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Thinking About Correlations

Various statistical concepts play a central role in Chapter 14, and perhaps the most important concept is that of correlation.

How do we know, for example, whether the IQ test is valid? The most common technique is to ask whether we can use the test to predict other things we care about. For example, if we know someone's IQ score, can we use this as a basis for predicting how well they'll do in school, or how successful they'll be in their jobs? If so, this is an indication that the IQ test is measuring something important.

As the chapter describes, the IQ test does seem valid by this criterion. For example, many studies show a correlation of roughly +.50 between IQ scores and students' grade point averages in college. Therefore, if we know someone's IQ, we can predict their grades with some reasonable accuracy; this suggests that IQ may be measuring some capacity that is helpful in attaining good grades — just as we would expect if the IQ test does (as planned) measure intelligence.

Many students, however, are skeptical about these claims. Often, they mention that they know people who do well on standardized tests (such as the IQ test), but who do poorly in school, or vice versa. These seem to be exceptions to the pattern that's being claimed; should these cases undermine our acceptance of the IQ test's validity?

What Does A Correlation of +.50 Signify?

One way to think about the validity of the IQ test is to consider what a correlation of +.50 means. Should we be impressed by how big this correlation is because it is certainly different from a correlation of zero? Or should we be impressed by how small this correlation is because it is certainly different from a (perfect) correlation of 1.00?

Let's approach this issue step by step. First, let's imagine (contrary to fact!) that there is a perfect (1.00) correlation between IQ and grade-point average. In this case, a listing of students might look like Table 1, below.

Press the "plot data" button to generate a scatter plot diagram of the data Table 1. Each student will then be represented by a dot on our diagram; the dot's left-right position indicates IQ, and its height indicates the corresponding GPA.

But of course the data are more complicated than that. Table 2 shows a correlation between IQ and GPA of .54 — just the sort of correlation we would expect to observe in real data.

Press the "Plot Data" button. How should we think about the resulting diagram?

Notice that there is a relationship here. Look at the general "shape" of the graph, rather than the individual points. Points on the right are generally higher than the points on the left. Click on the "Ellipse View" button. As you can see, the ellipse is plainly pointing upward.

Alternatively, you could divide the graph into quadrants, with a horizontal rule between the top and the bottom half of the points, and also a vertical rule between the left and the right half of the points. Click on the "Quadrant View" button. Notice that the points are not evenly distributed across the four quadrants of the graph. Points on the left tend to be lower rather than higher, while points on the right tend to be higher rather than lower.  This perspective merely confirms what the correlation statistic is telling you. As we said, this graph depicts a +.53 correlation — plainly a positive relationship.

By the same token, this is not at all an "exception-less" pattern, and that is why the correlation is as small as +.53 (i.e., is much less than +1.00). Said differently, it's guaranteed that there will be exceptions to the pattern. If there were no exceptions, the correlation would be higher than it is.

The exceptions, too, are visible in the graph. Press the "Exceptions" button. Consistent with the pattern, the highest score (circled in red) is all the way to the right. But the second highest score (circled in blue) is in the middle of the graph, a departure from the "further right = higher up" overall trend. And one of the highest scores is also way to the left (circled in yellow). This is certainly contrary to the overall trend!

What causes these exceptions? Or, said differently, why is the correlation "merely" +.53? The answer should be obvious: School grades do depend on your level of intelligence, but they also depend on many other factors. Some students, whether they are smart or not, work harder than others. Some students, smart or not, have good work habits, and some do not. Some students, smart or not, choose somewhat easier courses. Some students, smart or not, have good luck with the exam questions (e.g., the exam questions happen to line up nicely with the things the students emphasized in their studying). Some students stay healthy during the term, and so can work effectively; others become ill. Some students have quiet room-mates who don't disturb them while studying. Some students . . .

Conclusion

The obvious point here is that there are many factors (we have named only a few!) that contribute to success in college, and most of these factors are entirely separate from IQ. It's no wonder, therefore, that IQ is not any where close to being a perfect predictor of academic performance; it is, so to speak, only one ingredient in the mix of factors that determines this performance.

If IQ is just one factor of many, then why should we focus on IQ? Why not focus instead on the other factors? The answer is that IQ is probably all by itself the most important factor. 

To say this more precisely: IQ scores only account for part of the pattern. In fact, they account for roughly 25% of the pattern.* Why should we care about this "small piece" of the puzzle? The answer is that all of the other pieces of the puzzle each count for a much smaller percentage. Motivation counts only for a small bit, and likewise work habits, and luck, and so on. If we had to choose the single most important factor in shaping school performance, it would be IQ.

Where does all of this leave us? We have tried in this tutorial to draw your attention to two key points: The correlations often found in psychology are usually between +.30 and +.60. These are significantly greater than 0, and that tells us that there is indeed a relationship between the two variables being correlated.

As a concrete example, common sense observation tells us that, in general, men are somewhat taller than women. This isn't always true, but it's true frequently enough so that the height difference between the sexes is apparent to any observer. And, in quantitative terms, the correlation between sex and height is around +.42 — a strong enough pattern so that it is easily seen.

At the same time, though, correlations between +.30 and +.60 are low enough so that we guaranteed exceptions to the pattern; it is those exceptions that (in the calculations) lead to correlations that are, like the ones we are considered, moderate in strength. These exceptions don't mean there is no pattern. They simply tell us that the pattern is not perfectly reliable.

To continue the example, we all know short men and tall women. Therefore, the correlation between sex and height is, of course, not perfect. That doesn't change the fact that there is a reliable and detectable correlation.

The state of affairs just described — with a detectable pattern + exceptions — is extremely common in psychology. That is because very few of the variables we care about (whether it is height or school performance) are shaped by just one factor. But correlations, by their nature, look at just one factor at a time, and, with many other factors contributing to the observable data, we're virtually guaranteed only a moderate role for any single variable we examine.



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