Biserial Correlation

Biserial correlation (rbis) is a correlational index that estimates the strength of a relationship between an artificially dichotomous variable (X) and a true continuous variable (Y). Both variables are assumed to be normally distributed in their underlying populations. It is used extensively in item analysis to correlate a dichotomously scored item with the total score. Assuming that an artificially dichotomous item is scored 1 for a correct answer and 0 for an incorrect answer, the biserial correlation is computed from

where Y¯1 is the mean of Y for those who scored 1 on X, Y¯0 is the mean of Y for those who scored 0 on X; SY is the standard deviation of all Y scores; n1 is the number of observations scored 1 on X; n0 is the number of observations scored 0 on X; n. = n1 + n0, or the total number of observations; and u is the ordinate (i.e., vertical height or the probability density) of the standard normal distribution at the cumulative probability of p1 = n1/n. (i.e., the proportion of observations scored 1 on X).

Assuming that the following data set is obtained from 30 examinees who took the Science Aptitude Test, their total test scores and item scores on Item #13 are as shown in the following table.

Treating the total score as the Y variable and the Item #13 score as the X variable, one calculates Y¯0 = 12.93, Y¯1 = 29, SY = 11.92, n0 = 15, n1 = 15, n. = 15 + 15 = 30, and p1 = 15/30 = 0.50. Taking p1 to the standard normal distribution, one determines the ordinate, u, to be 0.3989. Substituting these values into the formula above, one obtains the biserial correlation as follows:

The above value for biserial correlation can be interpreted as a measure of the degree to which the total score (continuous variable) differentiates between correct and incorrect answers on Item #13 (artificially dichotomous variable). The value of 0.85 suggests that the total score correlates strongly with the performance on this item. If the 15 students who failed this item had scored the lowest on the total test, the biserial correlation would have been 1.00 (the maximum). Conversely, if “pass” and “fail” on Item #13 were matched randomly with total scores, the biserial correlation would be 0.00 (the absolute minimum).

Examinee	Item #13 score (X)	Total score (Y)	Examinee	Item #13 score (X)	Total score (Y)	Examinee	Item #13 score (X)	Total score (Y)
1	0	8	11	1	14	21	0	28
2	0	12	12	1	13	22	1	33
3	0	6	13	0	10	23	1	32
4	0	12	14	0	9	24	1	32
5	0	8	15	0	8	25	1	33
6	0	8	16	1	33	26	0	34
7	0	8	17	0	28	27	1	35
8	0	11	18	1	29	28	1	34
9	1	13	19	1	30	29	1	38
10	0	4	20	1	29	30	1	37

The biserial correlation is an estimate of the product-moment correlation, namely, Pearson's r, if the normality assumption holds for X and Y population distributions. Its theoretical range is from −1to +1. In social sciences research, rbis computed from empirical data may be outside the theoretical range if the normality assumption is violated or when n· is smaller than 15. Thus, rbis may be interpreted as an estimate of Pearson's r only if (a) the normality assumption is not [Page 73]violated, (b) the sample size is large (at least 100), and (c) the value of p1 is not markedly different from 0.5. Biserial correlation should not be used in secondary analyses such as regression.

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