What does this equation mean?

A critique earns the right to be taken seriously only once it can say how large the effect it is worried about would need to be, and Xiao-Li Meng’s 2018 identity for bias in self-selected big-data samples gives a way to say exactly that, using only numbers the paper and its data-availability statement already make public [ 11 ] . For a population of size N with a binary response indicator R (did this person’s data reach the sample) and an outcome Y (their realized value on some Moral-Machine-relevant preference indicator), Meng’s identity relates the sample mean to the true population mean as $\bar{Y}_n - \bar{Y}_N = \rho_{R,Y}\sqrt{\frac{1-f}{f}}\,\sigma_Y$. where $\rho_{R,Y}$ is the “data defect correlation” between selection and outcome across the full population, f = n/N is the sampling fraction, and $\sigma_Y$ is the outcome’s population standard deviation. The identity’s uncomfortable lesson for any self-selected big-data project is that the error term does not shrink as n grows unless f grows with it — and for a national population sampled through a viral website, f stays vanishingly small no matter how large n gets. Meng introduced this identity using the 2016 US presidential election’s Cooperative Congressional Election Study data as its worked example, and the pattern he documents there is the same “Law of Large Populations” this article is applying here: across US states, the larger a state’s voter population, the further the self-selected sample’s estimated Trump vote share tended to sit from the state’s true result relative to the sample’s own confidence interval — bigger population sizes made the bias worse, not better, because a fixed small defect correlation gets multiplied by a sampling-fraction term that shrinks fastest exactly where the population is largest [ 11 ] . Nothing about that mechanism is specific to elections; it is a property of the arithmetic, and a 39-million-decision survey spread across 130 national populations of wildly different sizes sits squarely inside the class of designs the identity describes.