Example - Correlation

This example covers four cases to demonstrate the concept of correlation using simple numbers and a small data set of 10 observations. The objective of this example is to demonstrate

• how perfect correlation, extreme values (+1 and -1), can occur,
• the impact of a single outlier to the value of the correlation coefficient, and the potential for misinterpretation of data,
• how the correlation coefficient measures the linear relationship only.

Let's look at the cases.

Case 1: Perfect Positive Correlation

The first two columns present the data for x and y followed by columns for manual computation using the Pearson Correlation formula. To the right of the manual computation columns are two small tables obtained using the Microsoft Excel program's statistical functions. The first table includes the correlations and the second table is the variance-covariance matrix.

You can easily verify using the variance-covariance matrix or the intermediate computations and the formula that the correlation coefficient attains its maximum value of r=+1 for this data set. In this case both variable x,y values increase proportionally and linearly suggesting a one-to-one strong relationship. Each five-unit increment in the value of x results into a one unit increment in the value of y. Please see the table, graph and animation below.

Case 2: Non-linear Relationship

The presentation of data and variables are as in case 1. You can see from the data, graph, and the animation that the relationship between the variables is strong but not linear. The correlation coefficient does not capture the non-linearity, but only measures the strength of the linear relationship.

Case 3: Outlier

The presentation of data and variables are as in case 1. You can see from the data, graph, and the animation that the relationship between the variables is strong with one significant outlier. The value of the correlation coefficient is significantly lowered by the outlier data point.

Case 4: Perfect Negative Correlation

The presentation of data and variables are as in case 1. You can see from the data, graph, and the animation, that the relationship between the variables is strong. However, when the values of x increase then the values of y decrease proportionally and linearly.

Please try to create this complete example on your own utilizing the capabilities of MS Excel or another software of your choice.