PhD programs in statistics and data science at major universities
differ in their preferences. I would say that a solid background
in calculus through multiple integration and infinite series is
expected by all. Real analysis and measure theory are clearly the more important than abstract algebra. Linear algebra is directly applicable. A post-calculus course in statistics and probability
will make the first year easier. Computation is of increasing importance
in statistical inference, probability modeling, and data science, so it
is a good idea to know the basics of one computer programming language.
You should start now to look at the web sites of various departments
to which you might apply. Some of them have specific information on the
undergraduate courses they prefer. Almost all PhD programs will start
with a measure theoretic course in probability and statistics that
involves at least modest computing. These courses are supposed to be
accessible to well-prepared math majors with no previous statistics
or computing. However, you will likely be more at ease in your first
year courses and get more from them if you have at least a little
prior background in measure theory, statistics, probability, and/or
Another reason to take an undergraduate statistics course is so you will
understand what you are getting into. Statistics is a 'mathematical
science' in that it consistently uses mathematical methods. But mathematics
is mainly deductive in nature. (Start with axioms and prove what you can.)
Statistics and data science are fundamentally inductive. (Start with
data from the real world and speculate in a structured way on what they say about reality.) Many statistical ideas and methods are driven by applications,
sometimes specific applications.
I know of no 'famous century-old unsolved problems' in statistics. But there is a
constant barrage of challenging and untidy real-world problems to be solved--or
at least better understood--right away.
You should plan to apply to at least half a dozen PhD programs, including
a couple that may be a reach and a couple were admission seems likely.
There is a lot of randomness in how many PhD students any one program
can afford to admit in any one year. In a statistics or data science PhD
program you should expect to get a commitment for full support, contingent
on your steady progress towards the degree. 'Full support' means just enough
money for simple food and lodging and for tuition. This will probably
require some teaching or assisting with instruction or research.
Not everyone should go on to a PhD program after a BS (or MS) degree.
You need to have the motivation and background to do well. You may be used
to being at the top of your undergraduate class, but so will most of the other
Some of my smart and well-prepared students who have entered PhD programs have used
words like 'brutal' and 'incredibly demanding' to describe the pace
of PhD coursework. Without strong motivation and an adequate background
you may not make it through the first year. You need to have clear
professional goals in mind from the start to sustain you through a PhD program.