The Lebesgue constant gives the measure of how much worse the interpolation polynomial can be computed to the best approximation. The Lebesgue constant depends only on the choice of the the points of interpolation. The best choice of points is the one that minimizes the Lebesgue constant. For some distribution of points which are of interest in interpolation theory, such as equally spaced points and zeros of orthogonal polynomials. In this paper, we study the lower bound for Lebesgue constant with equally spaced points. The idea is motivated by numerical example, which shows that the Lebesgue constant is larger near the end points.