Median of medians confusion -- the "approximate" median part
Median-of-medians is a recursive algorithm which solves the more general selection problem: given an array $A$ of length $n$ (which we assume, for simplicity, has distinct elements) and an integer $k$, find the $k$'th smallest element (where $1 \leq k \leq n$). It works as follows:

If $n \leq K$ (where $K$ is some constant), solve the problem by brute force.
Otherwise, find an approximate median $x$ of $A$ in $O(n)$. By approximate median, we mean that $x$ is the $m$'th smallest element in $A$, where $\alpha n \leq m \leq (1-\alpha) n$ for some constant $\alpha &lt; 1$.
Use $x$ to split $A$ into elements smaller than $x$, comprising an array $A_{&lt;x}$ of length $m-1$, and elements larger than $x$, comprising an array $A_{&gt;x}$ of length $n-m$.
If $m = k$, then output $x$. If $m &gt; k$ then output the $k$'th smallest element of $A_{&lt;x}$. If $m &lt; k$ then output the $(k-m)$'th smallest element of $A_{&gt;x}$.

The running time of the algorithm satisfies the recurrence $T(n) \leq T(\alpha n) + O(n)$, whose solution is $T(n) = O(n)$.
The algorithm finds the exact median, but it does so by repeatedly finding approximate medians.