    Chapter 4: Analysing the Data Part II : Descriptive Statistics # Sums of squares

Consider for a minute transforming the original data in Table 4.1 to deviations. That is, each score is converted to the difference between that score and the mean. So all 1s become 1 minus 1.864 or -0.864. All 0s become 0 minus 1.864 or -1.864. All 2s become 2 minus 1.864 or 0.136. It might be obvious to you that if the scores tended to differ a lot from the mean, then these differences would tend to be large (ignoring the sign), whereas these differences would tend to be small (ignoring sign) if the scores tended vary little from the mean.

The measure typically used to quantify variability in a distribution is based on the concept of average squared deviation from the mean.

LetÕs take each difference from the mean and square it. Then, letÕs add up these squared deviations. When you do this you have the sum of the squared deviations (which is then reduced to "Sums of Squares", or SS). Its formula is The left-hand side of the equation is the definitional formula and the right hand side is the computational formula. SPSS output does not give the Sums of Squares for a variable when you choose Frequencies. However, many later statistical procedures do give this as part of the output. ItÕs value lies in summarising the total amount of variability in the variable being examined. For the sex partners data SS = 848.74 (calculated by the method below). The size of this number depends on the size of the numbers in the data and how much data there is (i.e., the sample size). There are no units for SS.

Sometimes there is confusion about the terms variability and variance. Variability refers to the Sums of Squares for a variable, while variance refers to the Sums off Squares divided by N-1. Sums of Squares are widely used because they are additive. Once we divide by N-1, the additive property disappears. When we later talk about the "proportion of variance explained" we really mean the "proportion of variability explained". If a variable X explains 56% of the variability in variable Y it refers to the proportion of YÕs Sums of Squares that is attributable to variable XÕs Sums of Squares.

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