A vector space can be equipped with more than one $n$-norm. In such a case, an equivalence relation of $n$-norms is usually studied. Here we discuss and present some results on several equivalence relations of $n$-norms which may be defined on a vector space. In particular, our results correct an error that we found in B.S. Reddy, H. Dutta, [On equivalence of $n$-norms in $n$-normed spaces, preprint, {http://www.akamaiuniversity.us/PJST11\_1\_233.pdf}, March 7, 2011]. We also discuss an equivalence relation of $n$-norms on finite dimensional spaces.