Pearson Correlation Coefficient

What Is the Pearson Correlation?

Put simply, the Pearson correlation is a measure of the linear relationship between two variables, X and Y, giving a value between +1.0 and −1.0, where 1.0 is a perfect positive correlation, 0.0 (zero) is no correlation, and −1.0 is a perfect negative correlation. Examples of the possible data distributions associated with five Pearson correlations are illustrated in Figure 1.

Figure 1 Example of five Pearson product-moment correlation coefficients

Importantly, where correlational estimates are concerned, there is no attempt to establish one of the variables as independent and the other as dependent. Therefore, relationships identified using correlation coefficients should be interpreted for what they are: associations, not causal relationships. To arrive at a Pearson correlation value (r) between two variables of interest, a number of calculations and logical steps are made. To illustrate these steps, a fictional example of two educational variables of interest is now provided.

Calculation of the Pearson Correlation Coefficient

Suppose you are the head of curriculum at a small English as a Second or Other Language institute in Auckland, New Zealand. A new cohort of intermediate-level English as a Second or Other Language students arrives every 10 weeks to participate in your program. The cohort flies to Auckland from various spots in the Asia-Pacific region: nearby in Polynesia, farther away in Micronesia, [Page 1230]even farther in Southeast Asia, and at points beyond in East Asia. Being one of the teachers on the course, you notice a trend whereby, despite exhibiting equivalent levels of English fluency, the students originating from farther abroad tend to have more limited knowledge of New Zealand, its culture, and its customs, and often struggle with course material integrating such content. For the purpose of trying to better understand and tailor to the needs of particular student groups, you would like to explore the statistical relationship between (a) the distance that students travel to get to New Zealand and (b) their general knowledge of New Zealand.

To illustrate the steps taken to calculate a Pearson correlation, a fictional educational data set that includes a sample of one intake, namely 10 (N = 10) international students (Table 1, ID column) will be used. The time that it takes each student to fly directly to Auckland, New Zealand, the location of the course, can be used as a proxy measure of each student’s distance of travel to New Zealand. The flight times are presented in Table 1, flight time column (X). On the first day of the course, the students sit a 10-item general knowledge test about New Zealand. The results of the test, out of 10, are also presented in Table 1, test score column (Y).

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