Here we present the first quantitative model of the inhomogeneous solar corona, which we call the composite and elementary loops in a thermally inhomogeneous corona (CELTIC) model. We develop a self-consistent statistical model that quantifies the distributions of physical parameters, i.e., the distributions of loop widths, N(w,T_e), electron densities, N(n_e,T_e), electron temperatures, N(T_e), and statistical correlations between them. The parameterized distributions are constrained by the observed triple-filter fluxes of the EUV corona measured at some 18,000 loop positions with TRACE in the temperature range of T_e ≈ 0.7-2.7 MK, as well as by the individual loop parameters (w,n_e,T_e) measured at these positions in ≈240 detected loops, mostly sampled in active regions. The CELTIC model is inverted from the TRACE data and reproduces both the fluxes of the composite (active region and quiet Sun) background corona and the distributions of loop parameters in a self-consistent way. The best-fit values constrain a statistical correlation between the density and temperature, i.e., n_e T_e^α, with α = 0.9 ± 0.6, and between the loop width and temperature, i.e., w T_e^β, with β = 1.3 ± 0.7, which can be related to the thermal pressure in a regime with a high plasma-β parameter. A possible explanation is a heating process located in the lower transition region or the upper chromosphere (e.g., as reproduced in the recent MHD simulations of Gudiksen and Nordlund), which produces sufficiently high electron densities, high plasma-β parameters, and thermally homogeneous loop cross sections as observed in elementary loop strands.