, 10, 11, 12. If M = 10, what must the fifth score be for the mean to be M = 10? The four scores sum to 40; hence, to obtain M = 10, the five (n) scores must add be 50. In this case, fifth score must be equal to 10. As another example, say four of the scores are 1, 2, 3, 4; what must the fifth score be for the mean to be M = 10? The sum of five scores must be equal to 50 to have M = 10, and because the sum of the four scores was only 10, the fifth score must be equal to 40 (50 – 10 = 40).
In any set of sample data, exactly n – 1 scores can vary independently of each other and theoretically take on any conceivable value for that variable. But exactly one score is completely dependent on the values of the other n – 1 scores. Thus, degrees of freedom are the number of scores allowed to vary independently with the last score being completely dependent on the others. In calculating estimates of the population variance and standard deviation the degrees of freedom determine the accuracy of those population estimates. That is, as degrees of freedom increase the accuracy estimating the population parameters also increases. This means that larger samples generate better estimates of the population.
