The correlation coefficient formulas are used to find the strength of the association between two linear variables. With linear regression we are interested in direction — one variable is predicted and the other variable is the predictor; in correlation the interest is non-directional and the relationship is what is critical. The formulas return a value between -1 and 1, where results close to 1 indicate a strong positive correlation and results close to -1 indicate a strong negative correlation. A result of zero indicates no correlation at all.
Meaning of the Correlation Coefficient
A correlation coefficient of 1 means that for every positive increase of 1 in one variable, there is a positive increase of 1 in the other variable. A correlation coefficient of -1 means that for every positive increase of 1 in one variable, there is a negative decrease of 1 in the other variable. Zero means that for every increase, there is neither a positive or negative increase in the other variable — the two just aren’t related.
The absolute value of the correlation coefficient determines the strength of the relationship. For example, |-.75| = .75, which has a stronger relationship than a correlation coefficient of .65.
Types of correlation coefficient formulas
There are several types of correlation coefficient formula. While you can use a formula to calculate a correlation coefficient by hand, the calculations are quite involved and time-consuming; it’s recommended that you use a calculator such as the TI-89 to make the calculations for you.
One of the most commonly used formulas is Pearson’s correlation coefficient formula:
Two other formulas are commonly used: the sample correlation coefficient and the population correlation coefficient.
Sample correlation coefficient
Sx and sy are the sample standard deviations, and sxy is the sample covariance.
Population correlation coefficient
Similarly, the population correlation coefficient uses σx and σy are the population standard deviations, and σxy is the population covariance.