### Correlation

Scores can be reported in different ways. One example is the **standard score** or **z score**. Standard scores enable psychologists to compare scores that are initially on different scales. For example, a z score of 1 for an IQ test might equal 115, while a z score of 1 for the SAT I might equal 600.The mean score of a distribution has a standard score of zero. A score that is one standard deviation above the mean has a z score of 1. A standard score is computed by subtracting the mean raw score of the distribution from the raw score of interest, then dividing the difference by the standard deviation of the distribution of raw scores. Another type of score, the **percentile score**, indicates the percentage of scores at or below a particular score. Thus, if you score at the 90th percentile, 90% of the scores are the same or below yours. Percentile scores vary from 1 to 99.

A statistical measure of the degree of relatedness or association between two sets of data, *X* and *Y*, is called the **correlation coefficient**. The correlation coefficient (*r*) varies from –1 to +1. One indicates a perfect relationship between the two sets of data. If the correlation coefficient is –1, that perfect relationship is inverse; as one variable increases, the other variable decreases. If the correlation coefficient (*r*) is +1, that perfect relationship is direct; as one variable increases the other variable increases, and as one variable decreases, the other variable decreases. A correlation coefficient (*r*) of 0 indicates no relationship at all between the two variables. As the correlation coefficient approaches –1 or +1, the relationship between variables gets stronger. Correlation coefficients are useful because they enable psychologists to make predictions about *Y* when they know the value of *X* and the correlation coefficient. For example, if *r* = .9 for scores of students in an AP Biology class and for the same students in AP Psychology class, a student who earns an A in biology probably earns an A in psychology, whereas a student who earns a D in biology probably earns a D in psychology. If *r* = .1 for scores of students in an English class and scores of the same students in AP Calculus class, knowing the English grade doesn't help predict the AP Calculus grade.

*Correlation does not imply causation*. Correlation indicates only that there is *a relationship between variables, not how the relationship came about*.

The strength and direction of correlations can be illustrated graphically in **scattergrams** or **scatterplots** in which paired X and Y scores for each subject are plotted as single points on a graph. The slope of a line that best fits the pattern of points suggests the degree and direction of the relationship between the two variables. The slope of the line for a perfect positive correlation is *r* = +1, as in Figure 6.2a. The slope of the line for a perfect negative correlation is *r* = –1, as in Figure 6.2b. Where dots are scattered all over the plot and no appropriate line can be drawn, *r* = 0 as in Figure 6.2c, which indicates no relationship between the two sets of data.

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