We consider the difference between ensemble and volume average in cosmology. It is known that for sufficiently weak long-range correlations the root mean square of the difference, which we call ergodicity bias, decays like $R^{-3/2}$ in the limit of large volume $R^3$. We calculate the condition this imposes on the power spectrum of a Gaussian random field. We quantify the bias for finite $R$, and show that the $R\to\infty$ limit is of little relevance for cosmological observations when the measured scales and correlations extend to the size of the observable universe. We consider curvature, density, and velocity perturbations. On large scales the bias is important in all three cases. For the density perturbations, which are observationally the most relevant, the relative bias first exceeds 100% at the separation $r=177$ Mpc, and is larger than 100% for all $r>560$ Mpc. It should be taken into account when comparing ensemble and volume averages for large-scale structure. The bias is also large for the cosmic microwave background temperature perturbations on large angular scales, but this is not relevant for observations, as their analysis does not involve volume averaging.
Measurements of the muonic component of extensive air showers constrain cosmic-ray mass composition and hadronic interactions at energies beyond those accessible at accelerators. Arrays of segmented detectors with binary readout are widely used for this purpose: they sample the muon density at different distances from the shower core to reconstruct the muon lateral distribution function (LDF). Each detector response is summarized by the number of activated segments, $k$, whose probability distribution provides the likelihood relating the observation to the expected muon content. Signal pile-up, detector inefficiency, corner-clipping muons, and background signals shape this distribution, and neglecting them can bias the reconstruction. Existing analytical models include pile-up but otherwise assume an ideal detector response. In this work, we develop a unified statistical framework that incorporates inefficiency, corner clipping, and background through a small set of physically interpretable parameters. We derive exact expressions for the detector response and the likelihood required for muon-LDF reconstruction, together with a simple binomial approximation that preserves the main statistical properties of the exact distribution. Dedicated Monte Carlo simulations are used to assess the impact of the assumptions underlying the analytical treatment and show that it is negligible over the parameter range considered. They also show that the exact and approximate likelihoods yield similar performance in terms of estimator bias and confidence-interval coverage. Although motivated by the Underground Muon Detector of the Pierre Auger Observatory, the framework applies more broadly to segmented particle detectors with binary readout in which particle content is inferred from the number of activated segments.