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CHAPTER 12 Regression analysis 407
This diagram shows just one datapoint – if you try to draw a regression line, any line is as good as any
other line. This isn’t very useful! The degrees of freedom for regression are n - 1 (where n is the number
of datapoints). So here, DF = 0 (1 - 1 = 0). As Dr Yu says ‘. . . the data has no “freedom” to vary, and you
don’t have any “freedom” to conduct research with this data set’. In fact, if you try carrying out such an
analysis in SPSS, you will obtain an error message.
Now imagine you have two datapoints for your linear regression.
When there are only two points, a straight line will always be a perfect fi t. In this case, there is one degree
of freedom for estimation. DF = 1 (2 - 1 = 1). If you try this in SPSS, you will obtain output. However,
the results mean nothing because we already know the fi t is perfect. With just two datapoints, it couldn’t
be any other way.
Now look at this scatterplot:
Here the best fi tting line is shown between three datapoints, so the degrees of freedom are 2. This time
the line has more freedom to vary, since the line of best fi t is not restricted to the path between two
datapoints.
Dr Yu explains degrees of freedom as ‘the number of pieces of useful information’. When we had
DF = 0, there was no useful information. When we had DF = 1, there was not enough information to be
useful to us. Even with DF = 2, there was still not enough useful information for us to perform a useful
analysis.
Dr Yu has an online audio-visual explanation of degrees of freedom, with interactive activities. We
have provided you with his website address so that you can learn more while having fun (Yu, 2003).