We have examined the Mg– \sigma relation for early-type galaxies in the EFAR sample and its dependence on cluster properties .
A comprehensive maximum likelihood treatment of the sample selection and measurement errors gives fits to the global Mg– \sigma relation of \hbox { Mg$b ^ { \prime } $ } = 0.131 \log \sigma - 0.131 and \hbox { Mg$ { } _ { 2 } $ } = 0.257 \log \sigma - 0.305 .
The slope of these relations is 25 % steeper than that obtained by most other authors due to the reduced bias of our fitting method .
The intrinsic scatter in the global Mg– \sigma relation is estimated to be 0.016 mag in Mg b ^ { \prime } and 0.023 mag in Mg _ { 2 } .
The Mg– \sigma relation for cD galaxies has a higher zeropoint than for E and S0 galaxies , implying that cDs are older and/or more metal-rich than other early-type galaxies with the same velocity dispersion .

We investigate the variation in the zeropoint of the Mg– \sigma relation between clusters .
We find it is consistent with the number of galaxies observed per cluster and the intrinsic scatter between galaxies in the global Mg– \sigma relation .
We find no significant correlation between the Mg– \sigma zeropoint and the cluster velocity dispersion , X-ray luminosity or X-ray temperature over a wide range in cluster mass .
These results provide constraints for models of the formation of elliptical galaxies .
However the Mg– \sigma relation on its own does not place strong limits on systematic errors in Fundamental Plane distance estimates due to stellar population differences between clusters .

We compare the intrinsic scatter in the Mg– \sigma and Fundamental Plane ( FP ) relations with stellar population models in order to constrain the dispersion in ages , metallicities and M / L ratios for early-type galaxies at fixed velocity dispersion .
We find that variations in age alone or metallicity alone can not explain the measured intrinsic scatter in both Mg– \sigma and the FP .
We derive the joint constraints on the dispersion in age and metallicity implied by the scatter in the Mg– \sigma and FP relations for a simple Gaussian model .
We find upper limits on the dispersions in age and metallicity at fixed velocity dispersion of 32 % in \delta t / t and 38 % in \delta Z / Z if the variations in age and metallicity are uncorrelated ; only strongly anti-correlated variations lead to significantly higher upper limits .
The joint distribution of residuals from the Mg– \sigma and FP relations is only marginally consistent with a model having no correlation between age and metallicity , and is better-matched by a model in which age and metallicity variations are moderately anti-correlated ( \delta t / t \approx 40 % , \delta Z / Z \approx 50 % and \rho \approx - 0.5 ) , with younger galaxies being more metal-rich .