18.2.5—Spearman's and Pearson's correlation

Pearson’s linear correlation tests whether two quantitative variables show a linear relationship. The coefficient r ranges from -1 to +1: a value near +1 indicates a strong positive relationship, a value near -1 a strong negative relationship, and a value near 0 little or no linear correlation.

1. Pair each reading of variable x with the reading of variable y from the same sample or quadrat.
2. Plot the paired values on a scatter graph and check that a linear pattern is plausible.
3. State the null hypothesis that there is no linear correlation between the variables.
4. Check the data assumptions: both variables are quantitative, the relationship is approximately linear, and the data show a normal distribution.
5. Calculate the means, products and standard deviations required by the supplied Pearson equation, then substitute the values to obtain r.
6. Compare the result with the appropriate significance criterion or critical value for the sample, then state the strength and direction of the correlation and whether the null hypothesis is rejected.

Interpret the statistic with the scatter graph and the sample size, not from a single point. A statistically significant correlation supports an association in the sampled data under the test assumptions; it does not show that one variable causes the other.

Correlation is not causation: a third factor, sampling pattern or coincidence may explain an association. Pearson’s method is for the stated quantitative, approximately linear and normally distributed case; Spearman’s rank correlation and Simpson’s index are separate cards.