Top panel: T_e(OIII)–T_e(NII) relation for our nitrogen sub-sample of 53 systems (points). Fits to the T_e(OIII)–T_e(OII) relation from the literature are also shown. Centre panel: relation between direct Z_Te and the hyperbolic semi-axis, a. Red lines indicate the least-squares fit to our full calibration sample, using T_e(OII) (see Eq. (10)). Black lines indicate the least-squares fit to our nitrogen sub-sample when using T_e(NII). Bottom panel: relation between the semi-direct Z_Te deficit and [OIII]/[OII]. Again, red lines denote our least-squares [OII]-based fit (see Fig. 7), black lines denote our least-squares alternative [NII]-based fit.