If \(M\) is a mean on \(n\)-tuples, \({\mathbf x}=(x_1, \ldots ,x_n)\), and \(f\) a function of a real variable, then \(M\) is called invariant under \(f\) if \(M(f({\mathbf x}))= f(M({\mathbf x}))\), here \(f({\mathbf x})=(f(x_1),\ldots, f(x_n))\). If a mean is strictly monotonic, smooth and invariant under \(f(x) = x^r\), \(r>0\), and \(g(x) =\lambda x\), \(\lambda>0\), then it is the geometric mean, while if it is invariant under translation and \(g\), it is the arithmetic mean. The term invariant mean was introduced by \textit{A. Horwitz} [J. Math. Anal. Appl. 270, No. 2, 499--518 (2002; Zbl 1004.26020)], but the present paper uses the term in a more natural way and calls the invariance of Horwitz type 1 invariance. Various interesting results involving both these types of invariance are given.