The $𝑂(𝑁)$ non-linear sigma model (NLSM) is an example of field theory on a target space with nontrivial geometry. One interesting feature of NLSM is asymptotic freedom, which makes perturbative calculations interesting. Given the successes in Lattice Gauge Theories, Numerical Stochastic Perturbation Theory (NSPT) is a natural candidate for performing high-order computations also in the case of NLSM. However, in low-dimensional systems NSPT is known to display statistical fluctuations substantially increasing for increasing orders. In this work, we explore how for $𝑂(𝑁)$ NLSM this behaviour is strongly dependent on $𝑁$. As largely expected on general grounds, the larger is $𝑁$, the larger is the order at which a NSPT computation can be effectively performed.