A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent domination number of $G$, denoted by $γ_i(G)$, is the minimum size of all independent dominating sets of $G$. The independent domination stability, or simply $id$-stability of $G$ is the minimum number of vertices whose removal changes the independent domination number of $G$. In this paper, we investigate properties of independent domination stability in graphs. In particular, we obtain several bounds and obtain the independent domination stability of some operations of two graphs.