In the paper under review, cardinal invariants (also called cardinal characteristics) like \(\text{cov}(\mathcal{J})\), \(\text{cov}^*(\mathcal{J})\), \(\text{non}(\mathcal{J})\), of analytic \(P\)-ideals \(\mathcal{J}\subseteq\mathcal{P}(\omega)\) or \(\mathcal{J}\subseteq\mathcal{P}(\mathbb{R})\) are studied: Analytic \(P\)-ideals are for example the ideal \(\mathcal{N}\) of Lebesgue measure null subsets of \(\mathbb{R}\), or the ideal \(\mathcal{Z}=\big{\{}A\subseteq\omega: \lim_{n\to\infty}\frac{| A\cap n| }{n}=0\big{\}}\) of asymptotic density zero subsets of \(\omega\). With respect to the ideals \(\mathcal{N}\) and \(\mathcal{Z}\) respectively let \[ \begin{aligned}\text{cov}(\mathcal{N})&= \min\big{\{}| \mathcal{A}| :\mathcal{A}\subseteq\mathcal{N}\wedge \bigcup\mathcal{A}=\mathbb{R}\big{\}},\\ \text{non}(\mathcal{N})&= \min\big{\{}| Y| :Y\subseteq\mathbb{R}\wedge Y\notin\mathcal{N}\big{\}},\\ \text{and} \text{cov}^*(\mathcal{Z})&= \min\big{\{}| \mathcal{A}| :\mathcal{A}\subseteq\mathcal{Z}\wedge \forall X\in [\omega]^\omega \exists A\in\mathcal{A}(| A\cap X| =\aleph_0)\big{\}}.\end{aligned} \] Among other results it is shown that \(\min\{\mathfrak{b},\text{cov}(\mathcal{N})\}\leq \text{cov}^*(\mathcal{Z})\leq \max \{\mathfrak{b},\text{non}(\mathcal{N})\}\), where \(\mathfrak{b}\) is the bounding number (i.e., the least cardinality of an unbounded family of functions from \(\omega\) to \(\omega\)).