The Correlation Coefficient

The Pearson product-moment correlation coefficient for X and Y is most easily thought of as the mean of the summed cross-products of the z-scores of X and Y; that is,

rXY = iV "t ^^ where %i = standarddeviationofX'

although it can be written in many other forms. This coefficient takes on a value of +1 if all of the data points (X, Y) fall exactly on a straight line with a positive slope, so that Y increases as X increases, and -1 if they all fall on a straight line with a negative slope. If there is no linear component to the relationship between Y and X, the correlation coefficient will have a value of close to 0.

The size of the correlation coefficient stays the same if either or both ofX and Y undergo linear transformations in which each value is multiplied by a constant and another constant is then added to or subtracted from the product. Therefore, the correlation coefficient stays the same if units are changed, so that the correlation between height and weight would be the same if height was measured in inches or meters.

The correlation coefficient is often misinterpreted. Among issues to consider are the following: First, the correlation coefficient is an index of linear relationship, not relationship in general; therefore, a correlation of 0 does not rule out the existence of a systematic nonlinear relationship between the variables. Second, if X and Y are correlated, it does not necessarily follow that there is a direct causal relationship between them; the correlation could occur because of the influence of other variables. For example, among elementary school students, vocabulary size is strongly correlated with height because both are related to chronological age. Finally, although the correlation coefficient is an index of linear relationship, unless the variances ofX and Y are equal, r doesn't by itself provide information about the nature of the best-fitting linear function. In particular, the slope, or rate of change, of Y with X is given by b have been if one or both variables had not been artificially dichotomized. Suppose we wish to correlate math and verbal ability. Assume we have normally distributed scores on a verbal ability test but all we know about math ability is whether or not each student passed the test. If we assume that math ability is normally distributed, we can generate the biserial correlation coefficient, which is an estimate of what the correlation coefficient would be if we had continuous scores on both dimensions. The tetrachoric correlation coefficient results if we apply the same logic to two di-chotomous variables.

that is, the correlation coefficient multiplied by the ratio of the standard deviations. So if in two groups the rates of change of Y with changes in X are the same, the correlations may well be different if there is more variability in the values of X and Y in one group than in the other. For this reason, r is often referred to as a sample-specific measure.

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