<p>Afan, I cannot do the research for all universities, so I will just talk about Michigan, assuming that schools like Cal, UNC, UCLA, UVA, UTA and a couple more state schools have similar statistics. </p>
<p>1) Putnam: This competition has been dominated by Harvard, Caltech, Princeton and MIT, with 5-7 other universities featuring in a supporting role, among them Cal, Cornell, Duke, Michigan, Michigan State, Toronto, Washington University and Waterloo. Do you realize that schools like Dartmouth, Columbia, Penn, Chicago and Yale have never done well in that competition? Those universities all have awesome Math departments and produce some of the best mathematicians. The reason is simple; The Putnam is a tradition. Some universities have traditionally done well in the competition and attracted students who devote a lot of time to preparing for the event. It is not a measure of academic quality. </p>
<p>2) You say that private elites send more students to graduate school than the publics. I am not sure where you got that information from. Roughly 45%-50% of Michigan students go on to earn a graduate degree of some sort (most of those straight out of college). Roughly 25% go to Law school or Medical school and the remaining 20%-25% get MBAs, Masters in Architecture, Engineering, Pharmacy or Dentistry or a PhD. Can you name me many private universities that have more than 50% of their undergrads go on to earn a graduate degree? And those students don't go just anywhere. In fact, roughly 10% of those students end up getting their graduate degrees at one of the University of Michigan's graduate schools. I think we can both agree that Michigan's graduate schools are pretty solid. But Michigan also sends many of its students to other top graduate schools. In fact, another quarter go on to other top 25 graduate programs. In other words, a third of Michigan students who pursue graduate studies do so at elite graduate programs. Just from last year's class of 5,000, close to a thousand Michigan students enrolled into Law school and another 350 enrolled into Medical school. Of the thousand or so who enrolled into Law school, 65 chose Michigan, another 40 chose Harvard, Yale, Stanford, Columbia, Chicago and NYU and yet another 150 chose top 20 Law schools like UVa, Duke, Georgetown, UCLA, USC, UT-Austin etc...Just so that you get a feel for how many options those students have, top 10 Law schools like NYU, Georgetown, Duke, Penn, Cal have less than a 20% yield rate from successful Michigan applicants. The Medical school and MBA stats aren't as clear, but I know that each year, roughly 50 Michigan students chose to enroll into Michigan's Medical school and another 50 or so chose to enroll into top Medical schools like Penn, Harvard, Washington U, Johns Hopkins, Cornell, Columbia, Chicago, Northwestern etc... Also, 50 or so Michigan students chose to enroll into Michigan, and another 100+ Michigan students chose to enroll into top MBA programs like Wharton, Kellogg, Chicago, Columbia, Harvard etc...</p>
<p><a href="http://www.lsa.umich.edu/lsa/detail/0,2034,12364_html_690,00.html%5B/url%5D">http://www.lsa.umich.edu/lsa/detail/0,2034,12364_html_690,00.html</a></p>
<p><a href="http://www.med.umich.edu/medschool/admissions/life/diversity.htm%5B/url%5D">http://www.med.umich.edu/medschool/admissions/life/diversity.htm</a></p>
<p>I must admit that you were quite right in saying that more than 2%-4% of students at trop universities go on to earn PhDs. But my point stands, there isn't much of a difference between the elites. As I mentioned above, I knew that top LACs, Chicago, MIT, Caltech and Harvard produce many PhDs. However, there isn't much of a difference between Cal, Columbia, Cornell, Dartmouth, Duke, Michigan, Northwestern and Penn. </p>
<p>At any rate, even if Michigan's numbers weren't that impresive, which clearly isn't the case, one thing is clear. Any of the top state universities will provide a student with unlimited options and opportunities. You make it sound like a student will get more out of an education at a private elite like Penn or Johns Hopkins than at a public elite like UVa or Cal. That is not the case.</p>