A statistical analysis is carried out of the similarity of profiles of plasma parameters: electron temperature Te, density ne, and pressure Pe, at the stage of quasi-stationary plasma current (the so called flat-top stage) of about 9000 discharges in the JET tokamak, among the discharges from #84458 to #99419 that covers almost the entire database of successful discharges in the JET ITER-like wall machine. For all these parameters, the existence of universal profiles is shown as functions of the normalized minor radius of the plasma column ρ. The discharge universal profiles (DUPs) are obtained by dividing the space-time-dependent profile by its value at the center of the plasma column or by the space-averaged value of this parameter in the region ρ ⩽ ρmax = 0.5–1, and subsequent averaging over time at the flat-top stage of each discharge. The machine universal profiles (MUPs) are obtained by averaging over time of flat-top stage of all discharges. It is shown that for 86% of the discharges, the time-averaged relative root-mean-square deviation of Pe profile from the DUP is less than 20% at ρ ⩽ 0.8, and for the MUP, it is less than 17%. A similar picture is observed for the profiles of Te and ne/√Te. An auxiliary heating, Paux ⩾ 1 MW or 10 MW, was present at, respectively, 44% and 21% of the total duration of the flat-top stage of discharges. It is shown that large, ∼100% in magnitude, jumps of Te at ρ = 0, caused by the switch-on of auxiliary heating, can be described with a ∼20% accuracy by the jumps of the space-averaged temperature if the DUP is used to describe the stationary shape of the temperature spatial profile. The results obtained illustrate the degree of plasma self-organization in tokamaks.