Top panel: relation between Z_Te and the hyperbolic semi-axis a for our direct dataset. Linear fits to the data (Eq. (10)) are shown for our Bayesian analysis (dark grey solid line) and least squares fitting (black solid line). Dashed lines indicate the 1σ scatter around the distribution of Z_Te values for the least-squares fit. The grey area around the Bayesian best-fit indicates the range of possible fits considering the full covariance matrix. Bottom panels: Z_Te − T_e(OIII) and Z_Te–T_e(OII) relations for the same dataset. The large spread in the Z_Te − T_e(OII) relation leads to quite discrepant fits being returned by our two statistical fitting methods.