I'm following notes at onlinecourses and I got confused on transformation of sufficient statistics. For example, if $X$ is a sufficient statistic for $\mu$, why $Y=X^2$ is not a sufficient statistic for $\mu$? In another source Essentials of Statistical Inference by G. A. Young p. 92 $|X|$ is sufficient. Aren't $X^2$ and $|X|$ both many-to-one functions? Is this something to do with having single observation? Apologies in advance if this question was already asked.

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