Section 9.1 Introduction to Correlation
Correlation measures the extent to which two or more variables are related to each other and is typically expressed as a correlation coefficient. The correlation test is a basic but commonly used method for examining the relationship between variables by assessing how two variables change together. However, there’s a common misconception among students who interpret the warning about correlation: “correlation does not equal causation.” Some students mistakenly believe that two correlated variables can never be causally linked. This is an erroneous conclusion. A more accurate understanding of the warning is that correlation does not necessarily imply causation. In other words, while correlation between variables is necessary for causation, it is not always sufficient (Vogt & Johnson, 2011).
Subsection 9.1.1 Pearson Product-Moment Correlation Coefficient
Correlation measures the strength and direction of the linear relationship between two variables. There are various types of correlations. However, this chapter will focus on a Pearson product-moment correlation coefficient (aka Pearson correlation coefficient, Pearson’s \(r\text{,}\) or Pearson’s correlation) designed to evaluate the relationship between continuous variables (or one dichotomous variable and one continuous variable). It is calculated by dividing the covariance (a measure of how two variables covary together) by the product of their standard deviations.
The Pearson correlation coefficient ranges from \(-1\) to \(+1\text{,}\) indicating the strength and direction of the linear relationship between two variables. A positive correlation (value closer to \(+1\)) suggests that as one variable increases, the other also tends to increase. On the other hand, a negative correlation (value closer to \(-1\)) indicates that as one variable increases, the other tends to decrease, lowering the value of the other variable. Conversely, a correlation of 0 indicates no discernible relationship between the two variables. The absolute value of the correlation coefficient indicates the strength of the relationship between the variables, with larger absolute values representing stronger relationships.
What standards should we use to determine whether a relationship is strong or weak? Many people use Cohen’s (1988) guideline to interpret the magnitude of Pearson correlation coefficients. A small correlation, falling within the r = 0.1 to 0.29 range, suggests a relatively weak relationship between the variables under consideration. When the correlation coefficient falls between r = 0.3 and 0.49, it is classified as a medium correlation, indicating a moderate association between the variables. Conversely, a large correlation, defined as r ≥ 0.5, signifies a robust relationship between the variables.
Finally, a perfect correlation of 1 or -1 indicates that the value of one variable can be precisely determined by knowing the value of the other variable. Conversely, a correlation of 0 indicates no discernible relationship between the two variables.
