We present a new approach to counting the proportion of hyper- elliptic curves of genus g defined over a finite field Fq with a given a-number. In characteristic three this method gives exact probabilities for curves of the form y2 = f(x) with f(x) ∈ Fq[x] monic and cubefree. These results are sufficient to derive precise estimates (in terms of q) for these probabilities when restricting to squarefree f. As a consequence, for positive integers a and g we show that the non-empty a-number strata of the moduli space of hyperelliptic curves are of codimension 2a−1 which contrasts the analogous result for the moduli space of abelian varieties in which the codimensions of these strata are a(a + 1)/2.