Statistical Analysis for Education and Psychology Researchers

The sample size, n, does not vary. It is a fixed value for the sample. The scores xi may
well vary, and take on different individual values. Thus the individual scores, xi are free
to vary.
Each xi score could assume any possible value, and knowing the values of all but one
score does not tell us the value of the last score. If there are five scores and we know four
of them, (2, 5, 3, 7, ?), there is no way for us to determine the value of ‘?’. We must know
the individual values of the xi, and the total number, n, of all xi scores to calculate the
mean. In this sense the degrees of freedom for the statistic x is simply n. It is a fixed
value, the number of scores.
Now consider a more complicated example, degrees of freedom for the sample
variance. An explanatory formula for the sample variance S^2 is (we would usually use
formula 3.1 for computational purposes):

Again, consider which components in this formula can vary and which cannot. The

sample size, n, does not vary. It is a fixed value for the sample. The mean, does not
vary. It is also a fixed value for the sample. The deviations between each xi score and the

mean are free to vary and take on different values depending upon the

individual xi scores in the distribution. The question is, How many of these
scores are free to vary? The answer to this gives the degrees of freedom for the sample
variance.

The answer is all but one scores are free to vary. This is because the sum of

the deviations of scores about their own mean is always zero, If you
know n−1 of these deviations, you can always determine the last one because its value is
such that the sum of all deviations is zero.
Consider for example the distribution of scores (3, 7, 11). The mean is 7. Look what

happens if any two of the deviations are computed:

The third deviation must be +4 because the sum of the deviations from the mean is
always zero.
In this sense the last deviation is not free to vary. Knowing n−1 of the deviations, the
last deviation can be determined. Therefore only n−1 deviations are free to vary and take
on any value. Thus, the number of degrees of freedom for the sample variance (and the

Initial data analysis 71