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In our discussion of variance, we saw that if we wanted to use the sample variance as an estimator of the population variance, we needed to subtract 1 from the denominator. That is, S² was “corrected” into s²:

We say we lost a degree of freedom in the denominator of the statistic. But what are degrees of freedom? They are the number of independent units of information in a sample that are relevant to the estimation of some parameter (Everitt, 2002). In the case of the sample variance, s², one degree of freedom is lost since we are interested in using s² as an estimator of σ². We are losing the degree of freedom because the numerator, , is not based on n independent pieces of information since μ had to be estimated by . Hence, a degree of freedom is lost. Why? Because values of y_i are not independent of what is, since is fixed in terms of the given sample data. In general, when we estimate a parameter, it “costs” a degree of freedom. Had we μ, such that , we would have not lost a degree of freedom, since μ is a known (not estimated) parameter.

A conceptual demonstration may prove useful in understanding the concept of degrees of freedom. Imagine you were asked to build a triangle such that there was to be no overlap of lines on either side of the triangle. In other words, the lengths of the sides had to join neatly at the vertices. We shall call this the “Beautiful Triangle” as depicted in Figure 2.9. You are now asked to draw the first side of the triangle. Why did you draw this first side the length that you did? You concede that the length of the first side is arbitrary, you were free to draw it whatever length you wished. In drawing the second length, you acknowledge you were also free to draw it whatever length you wished. Neither of the first two lengths in any way violated the construction of a beautiful triangle with perfectly adjoining vertices.

However, in drawing the third length, what length did you choose? Notice that to complete the triangle, you were not free to determine this length arbitrarily. Rather, the length was fixed given the constraint that the triangle was to be a beautiful one. In summary then, in building the beautiful triangle, you lost 1 degree of freedom, in that two of the lengths were of your free choosing, but the third was fixed. Analogously, in using s² as an estimator of σ², a single degree of freedom is lost. If is equal to 10, for instance, and the sample is based on five observations, then y₁, y₂, y₃, y₄ are freely chosen, but the fifth data point, y₅ is not freely chosen so long as the mean must equal 10. The fifth data point is fixed. We lost a single degree of freedom.

Figure 2.9 The “Beautiful Triangle” as a way to understanding degrees of freedom.

Degrees of freedom occur throughout statistics in a variety of statistical tests. If you understand this basic example, then while working out degrees of freedom for more advanced designs and tests may still pose a challenge, you will nonetheless have a conceptual base from which to build your comprehension.