Correlation between cluster properties. Lower triangle: scatter plot of the residuals at fixed halo mass M_500c of log₁₀ gas fraction f_g, log₁₀ formation redshift z_f, log₁₀ concentration c, log₁₀ temperature, log₁₀ fossilness, log₁₀ richness n, log₁₀ X-ray luminosity, and log₁₀ corrected luminosity (as in Eq. (1)). The dotted lines show the slope for individual correlation coefficients. When it points in a direction different from the data elongation, it indicates that the data do not satisfy the assumption for the derivation of the correlation coefficient. The upper triangle shows the corresponding correlation coefficient ρ and the amplitude of the covariance Δ.