In a room with a hundred people, you can measure each person's height and compute the average. But if someone infinitely tall walks in, the average height ceases to exist: no matter how many more people you add, the sum grows without bound. Something similar has happened with wind speed measurements in the McMurdo Dry Valleys, where weather stations record gusts that defy ordinary statistics.
Researchers in the study by Spears et al. (2012) sought a link between the frequency of foehn days and the ENSO and SAM climate indices. They computed mean wind speeds and calculated correlations, obtaining significant results. However, checking the data themselves revealed that the speeds do not follow a normal distribution but a power law with an exponent less than two. In that case, the expected value of wind speed, that is, its true mean, diverges: it is infinitely large.
When the mean does not exist, any calculation of the arithmetic mean from a finite sample does not converge to a specific number: from sample to sample, the result will vary randomly. Hence, Pearson correlations, which rely on means, lose statistical meaning. The claim of significance at the 95% level is erroneous, because for such data there is no rigorous definition of correlation significance.
The study shows that before any correlation analysis of geophysical series, one must check the type of distribution. If the data follow a power law with an exponent less than 2, standard methods based on means and variances are inapplicable. This sets this paper apart from neighboring studies that use machine learning for prediction but do not question the existence of distribution moments. Here, for the first time, the divergence of the mean is demonstrated for winds in the Dry Valleys, casting doubt on all previous conclusions about links to ENSO and SAM.
Let us return to the room with the infinitely tall person: any attempt to compute the average height is doomed. Likewise, in the McMurdo Dry Valleys, the mean wind speed is a quantity that cannot be defined. Consequently, correlations built on it cannot reveal the true relationship between local winds and global climate.
If a distribution has an exponent less than 2, the mean is infinite, but the median remains finite. So in such cases, it is better to describe the typical wind speed with the median rather than the arithmetic mean.