Decision Points for Correlation Coefficient

Summary:  After you compute the linear correlation coefficient r of your sample, you may wonder whether this reflects any linear correlation in the population. By comparing r to a critical number or decision point, you either conclude that there is linear correlation in the population, or reach no conclusion. You can never conclude that there’s no correlation in the population.

See also:  This page gives a simple mechanical test, but a proper statistical test exists. The optional advanced handout Inferences about Linear Correlation explains how decision points are computed and the theory behind the test. You need to learn about t tests before you’ll understand all of it, but right now you can use the Excel spreadsheet that you’ll find there.

Procedure

The decision points are used to answer the question “From the linear correlation r of my sample, can I rule out chance as an explanation for the correlation I see? Can I infer that there is some correlation in the population?”

To answer that question, you temporarily disregard the sign of r. The result is written as | r |, read “the absolute value of r”. Then you compare | r | to the decision point, and obtain one of the only three possible results:

Here’s a table of decision points (also known as critical values of r). Find the sample size n, then read the decision point next to it. (If your sample size is not shown, either refer to the Excel workbook or use the next lower number that is shown in the table. Example: n = 35 is not shown, and therefore you will use the decision point for n = 30.)

Examples

Suppose you have a data set with 50 points, and you compute r = −.35. You look up n = 50 in the table and find .279 as the decision point. Comparing, .35 > .279, and therefore you conclude that the population is negatively correlated: as x increases, y tends to decrease.

Suppose you have a data set with 90 points, and you compute r = .20. n = 90 isn’t in the table; you use n = 80 to find the decision point of .220. Comparing, .20 ≤ .220; therefore you can’t make any conclusion about whether there is correlation in the population.

Interpretation

Be very careful with your interpretation, and don’t say more than the statistics will allow.

The question was simply whether there is some correlation in the population, not how much. The population might have stronger or weaker correlation than your sample; all you know is that it has some. (Though we won’t learn how to do it in this course, it is possible to estimate the correlation coefficient of the population.)

If you conclude there is some correlation in the population, it might not be true. From a completely uncorrelated population, there’s still one chance in 20 of drawing a sample with | r | greater than the decision point. Because 1/20 is .05, we call .05 the significance level.

Even if you conclude that there is some correlation in the population, that’s the start of your investigation, not the end. If there’s a correlation in the population, you can’t just assume that one variable drives the other: correlation is not causation. (Causation by Steve Simon gives some hints for investigating causation, using smoking and lung cancer as an example.)

Finally, note that there’s no way to reach the conclusion “there’s no correlation in the population." Either there (probably) is, or you can’t reach any conclusion. This will be a general pattern in inferential statistics: either you reach a conclusion of significance, or you don’t reach any conclusion at all. (In statistics, we say that you never accept the null hypothesis.) You can conclude “something is going on”, you can fail to reach a conclusion, but you can never conclude “nothing is going on”.