Understanding the Correlation Coefficient

In statistical data analysis we sometimes use a correlation coefficient to quantify the linear relationship between two variables.

The most commonly used correlation statistic is the Pearson correlation coefficient. This statistic measures both the strength and direction of the linear relationship between two variables.

Correlation Example

Suppose we want to look at the relationship between age and height in children. We select a group of children for study, and for each child we record their age in years and their height in inches. We could plot these values on a graph so that the child's age would be on the horizontal axis and the child's height would be on the vertical axis. Each dot on the plot represents a single child's age and height. This is called a scatter plot.

Since older children are generally taller than younger children, we would expect the dots on the plot to roughly approximate a straight line (a linear relationship between the variables) and that the line will slope upward (since age and height tend to increase at the same time).

Correlation Coefficient

The Pearson correlation coefficient is a number between -1 and +1 that measures both the strength and direction of the linear relationship between two variables.

The magnitude of the number represents the strength of the correlation. A correlation coefficient of zero represents no linear relationship (the scatter plot does not resemble a straight line at all), while a correlation coefficient of -1 or +1 means that the relationship is perfectly linear (all of the dots fall exactly on a straight line).

The sign (+/-) of the correlation coefficient indicates the direction of the correlation. A positive (+) correlation coefficient means that as values on one variable increase, values on the other variable tend to also increase; a negative (-) correlation coefficient means that as values on one variable increase, values on the other tend to decrease, that is, they tend to go in opposite directions.