We consider two commutativity ratios $\Pr(G)$ and $f(G)$ in a finite group $G$ and examine the properties of $G$ when these ratios are `large'. We show that if $\Pr(G) > \frac{7}{24}$, then $G$ is metabelian and we give threshold results in the cases where $G$ is insoluble and $G'$ is nilpotent. We also show that if $f(G) > \frac{1}{2}$, then $f(G) = \frac{n+1}{2n}$, for some natural number $n$.

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