We consider the evolution algebra of a free population generated by an $F$-quadratic stochastic operator. We prove that this algebra is commutative, not associative and necessarily power–associative. We show that this algebra is not conservative, not stationary, not genetic and not train algebra, but it is a Banach algebra. The set of all derivations of the $F$-evolution algebra is described. We give necessary conditions for a state of the population to be a fixed point or a zero point of the $F$-quadratic stochastic operator which corresponds to the $F$-evolution algebra. We also establish upper estimate of the $\omega$-limit set of the trajectory of the operator. For an $F$-evolution algebra of Volterra type we describe the full set of idempotent elements and the full set of absolute nilpotent elements.