Can we easily compute the percentile of a ratio, using the percentiles of the values in the ratio?

Question

Given:

• a (large) group of people $G$
• for each person in the group, attribute values $x,y$
• for each person in the group, a percentile for $x$ over the entire group
• for each person in the group, a percentile for $y$ over the entire group

Now, given that we can compute $z=\frac x y$, can we easily compute the percentile for a particular person's $z$ value, without precomputing the set of $z$ for every person?

Answer 1

If you mean that given one persons $x,y$ and their percentiles, can I compute the percentile of their $z$, you cannot. Let the person have $x=y=1$ and both their $x$ and $y$ percentiles are $50$. One distribution that supports this is have half the rest have $x=0.95, y=0.9$ and the other half have $x=2, y=1.1$ Our example person then has $z$ percentile of $0$. It should be easy to see how to make their $z$ percentile be $100$ as well.