<p>This paper studies a class of linear quadratic mean field games where the coefficients of quadratic cost functions depend on both the mean and the variance of the population’s state distribution through its quantile function. Such a formulation allows for modelling agents that are sensitive to not only the population average but also the population variance. The potential mean field game equilibria are identified. Their calculation involves solving two nonlinearly coupled differential equations: one is a Riccati equation and the other the variance evolution equation. Sufficient conditions for the existence and uniqueness of a mean field equilibrium are established. Finally, numerical results are presented to illustrate the behavior of two coupled differential equations and the performance of the mean field game solution.</p>