Addendum to Chapter 14 Tue, Apr 25, 2006 Degrees of Freedom Addendum to Chapter 14 Tue, Apr 25, 2006

Goodness-of-Fit Degrees of Freedom In the goodness-of-fit test for a fair die, we had the following data: Given that the total is 60, how many cells could we fill in arbitrarily and still make the total 60? 1 2 3 4 5 6 Total 8 10 14 12 9 7 60 1 2 3 4 5 6 Total ? 60

Goodness-of-Fit Degrees of Freedom In the goodness-of-fit test for a fair die, we had the following data: Given that the total is 60, how many cells could we fill in arbitrarily and still make the total 60? 1 2 3 4 5 6 Total 8 10 14 12 9 7 60 1 2 3 4 5 6 Total ? 60

Goodness-of-Fit Degrees of Freedom Use the spreadsheet DegreesOfFreedom1.xls. We could fill in all but one of the cells, i.e., n – 1 cells. That would force the value in the last cell to make the total equal to 60. Therefore, we have n – 1 degrees of freedom.

Independence Degrees of Freedom In the test of independence, we use the data Given the row and column totals, how many cells could we fill arbitrarily and still make the row and column totals the same as before? A B C D F 400 - 500 7 8 16 20 21 500 – 600 13 28 32 22 600 – 700 23 10 9 700 - 800 14 5

Independence Degrees of Freedom A B C D F Col Tot 400 - 500 ? 72 500 – 600 108 600 – 700 700 - 800 48 Row Totals 36 84 60 300

Independence Degrees of Freedom A B C D F Col Tot 400 - 500 ? 72 500 – 600 108 600 – 700 700 - 800 48 Row Totals 36 84 60 300

Independence Degrees of Freedom Use the spreadsheet DegreesOfFreedom2.xls. Thus, we can fill in all but one value in each row for all but one row. That is, we can fill in (cols – 1) values in each row for (rows – 1) rows. So we have (cols – 1)  (rows – 1) degrees of freedom.