What is the value of the product–moment correlation coefficient for the data set shown?
Taking a look at this set of data, we see that it is bivariate. That is, there are two variables involved, 𝑥 and 𝑦. Based on this graph, we want to solve for the value of the product–moment correlation coefficient. This is going to be a single number that represents how closely these variables 𝑥 and 𝑦 relate. By the way, there’s another name for the product–moment correlation coefficient. It’s also called the Pearson correlation coefficient. And again, this is a number that tells us how closely correlated two variables are.
Our graph shows us that all of the points in this data set lie along the same line. This is interesting because it means, for example, that if we were to take a given 𝑥-value and then double it, the 𝑦-value of that double point would be twice as great as the original. In other words, the relationship of any one of these data points to any other data point in the set is perfectly expressed by the slope of this black line.
Now, when we talk about the Pearson correlation coefficient, in general that can be a small as negative one and as great as positive one. Any correlation coefficient that is less than zero indicates a downward-sloping data set, where 𝑥 gets larger, 𝑦 gets smaller. On the other hand, a positive correlation coefficient indicates the opposite trend that 𝑥 and 𝑦 increase together.
In the data set shown, we see that indeed there is such a positive trend. We can say then that these two variables, 𝑥 and 𝑦, are directly or positively correlated. Now, the difference between a Pearson correlation coefficient of, say, 0.3 and 0.9 comes down to how uniformly a set of data follows a positive trend. For example, a correlation coefficient of 0.3 might represent this data set. While there is a positive correlation, the data are not tightly gathered around this line of best fit.
For a correlation coefficient of 0.9, however, we might get a graph that looks like this, where the data are very closely clustered around the best-fit line. Looking back at our graph, we see that here the data in the set follow this line of best fit perfectly. All of the points lie exactly along this line. And in this unusual case, we have an extreme value of the Pearson correlation coefficient. Because this set of data is positively correlated, that extreme value is positive one.
Our answer, then, is that the value of the product–moment correlation coefficient, also known as the Pearson correlation coefficient, for the data set shown is one.