In a series of papers we have presented an algorithm based on quantization for pricing American options. More generally, this amounts to solving numerically an obstacle problem for some semilinear partial differential equations. Our algorithm is based on a Monte Carlo method and so a statistical error results. In the present paper, we study this error: we prove the central limit theorem for the algorithm and we give evaluations of the variance. The difficulty comes from the fact that the algorithm is not linear. On the other hand, an interesting problem is to control the behaviour of the variance of the algorithm as the complexity increases. It turns out that the variance does not blow up if the time–discretization step and the space–discretization step tend to zero.