Essentials

74 ESSENTIALS OF STATISTICSz scores on both variables and multiply them together. Then we average thesecross products (i.e., add them up and divide by the number of cross products).This coefficient reaches its greatest magnitude when everyone has the same zscore on both variables. In that case the formula reduces to Σz 2 /N, which alwaysequals exactly 1.0 (the variance of a set of z scores is Σ(z-z 2)/N, which alwaysequals 1.0, but z is always zero, so Σz 2 /N 1.0). If everyone has the same z scoreon both variables, but with opposite signs, r equals –Σz 2 /N –1.0. For less thanperfect correlation the magnitude of r will be less than 1.0 and can be as small aszero when the two variables have no correlation at all.Computational FormulasFormula 4.1 is easy to understand, but it is not convenient for calculation. Substitutingthe corresponding z score formulas for the X and Y variables and rearrangingalgebraically yields the following convenient formula:∑ XY N x yr (4.2)Expressed verbally this formula tells us to average the cross products of thescores and subtract the product of the population averages, and then divide bythe product of the two population standard deviations. This formula is instructive.The numerator is called the covariance, and it determines the sign of the correlation.When a number that is relatively large for one variable tends to be pairedwith a relatively large number for the other variable (and, of course, small withsmall), the average of the cross products tends to be higher than the product ofthe averages, yielding a positive correlation. When relatively large numbers forone variable are consistently paired with relatively small numbers for the other;ΣXY/N will be smaller than x y, producing a negative correlation. When valuesfor the two variables are paired at random, the two terms in the numeratortend to be the same size, resulting in a zero or near-zero value for r. The denominatorof Formula 4.2, the product of the two standard deviations, tells you justhow large the covariance can get in either the positive or negative direction.When the covariance reaches its maximum negatively or positively, r will equal–1 or 1, respectively.The only aspect of Formula 4.2 that is not convenient is that it assumes thatyou are treating your set of scores as a population and have therefore calculatedthe standard deviations with N, rather than N – 1, in the denominator (as in Formula1.4). In most research settings, however, you will be treating your scores asa sample, and it is likely that you will want to calculate s rather than for each ofx y