Let $\theta$ be an equivalence relation in a set $Q$, and let $(Q,F)$ be an $m$-groupoid, $m\in N$. Then: a) $\theta$ is a congruence relation of the $m$-groupoid $(Q,F)$ iff for all $a,b\in Q$ and for every sequence $c^{m-1}_1$ over $Q$ (1.1) the following statement holds \[ \bigwedge^m_{i=1}(aheta b\Rightarrow F(c^{i-1}_1,a,c^{m-1}_i)heta F(c^{i-1}_1,b,c^{m-1}_i));\quadext{and} \] b) $\theta$ is a normal congruence of the $m$-groupoid $(Q,F)$ iff for all $a,b\in Q$ and for every sequence $c^m_1$ over $Q$ the following statement holds \[ \bigwedge^m_{i=1}(aheta beftrightarrow F(c^{i-1}_1,a,c^{m-1}_i)heta F(c^{i-1}_1,b,c^{m-1}_i));\quad (1.5). \] Further on, let $(Q,A)$ be an $n$-group (1.2), $\mathbf e$ its $\{1,n\}$-neutral operation (1.3) and $f$ its inversing operation (1.4). The main result of the paper is: If $\theta$ is a congruence relation of the $n$-groupoid $(Q,A)$, then: \begin{itemize} ıem[1)] $\theta$ is a normal congruence of the $n$-groupoid $(Q,A)$ for every $n\geq2$; ıem[2)] $\theta$ is a normal congruence of the $(n-2)$-groupoid $(Q,\mathbf e)$ for every $n\geq3$; ıem[3)] $\theta$ is a congruence of the $(n-1)$-groupoid $(Q,f)$ for every $n\geq2$; and ıem[4)] $\theta$ is a normal congruence of the $(n-1)$-groupoid $(Q,f)$ for $n=2$. \end{itemize}