We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting, where the input distribution consists of n unit-demand customers modeled as independent, identically distributed uniform random points in the two-dimensional plane. The objective is to visit every customer using a set of routes of minimum total length, such that each route visits at most k customers, where k is the capacity of a vehicle. All of the following results are in the random setting and hold asymptotically almost surely. The best known polynomial-time approximation for this problem is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan [15]. They showed that the ITP algorithm is near-optimal when k is either o(√ n) or ω(√ n), and they asked whether the ITP algorithm was "also effective in the intermediate range". In this work, we show that when k = √ n, the ITP algorithm is at best a (1 + c0)-approximation for some positive constant c0. On the other hand, the approximation ratio of the ITP algorithm was known to be at most 0.995 + α due to Bompadre, Dror, and Orlin [10], where α is the approximation ratio of an algorithm for the traveling salesman problem. In this work, we improve the upper bound on the approximation ratio of the ITP algorithm to 0.915 + α. Our analysis is based on a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest.