Nonnormality and Product Moment Correlation
Raymond C. Norris, Howard F. Hjelm · The Journal of Experimental Education · 1961
ALTHOUGH THE Pearson product moment correlation coefficient has been used as an index of r e 1 at ionship between two variables for more than 50 years, there continues to be considerable controversy concerning the circumstances under which its use is appropriate. As recently as 1957 and 1958 American Psychologist carried comments by Nefzger and Drasgow (8), Furfey (5), LaForge (6), and Milholland (7) on the ques tion of whether or not it is necessary to assume normality in the distributions on which the corre lations are based. Binder, in a more recent ar ticle in American Psychologist (2), attempt ed to show that much of the controversy has been misdirected because of a lack of understanding of the place of assumptions in the interpretation of the correlation coefficient. By developing three different mathematical models, Binder shows that different inferential inter pre t a t i o n s of the correlation coefficient grow out of the different mathematical models and concludes, The pref erence for the bivariate normal model to a more general model stems from its great deductive power and its usefulness in many empirical situ ations. . . . extent to which failure to satisfy the as sumption of bivariate normality distorts the in terpretation of the product moment coeffici ent was the subject of numerous studies in the period 1929 through 1932. Egon Pearson (9,10,11) con ducted a series of studies in which the score dis tributions deviated from bivariate normal in var ious ways and concluded that the sampling distri bution of the product moment coefficient was not seriously effected by failure to meet this theoret ical assumption. His findings were generally confirmed by other empirical investigations by Dunlap (4), Chesire, Oldis, and Pearson (3) and Rider (12). Baker (1), dealing with a bivariate distribution with markedly skewed marginal fre quencies, concluded that for samples of size forty the obtained sampling distribution was also skewed and that usual tests of significance would be mis leading. Marginal distributions in these studies deviat ed from the normal in both skewness and kurtosis. Population correlations ranged from zero to . 83, sample sizes from 5 to 52, and sampling distri butions from 50 coefficients to 1770. In most of the work sampling distributions were based on 500 or fewer correlations, and Baker's study (the only one which raised serious question about the dis tortion of the obtained sampling distribution) was based on only 50 samples of size 40. current study was undertaken in an attempt to pin down any empirical effects of nonnormality by taking larger numbers of samples than had been used previously and determining the obtained sampling distribution in each case. Sample sizes used were in the range of ones frequently en countered in educational and psychological re search. Several types of nonnormality frequently encountered in such research were used, and the population correlations of zero and . 83 represent ed the range of values usually encountered.