This paper gives the condition that a class of two-variable functions with iteration shown in the text are means, i.e. $M_f(x,y)=\lambda_1f(x)+\lambda_2f^2(x)+\mu_1f(y)+\mu_2f^2(y)$, where $f$ maps a real interval $I$ into itself, $f^2$ is the second iterate of $f$, and $\lambda_1,\lambda_2,\mu_1,\mu_2$ are real numbers. In addition, this study discusses the properties of $M_f(x,y)$ mean type, including their symmetry, equivalence, and the invariance of the quasi-arithmetic mean with respect to this mean type.
