What is a correlation coefficient?

What is a linear relation?

Describe a perfect correlation, including its possible coefficients.

What is the difference between a positive correlation and a negative correlation?

What magnitude of a correlation coefficient is large enough to be considered important, or worth talking about?

When we have a straight-line relation between two variables, we use a Pearson correlation coefficient. What does this coefficient describe?

Explain how the correlation coefficient can be used as a descriptive or an inferential statistic.

How are deviation scores used in assessing the relation between variables?

Explain how the sum of the product of deviations determines the sign of the correlation.

What are the null and research hypotheses for correlations?

What are the three basic steps to calculate the Pearson correlation coefficient?

Describe the third assumption of hypothesis testing with correlation.

What is the difference between test–retest reliability and coefficient alpha?

Why is a correlation coefficient never greater than 1 (or less than −1)?

Determine whether the data in each of the graphs provided would result in a negative or positive correlation coefficient.

Decide which of the three correlation coefficient values below goes with each of the scatterplots presented in Exercise 13.15.

Use Cohen’s guidelines to describe the strength of the following correlation coefficients:

For each of the pairs of correlation coefficients provided, determine which one indicates a stronger relation between variables:

Using the following data:

Using the following data:

Using the following data:

Calculate the degrees of freedom and the critical values, or cutoffs, assuming a two-tailed test with a p level of 0.05, for each of the following designs:

Calculate the degrees of freedom and the critical values, or cutoffs, assuming a two-tailed test with a p level of 0.05, for each of the following designs:

Which of the following is not a possible coefficient alpha: 1.67, 0.12, −0.88? Explain your answer.

A researcher is deciding among three diagnostic tools. The first has a coefficient alpha of 0.82, the second has one of 0.95, and the third has one of 0.91. Based on this information, which tool would you suggest she use and why?

Awe and correlation in the news: The New York Times reported on a study that examined the link between positive emotions and health. First citing previous research connecting negative moods with poor health, the reporter said: “Far less is known, however, about the health benefits of specific upbeat moods—whether contentment, say, might promote good health more robustly than joy or pride does. A new study singles out one surprising emotion as a potent medicine: awe” (Reynolds, 2015). What is awe? The reporter interviewed one of the researchers, Dacher Keltner, who said that awe is something that “will pass the goose-bumps test.” Keltner explained that “Some people feel awe listening to music, others watching a sunset or attending a political rally or seeing kids play.”

If this study were an experiment, how might the researchers have studied the emotion of awe as a medicine?

Debunking astrology with correlation: The New York Times reported that an officer of the International Society for Astrological Research, Anne Massey, stated that a certain phase of the planet Mercury, the retrograde phase, leads to breakdowns in areas as wide-ranging as communication and travel (Newman, 2006). The Times reporter, Andy Newman, documented the likelihood of breakdown on a number of variables, and discovered that, contrary to Massey’s hypothesis, New Jersey Transit commuter trains were less likely to be late, although by just 0.4%, during the retrograde phase. On the other hand, consistent with Massey’s hypothesis, the rate of baggage complaints at LaGuardia airport increased a tiny amount during retrograde periods. Newman’s findings were contradictory across all examined variables—rates of theft, computer crashes, traffic disruptions, delayed plane arrivals—with some variables backing Massey and others not. Transportation statistics expert Bruce Schaller said, “If all of this is due to randomness, that’s the result you’d expect.” Astrologer Massey counters that the pattern she predicts would only emerge across thousands of years of data.

Obesity, age at death, and correlation: In a newspaper column, Paul Krugman (2006) mentioned obesity (as measured by body mass index) as a possible correlate of age at death.

Exercise, number of friends, and correlation: Does the amount that people exercise correlate with the number of friends they have? The accompanying table contains data collected in some of our statistics classes. The first and third columns show hours exercised per week and the second and fourth columns show the number of close friends reported by each participant.

Externalizing behavior, anxiety, and correlation: As part of their study on the relation between rejection and depression in adolescents (Nolan, Flynn, & Garber, 2003), researchers collected data on externalizing behaviors (e.g., acting out in negative ways, such as causing fights) and anxiety. They wondered whether externalizing behaviors were related to feelings of anxiety. Some of the data are presented in the accompanying table.

Externalizing behavior, anxiety, and hypothesis testing for correlation: Using the data in the previous exercise, perform all six steps of hypothesis testing to explore the relation between externalizing and anxiety.

Direction of a correlation: For each of the following pairs of variables, would you expect a positive correlation or a negative correlation between the two variables? Explain your answer.

Cats, mental health problems, and the direction of a correlation: You may be aware of the stereotype about the “crazy” person who owns a lot of cats. Have you wondered whether the stereotype is true? As a researcher, you decide to assess 100 people on two variables: (1) the number of cats they own, and (2) their level of mental health problems (a higher score indicates more problems).

Cats, mental health problems, and scatterplots: Consider the scenario in the previous exercise again. The two variables under consideration were (1) number of cats owned, and (2) level of mental health problems (with a higher score indicating more problems). Each possible relation between these variables would be represented by a different scatterplot. Using data for approximately 10 participants, draw a scatterplot that depicts a correlation between these variables for each of the following:

Trauma, femininity, and correlation: Graduate student Angela Holiday (2007) conducted a study examining perceptions of combat veterans suffering from mental illness. Participants read a description of either a male or female soldier who had recently returned from combat in Iraq and who was suffering from depression. Participants rated the situation (combat in Iraq) with respect to how traumatic they believed it was; they also rated the combat veterans on a range of variables, including scales that assessed how masculine and how feminine they perceived the person to be. Among other analyses, Holiday examined the relation between the perception of the situation as traumatic and the perception of the veteran as being masculine or feminine. When the person was male, the perception of the situation as traumatic was strongly positively correlated with the perception of the man as feminine but was only weakly positively correlated with the perception of the man as masculine. What would you expect when the person was female? The accompanying table presents some of the data for the perception of the situation as traumatic (on a scale of 1–10, with 10 being the most traumatic) and the perception of the woman as feminine (on a scale of 1–10, with 10 being the most feminine).

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Trauma, femininity, and hypothesis testing for correlation: Using the data and your work in the previous exercise, perform the remaining five steps of hypothesis testing to explore the relation between trauma and femininity. In step 6, be sure to evaluate the size of the correlation using Cohen’s guidelines. [You completed step 5, the calculation of the correlation coefficient, in 13.35(b).]

Trauma, masculinity, and correlation: See the description of Holiday’s experiment in Exercise 13.35. We calculated the correlation coefficient for the relation between the perception of a situation as traumatic and the perception of a woman’s femininity. Now let’s look at data to examine the relation between the perception of a situation as traumatic and the perception of a woman’s masculinity (on a scale of 1–10, with 10 being the most masculine).

Trauma, masculinity, and hypothesis testing for correlation: Using the data and your work in the previous exercise, perform the remaining five steps of hypothesis testing to explore the relation between trauma and masculinity. In step 6, be sure to evaluate the size of the correlation using Cohen’s guidelines. [You completed step 5, the calculation of the correlation coefficient, in 13.37(b).]

Traffic, running late, and bias: A friend tells you that there is a correlation between how late she’s running and the amount of traffic. Whenever she’s going somewhere and she’s behind schedule, there’s a lot of traffic. And when she has plenty of time, the traffic is sparser. She tells you that this happens no matter what time of day she’s traveling or where she’s going. She concludes that she’s cursed with respect to traffic.

IQ-boosting water and illusory correlation: The trashy tabloid Weekly World News published an article—“Water from Mountain Falls Can Make You a Genius”—stating that drinking water from a special waterfall in a secret location in Switzerland “boosts IQ by 14 points—in the blink of an eye!” (exclamation point in the original). Hans and Inger Thurlemann, two hikers lost in the woods, drank some of the water, noticed an improvement in their thinking, and instantly found their way out of the woods. The more water they drank, the smarter they seemed to get. They credited the “miracle water” with enhancing their IQs. They brought some of the water home to their friends, who also claimed to notice an improvement in their thinking. Explain how a reliance on anecdotes may have led the Thurlemanns to perceive an illusory correlation (Chapter 5).

Driving a convertible, correlation, and causality: How safe are convertibles? USA Today (Healey, 2006) examined the pros and cons of convertible automobiles. The Insurance Institute for Highway Safety, the newspaper reported, determined that, depending on the model, 52 to 99 drivers of 1 million registered convertibles died in a car crash. The average rate of deaths for all passenger cars was 87. “Counter to conventional wisdom,” the reporter wrote, “convertibles generally aren’t unsafe.”

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Standardized tests, correlation, and causality: A New York Times editorial (“Public vs. Private Schools,” 2006) cited a finding by the U.S. Department of Education that standardized test scores were significantly higher among students in private schools than among students in public schools.

Arts education, correlation, and causality: The Broadway musical Annie and the Entertainment Industry Foundation teamed up to promote arts education programs for underserved children. In an ad in the New York Times, they said, “Students in arts education programs perform better and stay in school longer.”

Facebook likes and correlation: Be careful what you “like.” Researchers examined the relations between the number of Facebook “likes” a person has posted and the researchers’ ability to correctly identify various characteristics of the person, including gender, age, sexual orientation, ethnicity, religion, political beliefs, personality traits, and intelligence (Kosinski, Stillwell, & Graepel, 2013). The graph at top right shows the relations between number of likes and accuracy of identifying gender, age, and the personality characteristic of openness, respectively.

Athletes’ grades, scatterplots, and correlation: At the university level, the stereotype of the “dumb jock” might be strong and ever present; however, a fair amount of research shows that athletes maintain decent grades and competitive graduation rates when compared to nonathletes. Let’s play with some data to explore the relation between grade point average (GPA) and participation in athletics. Data are presented here for a hypothetical basketball team, including the GPA on a scale of 0.00 to 4.00 for each athlete and the average number of minutes played per game.

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Romantic love, brain activation, and reliability: Aron and colleagues (2005) found a correlation between intense romantic love [as assessed by the Passionate Love Scale (PLS)] and activation in a specific region of the brain [as assessed by functional magnetic resonance imaging (fMRI)]. The PLS (Hatfield & Sprecher, 1986) assessed the intensity of romantic love by asking people in romantic relationships to respond to a series of questions, such as “I want ______ physically, emotionally, and mentally” and “Sometimes I can’t control my thoughts; they are obsessively on ______,” replacing the blanks with the name of their partner.

A biased exam question, validity, and correlation: New York State’s fourth-grade English exam led to an outcry from parents because of a question that was perceived to be an unfair measure of fourth graders’ performance. Students read a story, “Why the Rooster Crows at Dawn,” that described an arrogant rooster who claims to be king, and Brownie, “the kindest of all the cows,” who eventually acts in a mean way toward the rooster. In the beginning the rooster does whatever he wants, but by the end, the cows, led by Brownie, have convinced him that as self-proclaimed king, he must be the first to wake up in the morning and the last to go to sleep. To the cows’ delight, the arrogant rooster complies. Students were then asked to respond to several questions about the story, including one that asked: “What causes Brownie’s behavior to change?” Several parents started a Web site, http://browniethecow.org, to point out problems with the test, particularly with this question. Students, they argued, were confused because it seemed that it was the rooster’s behavior, not the cow’s behavior, that changed. The correct answer, according to a quote on the Web site from an unnamed state official, was that the cow started out kind and ended up mean.

Holiday weight gain, reliability, and validity: The Wall Street Journal reported on a study of holiday weight gain. Researchers assessed weight gain by asking people how much weight they typically gain in the fall and winter (Parker-Pope, 2005). The average answer was 2.3 kilograms. But a study of actual weight gain over this period found that people gained, on average, 0.48 kilogram.

Health care spending, longevity, and correlation: New York Times columnist Paul Krugman (2006) used the idea of correlation in a newspaper column when he asked, “Is being an American bad for your health?” Krugman explained that the United States has higher per capita spending on health care than any country in the world and yet is surpassed by many countries in life expectancy [Krugman cited a study by Banks, Marmot, Oldfield, and Smith (2006), published in the Journal of the American Medical Association].

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Availability of food, amount eaten, and correlation: Did you know that sometimes you eat more just because the food is in front of you? Geier, Rozin, and Doros (2006) studied how portion size affected the amount people consumed. They discovered interesting things such as that people eat more M&M’s when the candies are dispensed using a big spoon as compared with when a small spoon is used. They investigated whether people eat more when more food is available. Hypothetical data are presented below for the amount of candy presented in a bowl for customers to take and the amount of candy taken by the end of each day of the study:

High school athletic participation and correlation: Researchers examined longitudinal data to explore the long-term effects of high school athletic participation (Lutz, Cornish, Gonnerman, Ralston, & Baker, 2009). They reported three findings. First, they found that high school athletic participation was related to a number of positive outcomes. These included increases in high school GPA, college completion, earnings as an adult, and various positive health behaviors. Second, they found that a number of other variables affected the relation between high school athletic participation and these positive outcomes; these other variables included race, gender, and type of school (e.g., public, private). Third, they found that high school athletic participation was related to several negative outcomes among male athletes; these included increases in alcohol consumption, sexist and homophobic attitudes, and violence.