Inside the Subspace Where Spurious Correlations Are Born

Spurious correlations can arise by chance, particularly in studies with small sample sizes, even when the true population correlation is zero. The Pearson correlation coefficient can be understood geometrically through a two-step process of centering and normalization. Centering the data vectors constrains them to an (n-1)-dimensional hyperplane, while normalization further restricts them to a unit sphere within that space. For independent Gaussian variables, this process results in a specific sampling distribution for the correlation that is highly dependent on the sample size, n. For instance, with only three subjects the distribution is U-shaped, making extreme correlations more likely than those near zero, while with four subjects the distribution is uniform.