There are multiple statistical tests that measure relationships by calculating correlation coefficients. In subsequent chapters, we’ll cover the Spearman rank-order correlation coefficient and the chi-square test of independence. But, the focus of this chapter is the head of the relationship test household, the most commonly used relationship test, the Pearson correlation coefficient. Usually, it is referred to as the Pearson r, or simply r.

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Here is what we’ll cover in the first part of the chapter as we introduce r:

Measures of association are statistics that quantify the degree of relationship between two variables. If two variables are related, they are said to be correlated. This means that the variables vary together systematically, that a change in one variable is associated with a change in the other. For example, alcohol consumption and blood alcohol level vary together systematically—the more alcohol a person consumes, the higher the level of alcohol in his or her blood. Depending on sex and weight as shown in Table 13.1, the relationship between the number of drinks consumed and blood alcohol level is well established.

The Pearson r is a specific measure of association. A Pearson correlation coefficient quantifies the degree of linear relationship between two interval and/or ratio variables. Because the Pearson r uses interval and/or ratio variables, distances between points can be measured, z scores calculated, and graphs made. The graphs, called scatterplots, can be examined for the degree to which the relationship between the two variables takes the form of a straight line because the Pearson r measures the degree of linear relationship. (We’ll discuss nonlinear relationships, and why Pearson r is not appropriate to use in those cases, later in this chapter.)

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If two variables correlate, the relationship may be a cause-and-effect relationship, but it doesn’t have to be. Statisticians love to say, “Correlation is not causation.” That’s because correlation between two variables does not guarantee a cause-and-effect relationship between them. A correlation between two variables only guarantees there is an association between the two variables, not that one causes the other.

To understand this, look at Figure 13.1. This type of graph, called a scatterplot, displays the relationship between two variables. For a random sample of 24 states, this graph shows the association between the number of community hospitals in a state (on the X-axis) and the number of deaths per year in the state (on the Y-axis). Each point represents a state. For example, Delaware, the state at the bottom left of the graph, has 6 community hospitals and about 7,000 deaths per year; New York, the state at the top right, has 206 community hospitals and about 153,000 deaths.

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The scatterplot shows clear evidence of an association between the two variables. The two variables vary together systematically: states like Delaware, with few hospitals have few deaths, and states like New York, with a lot of hospitals, experience a lot of deaths.

The relationship can be read in either direction. So far, this relationship has been viewed as states with more hospitals have more deaths. But, the scatterplot can be viewed just as legitimately as showing that states with more deaths have more hospitals.

If two variables are correlated, it simply means that they vary together systematically. A cause-and-effect relationship may exist between them, but there does not have to be. The number of deaths in a state and the number of community hospitals are correlated, but it doesn’t seem plausible that there is a cause-and-effect relationship between them. If such a relationship did exist, then a state could reduce the number of people who die each year by closing down hospitals. That doesn’t seem to be a course of action likely to work.

The number of hospitals per state and the number of deaths vary together systematically, but they don’t cause each other. Their correlation most likely results because each is associated with a third variable, population. States with more people, like New York, have more of everything than states with fewer people, like Delaware. New York, with a population of almost 20 million, has more pencils, cars, barbers, and murders than Delaware, with under a million residents. New York also has more community hospitals and more deaths, simply because more people live there.

Correlation just means two variables systematically vary together. If the goal of science is to understand cause and effect, it may seem that drawing a conclusion that two variables are associated represents a second-place finish. Not so. Correlation does not have to mean causation, but it may mean causation. If there is a cause-and-effect relationship between two variables, then they are correlated. An association exists between how much alcohol a person consumes and what his or her blood alcohol level is because consuming alcohol causes the alcohol concentration in the blood to rise.

Correlation may not prove cause and effect, but it can suggest cause. For example, finding that children who watch more violent TV tend to exhibit more aggressive behaviors doesn’t prove that TV is the culprit, but it certainly raises such a question and leads to more research. Thus, correlations are not proof of cause and effect, but they often serve as a jumping off point for using experimental techniques to explore cause and effect.

Studies in which the variables in a relationship test are not manipulated by the researcher do not address cause and effect. In a relationship test, it is rare that the explanatory variable is an independent variable and the outcome variable is called a dependent variable. Most commonly, they are just called X and Y and that’s what we’ll do here. However, if the researcher believes there is an order to the relationship, then the explanatory variable is called the predictor variable and the outcome variable is called the outcome variable. The language is meant to be straightforward—the variable that comes first, that is thought of as leading to or influencing the other variable, is the predictor variable. The one that comes second, and is influenced by the first variable, is the outcome variable.

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The degree to which two variables vary together determines whether the relationship is strong or weak. The strength of the relationship can be visualized in a scatterplot. Figure 13.2 shows the relationship between randomly generated numbers. If X values are randomly selected and each X is paired with a randomly selected Y, then the two do not vary together systematically. Figure 13.2 shows a rectangular scatterplot where there is no relationship, what is called a zero correlation, between two variables. Cases with low values on X could have low, medium, or high values on Y. And, the same is true for cases with medium or high values on X.

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Though the rectangular scatterplot shows a zero correlation clearly, that is not how statisticians illustrate a zero correlation. Statisticians use a circular scatterplot to demonstrate no relationship between two variables. If both variables are normally distributed, which is one of the assumptions for the Pearson r, then the scatterplot has a circular shape when no relationship exists. Figure 13.3 uses the traditional circular shape to illustrate the lack of relationship between two normally distributed variables.

What does a scatterplot look like when there is a relationship between two variables and they vary systematically? Figure 13.4 illustrates the relationship between the temperatures of objects measured in both degrees Fahrenheit and Celsius. This figure is an example of the strongest possible linear relationship between two variables, a perfect relationship. In a perfect relationship, all the data points fall along a straight line. Figure 13.4 shows that knowing an object’s temperature in Fahrenheit tells one exactly what it is in Celsius.

A strong relationship is one in which cases’ scores on variable X are closely allied with their scores on variable Y. When looking at a scatterplot, strength is shown by how well the points fall along a straight line. The more the points fall in a straight line, the stronger the association. Figure 13.5 gives four scatterplots. As the points form less of a line and more of a blob, as they move from a line to an oval or a circle, the relationship grows weaker.

Two situations exist in which the points in a scatterplot form a straight line, but the relationship is not a perfect one. These occur when the line is horizontal or vertical. When the line is horizontal, for example, every value of X is paired with the same value of Y. There is no variability in Y. With a vertical line, the opposite is true and there is no variability in X. The Pearson r formula needs both variables to have variability, and if either variable has none, Pearson r can’t be calculated. In these situations, r is said to be undefined.

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A scatterplot can also be drawn using z scores, not raw scores. z scores, also called standard scores, were introduced in Chapter 4. They are a transformation of raw scores into scores that reveal how far away from the mean the scores are in standard deviation units. z scores can be positive or negative, which means they fall above the mean or below the mean.

Figure 13.6 offers scatterplots for the community hospital/number of deaths data. The panel on the left in 13.6, like Figure 13.1, illustrates the relationship with raw scores, and the panel on the right shows what the scatterplot looks like when the same variables are transformed into z scores. Note that transforming from raw scores to z scores doesn’t change the relationship. The pattern of the dots, the shape, is exactly the same, whether raw scores or z scores are being plotted.

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Figure 13.7 shows the scatterplot for the relationship between temperature as measured in Fahrenheit and Celsius both with raw scores (the panel on the left) and with z scores (the panel on the right). These z scores give a new perspective on what a correlation means. Look at the dot on the upper right, which is for boiling water. In the panel on the left, the X value for this point is 212° (Fahrenheit) and the Y value is 100° (Celsius). To a person who doesn’t know much about Fahrenheit and Celsius, those scores don’t sound similar as they are 112 points apart. In the panel on the right, the z scores for this point are z_X = 1.63 and z_Y = 1.63. That is, the boiling point of water is exactly as far above the mean when measured in Fahrenheit as when it is measured in Celsius. That’s what a perfect correlation means.

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As correlations get weaker, the similarity between z_X and z_Y lessens. The community hospital/number of deaths data show a strong correlation, but not a perfect one. The point on the upper right is New York with 206 hospitals and 153,000 deaths. Converted, those are z scores of 2.2 and 2.8. The two z scores are not the same, but they are in the same ballpark, both far above the mean. As correlations grow weaker and weaker, the similarity between the two z scores for each data point becomes less and less.

In fact, the similarity–dissimilarity of the pairs of z scores is one way of calculating what a Pearson correlation coefficient is. This formula, called the definitional formula, calculates a Pearson r as the average of multiplied-together pairs of z scores. Each case’s raw score on X is transformed into a z score and its raw score on Y is transformed into a z score. The two z scores for each case are multiplied together and the mean of these products is calculated. The mean of all such multiplied-together scores is a Pearson correlation coefficient. Thus, the stronger the correlation, the more similar—on average—the z scores are.

Here is a quick example to illustrate the benefit of the z score method. Many years ago, the United States Postal Service ran an ad showing that they, compared to the United Parcel Service and FedEx, had the largest fleet of trucks. The totals were, respectively, 200,000, 130,000, and 35,000. The USPS also had the lowest price for overnight delivery among the three shippers—respectively, $3, $6, and $12. As the scatterplot in Figure 13.8 makes clear, this is a strong relationship: more trucks is associated with lower cost.

Table 13.2 shows the data, both as raw scores and transformed into z scores. Note how much clearer the relationship between the two variables is when expressed in z scores. The number of trucks the United States Postal Service has is as far above the mean, z = 0.95, as its price is below the mean, z = –0.87. This, the strength of the relationship between variable X expressed as a z score and variable Y expressed as a z score, is what the Pearson r expresses.

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Scatterplots are great for visualizing relationships, but their interpretation is subjective. For example, can the owner of a baseball team buy his or her way to the World Series? Figure 13.9 is a scatterplot showing the relationship between the payrolls of Major League Baseball teams and their winning percentages. How strong is the association? Just by looking at the graph, it is difficult to tell how strong the relationship between these two variables is. Deciding how strong a relationship is on the basis of a scatterplot is subjective and different people can have different—and valid—opinions. One person might interpret the points on Figure 13.9 as roughly linear and conclude that the payroll is strongly related to a team’s performance. Another may see the circular clump of scores in the middle of the graph and conclude that there doesn’t appear to be much of a relationship.

The Pearson r gets around this problem of how specific the prediction is by calculating a statistic called a correlation coefficient. A correlation coefficient is a number that summarizes the strength of the linear relationship between two variables. For the Pearson r, the correlation coefficient is abbreviated as r (think of r as being short for “relationship”). For a Pearson correlation, r is a value that ranges from –1.00 to +1.00. An r value of zero is less than a weak relationship; it means that there is no relationship between the two variables. Figure 13.2, where the dots form a rectangle, and Figure 13.3, where the dots form a circle, have Pearson r values of zero.

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As the r value moves further away from zero, in either a negative or positive direction, it represents a stronger relationship between X and Y. For example, an r of –.60 represents a stronger relationship than an r of .30. Pearson r values of –1.00 and 1.00, though they differ in sign, both represent perfect relationships.

Though Pearson r values of –1.00 and 1.00 both represent perfect relationships, the two indicate different types of relationships because their signs differ. The sign of a Pearson r, either positive or negative, gives information about the direction of the relationship. There are two options for direction: positive or negative.

There are three conditions that affect a Pearson r: (1) nonlinearity, (2) outliers, and (3) restriction of range.

Nonlinearity

The Pearson r is used to measure the degree of linear relationship between two variables. A linear relationship exists when the points in the scatterplot for the relationship between X and Y fall along a straight line. Figure 13.12 illustrates a nonlinear association, what is called the Yerkes-Dodson law. The Yerkes-Dodson law states that arousal affects performance in an orderly way—low arousal leads to poor performance, moderate arousal leads to optimal performance, and high arousal impairs performance. This scatterplot shows a relationship between X and Y, but the points in the scatterplot don’t fall on a straight line. Using a Pearson correlation to quantify this relationship would provide a value close to zero, meaning there is little linear relationship between the two variables. The Pearson r only measures how much linear relationship exists between two variables. If the relationship appears nonlinear, do not use a Pearson r to measure it.

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Outliers

Outliers are cases with extreme values, values that fall far away from the values that other cases have. One outlier in a data set can dramatically change the value of a correlation. The left panel in Figure 13.13 shows a scatterplot for a small data set. The points form a rough circle and the correlation between X and Y is zero.

The right panel in Figure 13.13 adds one data point, an outlier with extreme values on both X and Y. This data point has an X value and a Y value that are dramatically higher than any other case. As a result of adding this one case, the correlation between X and Y changes from r = .00 to r = .63.

Outliers inflate the strength of a relationship between two variables. Because outliers may exist, it is always a good idea to create and inspect a scatterplot before calculating a correlation coefficient.

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Restriction of Range

While outliers inflate the value of a correlation, restriction of range tends to deflate the value of a correlation. This means that a restricted range could lead a researcher to conclude that there is less of a relationship between two variables than actually exists.

What is restriction of range? Let’s consider an unrestricted range first. Look at the left panel of Figure 13.14, a scatterplot showing the hypothetical relationship between IQ and GPA in a sample of high school students. There is a full (unrestricted) range of IQ scores, ranging from 70 to 130. And, there is a full (unrestricted) range of GPA, from 0 to 4. Judging by the noncircular shape of the scatterplot, the correlation between these two variables is strong.

Suppose a researcher decided to restrict her examination of the relationship between IQ and GPA to students likely to be accepted at Ivy League colleges. Thus, she restricted her sample to students with IQs above 115 and GPAs above 3.00. The two lines in the top panel are the cut-off values for her sample and the few cases that fall in the upper right quadrant are the new sample.

This subsample has a restricted range on both the X variable and the Y variable. The new sample is shown in a scatterplot all by itself, in the right panel of Figure 13.14. What is the shape of this scatterplot? The points fall roughly in a circle, meaning that little correlation will be found between X and Y. Restricting the range of one or both variables in a correlation deflates its value.

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