Explicitly using the block structure of the unknown signal can achieve better reconstruction performance in compressive sensing. An unknown signal with block structure can be accurately recovered from under-determined linear measurements provided that it is sufficiently block sparse. However, in practice, the block sparsity level is typically unknown. In this paper, we propose a soft measure of block sparsity kα(x) = (||x||2,α/||x||2,1) α/(1−α) with α ∈ [0,∞], and present a procedure to estimate it by using multivariate centered isotropic symmetric α-stable random projections. The limiting distribution of the estimator is given. Simulations are conducted to illustrate our theoretical results.