Let d and g be positive integers with 1<d<g. If d is odd, we prove there exists B(d)>0 such that a positive proportion of odd genus g hyper elliptic curves over Q have at most B(d) points of degree d. If d is even, we similarly bound the degree d points not pulled back from degree d/2 points of the projective line. Our proof proceeds by refining Park’s recent application of tropical geometry to symmetric power Chabauty, and then applying results of Bhargava and Gross on average ranks of Jacobians of hyperelliptic curves.