Essentials

84 ESSENTIALS OF STATISTICSCAUTIONSome Reasons for a LowPearson’s r1. The true relation between the twovariables may be curvilinear.2. Most of the sample follows a fairlylinear relationship, but the samplecontains one or more bivariateoutliers.3. One or both of the variables mayhave a restricted (truncated) rangein your sample.4. There is very little relation betweenthe two variables; the pointsin the scatterplot do not exhibitany clear trend.ticipants except the new one. In thiscase a bivariate outlier could greatlyincrease the magnitude of the correlation.The problem depicted in Figure4.5 is called a truncated or restrictedrange. If you erect a vertical line threequarters of the way to the right in Figure4.1, the points to the right of thatline will not exhibit a strong trend andtherefore will yield a rather small r. Ingeneral, the trend of the points has tooutweigh the scatter of the points toattain a decent sample r. Because insocial science research there is usuallya good deal of scattering of points,you generally want to include a largerange on both variables so the trend,if any, will be prominent. Of course, you don’t want the range to be so large thatthe trend becomes curvilinear as in Figure 4.3. You can often use past research asyour guide in deciding on the optimum range of your variables. Unfortunately,sometimes a researcher can be stuck with a sample of convenience that has a restrictedrange on a variable of interest (e.g., all of the available participants are welleducated). In that case you may need to sample a large number of participants:The larger sample won’t make r any higher, but it can increase your chances of gettingstatistical significance with a fairly small r.REGRESSIONWhen two variables are highly correlated (whether positively or negatively),knowing a person’s value on one variable allows you to make a fairly accurate predictionconcerning his or her value on the other variable. Even with a moderatedegree of correlation, useful predictions can be made. This is the logic behind usingstandardized aptitude tests (like the Scholastic Aptitude Test [SAT] or AmericanCollege Test [ACT]) as one of the criteria for selecting applicants to a college(such tests yield a moderate r when correlated with a student’s college GPA upongraduation). Although social scientists rarely make predictions, the most commonmethod that is used for making them, linear regression, has other uses, as youwill see. In this chapter we will deal only with simple (i.e., bivariate) linear regression:the prediction of one variable by only one other based on the linear relation