Pearson’s product–moment correlation is an appropriate statistical tool for cases involving a comparison between two variables, X and Y , with metric data. It produces a coefficient, called r , which results from dividing the covariance of two variables by the product of their standard deviations. It measures the degree of a linear relationship between two variables, indicating this as a range of +1 to −1.

Statisticians may thank an English professor named Karl Pearson (1857–1936) for this calculation. In 1911, Pearson founded the Department of Applied Statistics at University College in London, England, the first academic statistics department in the world. Since then, researchers in many fields, especially medicine, have relied on Pearson’s to finetune their data, as the following medical example vividly shows.

Chronic Fatigue Syndrome Studies 

Researchers at the Human Performance Laboratory and Department of Internal Medicine at Vrije Universiteit Brussel in Brussels, Belgium, assembled a test group of 427 women with clinically diagnosed chronic fatigue syndrome (CFS)
and a control group of 204 age-matched women with sedentary lifestyles to participate in an exercise program to test heart rate. The women used a stationary ergometric bicycle and pedaled for 8 to 12 minutes as their oxygen levels, heart rate, and other physical parameters were monitored.

The researchers used the Pearson’s calculation to study variations in the associations between variables within each group (test and control). They also employed Pearson’s to discern any differences in the exercise parameters between the two groups. The purpose was to yield a better understanding of how exercise capacity varied between both groups. As the data show, Pearson’s provided clear-cut correlations.

The calculation showed, for example, the correlation differences in terms of workload at anaerobic threshold (WAT, reduced oxygen intake) for maximum heart rate was 0.37 for CFS women compared to 0.70 for control group sedentary women. It also showed that the maximum respiratory quotient for CFS women was positively associated with WAT (r = 0.26; P <.001 [P = difference]) and negatively associated with HRAT (heart rate at anaerobic threshold: r =0.15 ; P <.01). Among women in the control group, the maximum respiratory quotient was positively correlated with the resting heart rate (r =0.17; P <.02), and negatively correlated with WAT (r = 0.24; P <.002).

Questions

1. Explain the statistical operating principle that accounts for Pearson’s ability to clearly show differences between the two groups as reflected in the r values.

2. Is there an r value, possibly on the high side, that would seem suspect if reported for this study?