Sanderson M. Smith

DEGREES OF FREEDOM (INFORMAL PRESENTATION)

These examples attempt to provide an informal understanding of the phrase "degrees of freedom."

Example 1:

The table contains a set of numbers {a,b,c,d,e,f} whose sum is 100.

 SUM a b c d e f 100

How many of the six numbers would have to be known before the remaining numbers could be determined?

Response: If you knew any five of the six numbers in the set {a,b,c,d,e,f}, the sixth number would be determined. You could, for instance, substitute any values you desire for a,b,c,d, and e. The number f would then be determined. Given the sum of 100, the six numbers contain five independent numbers. In this example, we have 5 degrees of freedom.

Example 2:

In this table, the row and column sums are provided.

 Column 1 Column 2 ROW TOTALS Row 1 x y 70 Row 2 z w 30 COLUMN TOTALS 80 20 100

How many numbers in the set {x,y,z,w} must be known before the remaining numbers are determined?

Response: You need only know one of the numbers in the set to determine the others. For instance, if x = 60, then we must have y = 10, z = 20, and w = 10. There is only 1 degree of freedom.

Example 3:

This 4-by-3 table (4 rows, 3 columns) provides row and column sums.

 Column 1 Column 2 Column 3 ROW TOTALS Row 1 a b c 80 Row 2 d e f 70 Row 3 g h w 50 Row 4 x y z 60 COLUMN TOTALS 90 100 70 260

Response: You need to know six of the numbers, as illustrated in the table below with values substituted for a, c, e, w, x, and y.

 Column 1 Column 2 Column 3 ROW TOTALS Row 1 10 b 5 80 Row 2 d 15 f 70 Row 3 g h 20 50 Row 4 30 25 z 60 COLUMN TOTALS 90 100 70 260

With the six numbers provided, you can conclude that b = 65, h = -5, g = 35, d = 15, f = 40, and z = 5. In this 4-by-3 table, there are 6 degrees of freedom. (You can try substituting five values in the original table. This should convince you that the table is not determined if just five numbers are provided.)

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NOTE: If you have a table with r rows and c columns, where r > 1 and c>1, the degrees of freedom is (r-1)(c-1). In example #2, r = 2 and c = 2. The degrees of freedom is (2-1)(2-1) = 1. In example #3, r = 4 and c = 3. The degrees of freedom is (4-1)(3-1) = 6.