Bell Curve

560 Appendix Iaveraged over a period of years?The value of one's savings or possessions?And wealth, compared to many of the other things social science wouldlike to understand, is easy, reducible as it is to dollars and cents.But beyond the problem of measurement, social science must copewith sheer complexity. Our physical scientist colleagues may not agree,but we believe it is harder to do science on human affairs than on inanimateobjects-so hard, in fact, that many people consider it impossible.We do not believe it is impossible, but it is rare that any human orsocial relationship can be fully captured in terms of a single pair of variables,such as that between the temperature and volume of a gas. In socialscience, multiple relationships are the rule, not the exception.For both of these reasons, the relations between social science variablesare typically less than perfect. They are often weak and uncertain.But they are nevertheless real, and, with the right methods, they can berigorously examined.Correlation and regression, used so often in the text, are the primaryways to quantify weak, uncertain relationships. For that reason, the advancesin correlational and regression analysis since the late nineteenthcentury have provided the impetus to social science. To understandwhat this kind of analysis is, we need to introduce the idea of a scatterdiagram.Scatter DiagramsWe left your male high school classmates lined up by height, with youlooking down from the rafters. Now imagine another row of cards, laidout along the floor at a right angle to the ones for height. This set ofcards has weights in pounds on them. Start with 90 pounds for the classshrimp, and in 10-pound increments, continue to add cards until youreach 250 pounds to make room for the class giant. Now ask your classmatesto find the point on the floor that corresponds to both their heightand weight (perhaps they'll insist on a grid of intersecting lines extendingfrom the two rows of cards). When the traffic on the gym floorceases, you will see something like the figure below. This is a scatter diagram.Some sort of relationship between height and weight is immediatelyobvious. The heaviest boys tend to be the tallest, the lightestones the shortest, and most of them are intermediate in both height andweight. Equally obvious are the deviations from the trend that linkheight and weight. The stocky boys appear as points above the mass,Weight in poundsA scatter diagram.100 - .Appendix J 5618 0 1 ~ , ~ , ~ , ~ 1 1 , ~ , ~ 1 ~ 160 62 64 66 68 70 72 74 76 78 80Height in inchesthe skinny ones as points below it. What we need now is some way toquantify both the trend and the exceptions.Correlations and regressions accomplish this in different ways. But beforewe go on to discuss these terms, be reassured that they are simple.Look at the scatter diagram. You can see by the dots that as height increases,so does weight, in an irregular way. Take a pencil (literally orimaginarily) and draw a straight, sloping line through the dots in a waythat seems to you to best reflect this upward-sloping trend. Now continueto read, and see how well you have intuitively produced the resultof a correlation coefficient and a regression coefficient.The Cowelation CoefficientModem statistics provides more than one method for measuring correlation,but we confine ourselves to the one that is most important inboth use and generality: the Pearson product-moment correlation coefficient(named after Karl Pearson, the English mathematician and biometrician).To get at this coefficient, let us first replot the graph of theclass, replacing inches and pounds with standard scores. The variablesare now expressed in general terms. Remember: Any set of measurementscan be transformed similarly.