Illuminating Statistical Analysis Using Scenarios and Simulations

38What Correlations Happen Just by Chance?

Statistical Scenario—Sample Correlations Distribution

What will the distribution of sample correlations look like?

How do we determine -values?

How likely is it that a given sample correlation coefficient value (0.81 in this case) could arise by chance? Let's do a simulation involving pure chance. We'll do 10,000 correlations using nothing but purely random numbers. We'll use the same sample size as the statistical scenario and correlate 120 pairs of random numbers 10,000 separate times. Figure 38.1 is the histogram of simulation results. Notice that it is centered on zero and has a normal bell shape. This shows us what to expect when sampling from a population with a population correlation equal to zero, which is the null hypothesis.¹

Figure 38.1

You can see that sample correlations less than −0.2 or greater than +0.2 are unlikely to happen by chance with a sample size of 120 number pairs. Our statistical scenario sample correlation of 0.81 is almost impossible by chance. Its -value will be extraordinarily small.

What if we had a smaller sample size, say 30 number pairs? With smaller sample sizes, more extreme correlations should ...