Summary: For an edge coloring of a connected graph \(G\) of order 3 or more with positive integers, the chromatic mean of a vertex \(v\) of \(G\) is the sum of the colors of the edges incident with \(v\) divided by the degree of \(v\). Only edge colorings care considered for which the chromatic mean of every vertex is a positive integer. If adjacent vertices have distinct chromatic means, then \(c\) is a proper mean coloring of \(G\). The maximum vertex color in a proper mean coloring \(c\) of \(G\) is the proper mean index of \(c\) and the proper mean index \(\mu(G)\) of \(G\) is the minimum proper mean index among all proper mean colorings of \(G\). The proper mean index is determined for complete graphs, cycles, stars, double stars, and paths. The non-leaf minimum degree \(\delta^{\ast}(T)\) of a tree \(T\) is the minimum degree among the non-leaves of \(T\). It is shown that if \(T\) is tree with \(\delta^{\ast}(T) \geq 10\) or a caterpillar with \(\delta^{\ast}(T) \geq 6\), then \(\mu(T) \leq 4\). Furthermore, it is conjectured that \(\chi(G) \leq \mu(G) \leq \chi(G) + 2\) for every connected graph \(G\) of order 3 or more.