I have to look up some of the harder numbers (which do exist!), but the basic idea is that the number increased pretty dramatically with the Song dynasty, then stayed high (though as a percentage of total population, still pretty low <10%) throughout the imperial period. The numbers were too high in fact, and there were several periods (particularly in the Qing) in which degree holders flooded the market and found it difficult to find work. There were also far more lower degree holders than Jinshi (highest) holders. I will try to dig up some figures.

I have some data for Ming dynasty from Cambridge History of Ming Dynasty

In all, the sequence[metroplitan exam] was conducted ninty times, producing a total of 24,874 metropolitan graduates.There was no general pass quote at this highest level; graduates fluctated from a low of thirty-two to a high of 472. In general, a separate quota was established for each metropolitan examnination. The average number of successful candidates was 276 per examination. This works out to an average of about ninety new metropolitan graduates per year over the course of the dynasty. a figure lower than comparable averages for both the ealier Sung dynasty and the later Chi'ng dynasty. If the various provinces regularly filled their quotas for provincial graduates, it would mean that the dynasty produced a cumulative total of provincial graduates exceeding 100,000; that between 80 and 90 percent of them never became metropolitan graduates; and that, in any one year after the mid-1400s, on average, there were on active civil service eligibility lists some 12,000 provincial graduates and some 3,000 metropolitan graduates available to fill the estimated 15,000 to 25,000 posts for the civil service.

Metropolitan graduate = jinshi degree
Provincial graduate = juren

No data is available for xiucai, after 1400s the cash-strapped Ming government begun to sell this title to raise money. If we make an assumption that less than 10 percent became provincial graduate there could be a million xiucai over the course of the 276 years Ming dynasty. I don't know what was the passage rate for xiucai, but it's unlikely to be high.

The population of Ming was 65 millions in late 1300s to almost 150-200 millions by late 1500s.