On a curved surface like Earth, 'averaging' your data can backfire so hard that more info actually makes the result messier.
March 24, 2026
Original Paper
Support of Continuous Smeary Measures on Spheres
arXiv · 2603.20974
The Takeaway
We usually assume that collecting more data points leads to a more precise average. This study proves that on a sphere, if your data is spread out past a specific 'point of no return,' the average refuses to converge to a single point and remains a permanent 'smear' no matter how much data you add.
From the abstract
We investigate the support of smeary, directionally smeary, and finite sample smeary probability measures $\mu$ with density $\rho$ on spheres $\mathbb{S}^m$.First, in the rotationally symmetric case, we show that a distribution is not smeary, or equivalently, not directionally smeary whenever its support lies in a geodesic ball centered at the Fréchet mean of radius $R_m>\pi/2$, where $R_m=\pi/2+O(1/m)$. In the general case, we show that neither directional nor full smeariness holds whenever th