Let \(G\) be a finite group. The character degree frequency \(m_ G: \mathbb{N} \to \mathbb{Z}\) is defined \(m_ G(n) = |\{\chi \in \text{Irr }G\mid\chi(1) = n\}|\) and the class size frequency function \(w_ G: \mathbb{N} \to \mathbb{Z}\) by \(w_ G(n) = (1/n)|\{g \in G\mid| G: C_ G(g)| = n\}|\) which is the number of conjugacy classes of \(G\) with \(n\) elements. \textit{I. M. Isaacs} proved [Arch. Math. 47, 293-295 (1986; Zbl 0604.20005)] that for a pair of finite groups \(G\) and \(H\) such that \(m_ G = m_ H\) if \(G\) is nilpotent then so is \(H\). The aim of this note is to prove the following analogous Theorem: If \(G\) is a nilpotent group and \(H\) is a group with \(w_ H = w_ G\) then \(H\) is nilpotent. Previously the authors obtain the following Proposition: Let \(p\) be a prime and \(G\) a group. Denote by \({\mathcal S}_ p(G)\) the union of those conjugacy classes of \(G\) whose cardinality is a power of \(p\). Then it follows: \(| Z_ \infty(G)|_ p = |{\mathcal S}_ p(G)|_ p\). Finally they provide some examples to note that \(w_ G\) does not determine the orders of the intermediate terms of the upper central series of \(G\).