Biases in the Determination of Correlations Between Underground Muon Flux and Atmospheric Temperature
The underground rates of cosmic-ray muons exhibit seasonal variations correlated with effective atmospheric temperature, quantified via a single coefficient. We compare two analysis methods for studying the correlation: the standard Unbinned Method, where all rate-temperature data points are fit simultaneously via linear regression, and the Binned Method, where data points with similar temperatures are first grouped into bins before fitting. We find that while both methods are unbiased in the limit of negligible temperature uncertainties, the Binned Method develops significant bias when temperature uncertainties are present, due to binning-induced distortions. In contrast, the Unbinned Method remains robust if the uncertainties are accurately known. To address the widely encountered issue of imprecise uncertainty estimation, we propose a novel procedure that assesses correlation stability by varying the time intervals and their assigned uncertainties. This approach resolves methodological tensions in studies of seasonal modulation of the muon rate and provides a practical framework for robust correlation estimation under real-world conditions.