Reputation is not just a simple opinion that an individual has about another but a social construct that emerges through communication. Despite the huge importance in coordinating human behavior, such a communicative aspect has remained relatively unexplored in the field of indirect reciprocity. In this work, we bridge the gap between private assessment and public reputation: We begin by clarifying what we mean by reputation and argue that the formation of reputation can be modeled by a bi-stochastic matrix, provided that both assessment and behavior are regarded as continuous variables. By choosing bi-stochastic matrices that represent averaging processes, we show that only four norms among the leading eight, which judge a good person's cooperation toward a bad one as good, will keep cooperation asymptotically or neutrally stable against assessment error in a homogeneous society where every member has adopted the same norm. However, when one of those four norms is used by the resident population, the opinion averaging process allows neutral invasion of mutant norms with small differences in the assessment rule. Our approach provides a theoretical framework for describing the formation of reputation in mathematical terms.