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We test statistically the hypothesis that radio pulsar glitches result from an avalanche process, in which angular momentum is transferred erratically from the flywheel-like superfluid in the star to the slowly decelerating, solid crust via spatially connected chains of local, impulsive, threshold-activated events, so that the system fluctuates around a self-organized critical state. Analysis of the glitch population (currently 285 events from 101 pulsars) demonstrates that the size distribution in individual pulsars is consistent with being scale invariant, as expected for an avalanche process. The measured power-law exponents fall in the range 0:13 a 2:4, with a % 1:2 for the youngest pulsars. The waiting-time distribution is consistent with being exponential in seven out of nine pulsars where it can be measured reliably, after adjusting for observational limits on the minimum waiting time, as for a constant-rate Poisson process. PSR J05376910 and PSR J08354510 are the exceptions; their waiting-time distributions show evidence of quasi-periodicity. In each object, stationarity requires that the rate k equal /hi, where is the angular acceleration of the crust, hi is the mean glitch size, and is the relative angular acceleration of the crust and superfluid. Measurements yield 7 ; 10 5 for PSR J0358+5413 and 1 (trivially) for the other eight objects, which have a < 2. There is no evidence that k changes monotonically with spin-down age. The rate distribution itself is fitted reasonably well by an exponential for k ! 0:25 yr 1 , with hki 1:3 0:7 0:6 yr 1 . For k < 0:25 yr 1 the exact form is unknown; the exponential overestimates the number of glitching pulsars observed at low k, where the limited total observation time exercises a selection bias. In order to reproduce the aggregate waiting-time distribution of the glitch population as a whole, the fraction of pulsars with k > 0:25 yr 1 must exceed $70%.