Sum of the Squares, SS

Description

In measuring how spread out a set of measures are, the sum of the squares, often indicated as SS, gives a measure that is simple to calculate and use.

It is calculated as the square of the sum of differences between each measure and the average.

SS = SUM(X_i - AVERAGE(X))

The average of a set of x's may be written as x-bar (or x with a horizontal line above it).

Example

In a set of measure, SS is calculated as below. Column B, below is the gap between x in Column A and the average of Column A. This is squared in Column C to get rid of the minus signs (otherwise summing these would be close to zero).

	Column A	Column B	Column C

	x	x - x-bar	(x-x-bar)^2
	5	0.25	0.0625
	6	1.25	1.5625
	2	-2.75	7.5625
	8	3.25	10.5625
	3	-1.75	3.0625
	5	0.25	0.0625
	2	-2.75	7.5625
	7	2.25	5.0625

number of measures, n:	8
sum of measures:	38
average, x-bar:	4.75

Sum of squares, SS:			35.5

Discussion

Subtracting each number from the average (or mean) gives an indicate of individual spread. Adding these up, however, could well result in something close to zero. If the differences are squared, this gets rid of the minus signs, allowing the sum to add up to something sensible.

Whilst SS may be adequate for comparing the spread of two similar lists of numbers, it increases with the size of a list, and hence makes comparison of unequal length lists invalid.

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