{"id":397,"date":"2021-12-15T23:08:31","date_gmt":"2021-12-15T23:08:31","guid":{"rendered":"https:\/\/pressbooks.palomar.edu\/introtostats\/chapter\/chapter-12\/"},"modified":"2025-11-05T23:18:10","modified_gmt":"2025-11-05T23:18:10","slug":"chapter-12","status":"publish","type":"chapter","link":"https:\/\/pressbooks.palomar.edu\/introtostats\/chapter\/chapter-12\/","title":{"raw":"Chapter 12: Correlations","rendered":"Chapter 12: Correlations"},"content":{"raw":"
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Key Terms<\/h3>\r\n<\/header>\r\n
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confound variables<\/span><\/a><\/p>\r\n

correlation matrices<\/span><\/a><\/p>\r\n

covariance<\/span><\/a><\/p>\r\n

curvilinear relationship<\/span><\/a><\/p>\r\n

inverse relationship<\/span><\/a><\/p>\r\n

linear relationship<\/span><\/a><\/p>\r\n

magnitude<\/span><\/a><\/p>\r\n

negative relationship<\/span><\/a><\/p>\r\n

no relationship<\/span><\/a><\/p>\r\n

outlier<\/span><\/a><\/p>\r\n

positive relationship<\/span><\/a><\/p>\r\n

range restriction<\/span><\/a><\/p>\r\n

Spearman\u2019s rho<\/span><\/a><\/p>\r\n

sum of products<\/span><\/a><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Introduction: Seeing Connections in Social Inequality<\/b><\/p>\r\n

Correlation analysis allows us to see how social realities move together. From a social justice perspective, correlation helps reveal the relationships between privilege, power, and marginalization\u2014showing, for example, how income relates to educational opportunity, how policing intensity relates to neighborhood racial composition, or how access to health care tracks with poverty rates. Understanding correlation reminds us that social problems rarely exist in isolation; they are interconnected and systemic. By quantifying the strength and direction of these connections, correlation analysis helps us move beyond assumptions and anecdotes toward evidence-based arguments for equity and reform.<\/p>\r\n

Thus far, all of our analyses have focused on comparing the value of a continuous variable across different groups via mean differences. We will now turn away from means and look instead at how to assess the relationship between two continuous variables in the form of correlations. As we will see, the logic behind correlations is the same as it was behind group means, but we will now have the ability to assess an entirely new data structure.<\/p>\r\n\r\n

Variability and Covariance<\/h3>\r\n

A common theme throughout statistics is the notion that individuals will differ on different characteristics and traits, which we call variance<\/span>. In inferential statistics and hypothesis testing, our goal is to find systematic reasons for differences and rule out random chance as the cause. By doing this, we are using information on a different variable\u2014which so far has been group membership like in ANOVA\u2014to explain this variance. In correlations, we will instead use two continuous variables to account for the variance.<\/p>\r\n

Because we have two continuous variables, we will have two characteristics or scores on which people will vary. What we want to know is whether people vary on the scores together. That is, as one score changes, does the other score also change in a predictable or consistent way? This notion of variables differing together is called [pb_glossary id=\"705\"]<\/a>[\/pb_glossary]covariance<\/span> (the prefix co-<\/span> meaning \u201ctogether\u201d).<\/p>\r\n

Let\u2019s look at our formula for variance on a single variable:<\/p>\r\n

\"\"<\/p>\r\n

We use X<\/span> to represent a person\u2019s score on the variable at hand, and \"Upper to represent the mean of that variable. The numerator of this formula is the sum of squares, which we have seen several times for various uses. Recall that squaring a value is just multiplying that value by itself.<\/p>\r\n\r\n