# Shiken: JLT Testing & Evlution SIG Newsletter. 5 (3) October 2001 (pp )

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1 Statistics Corner: Questions and answers about language testing statistics: Point- biserial correlation coefficients James Dean Brown (University of Hawai'i at Manoa) QUESTION: Recently on the forum LTEST-L, there was a discussion about point-biserial correlation coefficients, and I was not familiar with this term. Could you explain what point-biserial correlation coefficients are and how they are important for language testers? ANSWER: To adequately explain the point-biserial correlation coefficient, I will need to address four questions: (a) What is the point-biserial correlation coefficient? (b) How is the point-biserial correlation coefficient related to other correlation coefficients? (c) How is the point-biserial correlation coefficient calculated? And, (d) how is the point-biserial correlation coefficient used in language testing? What Is the Point-Biserial Correlation Coefficient? As I defined it in Brown (1988, p. 150), the point-biserial correlation coefficient (symbolized as rpbi) is a statistic used to estimate the degree of relationship between a naturally occurring dichotomous nominal scale and an interval (or ratio) scale. For example, a researcher might want to investigate the degree of relationship between gender (that is, being male or female - a naturally occurring dichotomous nominal scale) and achievement in English as a second language as measured by scores on the end-of-theyear departmental examination (an interval scale). Aside from the types of scales involved, the interpretation of the resulting coefficient is very similar to that for the more commonly reported Pearson product-moment correlation coefficient (sometimes referred to as Pearson r, or simply r). In brief like the Pearson r, the rpbi can range from 0 to if the two scales are related positively (that is, in the same direction) and from 0 to if the two scales are related negatively (that is, in opposite directions). The higher the value of rpbi (positive or negative), the stronger the relationship between the two variables. [For more detailed explanations of the interpretation and assumptions of Pearson r and rpbi, see Brown, 1996, 1999.] How Is the Point-Biserial Correlation Coefficient Related to Other Correlation Coefficients? In distinguishing the point-biserial from other correlation coefficients, I must first point out that the point-biserial and biserial correlation coefficients are different. The biserial correlation coefficient (or rbi) is appropriate when you are interested in the degree of relationship between two interval (or ratio) scales but for some logical reason one of the two is more sensibly interpreted as an artificially created dichotomous nominal scale. For instance, you might be interested in determining the degree of relationship between passing or failing a first year university ESL course and language aptitude test scores. To do this, grades at the end of the course (A, B, C, D and F, often converted to a 4.00, 3.00, 2.00, 1.00, & 0.00 interval scale) might be artificially separated into a nominal scale made up of two groups: pass (A to D, or 1.00 to 4.00) and fail (F or 0.00). The degree of relationship between this new, artificially created dichotomy and the interval scores on the language aptitude test could then be determined by using the rbi coefficient. Thus the biserial correlation coefficient is appropriately applied when the nominal variable is artificially created (as in the pass-fail variable created from grade points), while the pointbiserial correlation coefficient is appropriately applied when the nominal variable occurs naturally (as in the naturally occurring male-female gender distinction). 13

2 A variety of different correlation coefficients have been developed over the years for various combinations of scale types, as summarized in Table 1. The point-biserial is just one of these statistical tools (see the fifth row of correlation coefficients). How Is the Point-Biserial Correlation Coefficient Calculated? The data in Table 2 are set up with some obvious examples to illustrate the calculation of rpbi between items on a test and total test scores. Notice that the items have been coded 1 for correct and 0 for incorrect (a natural dichotomy) and that the total scores in the last column are based on a total of 50 items (most of which are not shown). Table 2. Example Student Data 14

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