Context. Our understanding of stellar systems depends on the adopted interpretation of the initial mass function, IMF φ(m). Unfortunately, there is not a common interpretation of the IMF, which leads to different methodologies and diverging analysis of observational data. Aims. We study the correlation between the most massive star that a cluster would host, m max , and its total mass into stars, M, as an example where different views of the IMF lead to different results. Methods. We assume that the IMF is a probability distribution function and analyze the m max − M correlation within this context. We also examine the meaning of the equation used to derive a theoretical M −m max relationship, N × mup mmax φ(m) dm = 1 with N the total number of stars in the system, according to different interpretations of the IMF. Results. We find that only a probabilistic interpretation of the IMF, where stellar masses are identically independent distributed random variables, provides a self-consistent result. Neither M nor the total number of stars in the cluster, N, can be used as IMF scaling factors. In addition,m max is a characteristic maximum stellar mass in the cluster, but not the actual maximum stellar mass. A M −m max correlation is a natural result of a probabilistic interpretation of the IMF; however, the distribution of observational data in the N (or M) − m max plane includes a dependence on the distribution of the total number of stars, N (and M), in the system, Φ N (N), which is not usually taken into consideration. Conclusions. We conclude that a random sampling IMF is not in contradiction to a possible m max − M physical law. However, such a law cannot be obtained from IMF algebraic manipulation or included analytically in the IMF functional form. The possible physical information that would be obtained from the N (or M) − m max correlation is closely linked with the Φ M (M) and Φ N (N) distributions; hence it depends on the star formation process and the assumed definition of stellar cluster.