Summary: For a given graph \(G = (V, E)\), the degree mean rate of an edge \(u v \in E\) is a half of the quotient between the geometric and arithmetic means of its end-vertex degrees \(d(u)\) and \(d(v)\). In this note, we derive tight bounds for the Randić index of \(G\) in terms of its maximum and minimum degree mean rates over its edges. As a consequence, we prove the known conjecture that the average distance is bounded above by the Randić index for graphs with order \(n\) large enough, when the minimum degree \(\delta\) is greater than (approximately) \(\varDelta^{\frac{1}{3}}\), where \(\varDelta\) is the maximum degree. As a by-product, this proves that almost all random (Erdős-Rényi) graphs satisfy the conjecture.