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

Error bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models