  This paper gives the Kolmogorov and Wasserstein bounds in normal
approximation for the squared-length of total spin in the mean field classical
$N$-vector models. The Kolmogorov bound is new while the Wasserstein bound
improves a result obtained recently by Kirkpatrick and Nawaz [Journal of
Statistical Physics, \textbf{165} (2016), no. 6, 1114--1140]. The proof is
based on Stein's method for exchangeable pairs.
