In this paper we study a family of Legendre sequences and its pseudo-randomness in terms of their family complexity. We present an improved lower bound on the family complexity of a family based on the Legendre symbol of polynomials over a finite field. The new bound depends on the Lambert W function and the number of elements in a finite field belonging to its proper subfield. Moreover, we present another lower bound which is a simplified version and approximates the new bound. We show that both bounds are better than previously known ones.