Since in many cases PhD is basically free, it is by far more competitive than Bachelors.
Since in many cases PhD is basically free, it is by far more competitive than Bachelors.
I never got anywhere trying to get into Princeton as an undergraduate, with average SAT scores over 700, but did get accepted in the math (or statistics) department as a graduate student. I'd read lots of John Tukey's work -- uniformity in topology (in the back of Kelley's 'General Topology'), Tukey's lemma in axiomatic set theory, stepwise regression, power spectral estimation (Blackman and Tukey), the FFT (Cooley and Tukey), and 'Exploratory Data Analysis', had been working in applied math and computing, wrote G. Watson, and got accepted. I also got accepted in the Division of Applied Math at Brown and in Operations Research at Cornell. I went to Hopkins.
Kudos for being accepted by all the big names!
Your point about graduate school is spot on, though. The game completely changes (though I think it's still mostly a game). I especially love the "don't buy this muffler..." quote!
In many cases an Ivy League Bachelors is basically free as well.
Between tuition waivers for lower income students, and parents paying for higher income students, an Ivy League Bachelors is effectively free to most of the time.
For PhD, As sidww2 pointed out, and what I have experienced as well, the admission pool becomes much more international; and now you are competing with the world. There is a phrase about China saying that underlines the scale well: "in China when you are one in a million -- there are 1,300 other people just like you".
Moreover, overall the number of spots in graduate schools is by far smaller than for undergrad; making competition even tougher.