How much is enough? Applying the law of large numbers to the measurement of interactions between ecosystem services

Ecosystem services (ES) are at the forefront of the scientific literature, finding themselves in the research profiles of the National Science Foundation and European Research Council, as well as many other national research agencies. Yet despite many publications on the topic, issues of data availability, quality and quantity, and uncertainty still remain limitations to the field. In a recent analysis, Shanafelt et al. (2023) found a general trend in the interactions between ES when sampling a landscape: sampling ten percent of the landscape was sufficient to recover the mean correlation between ES measured at the landscape scale. In this paper, we delve deeper into this finding. Specifically, we apply Chebyshev's inequality and the law of large numbers to show that as the sample size increases, the sample correlation between any two ES approaches the "true" value measured from the underlying statistical distributions of those services across the landscape. Furthermore, there exists a sample size in which the difference between the sample correlation and the true value is tolerably null-the "ten's rule" from Shanafelt et al. (2023). We hypothesize that this sample size depends on the underlying correlation strength between those ES and the similarity between their spatial distributions, and test this hypothesis using regression analysis in theoretically-generated landscapes. Finally, we test our ability to predict this sample size in the actual Shanafelt et al. (2023) data. Our findings have applications for sample and experimental design, as well as for devising and implementing policy.