Abstract
Every averaging function can be looked at from the perspective of minimizing some sort of a penalty, a price paid for the deviation of the output from the inputs. Penalty functions are formally defined and their special classes are presented. The quasi-arithmetic means, OWA and Choquet integral are obtained as special cases of the penalty based averages. Other classes of averages such as the deviation and entropic means are also presented. This chapter introduces new penalty based averages and relates the averages to the maximum likelihood principle.