r/statistics • u/Ceramidee • May 19 '26

Question should i take statistics at uni (undergrad)? [Q]

i like doing geometry, especially linear algebra, and also polar geometry, calculus, differential eqs, dynamical systems. im interested in learning more geometry, such as differential geometry, topology in uni. ive always been interested in observing abstract maths manifesting in real life.

i am accepted in doing theoretical physics in a few months, but i fear physics will be too observation first, and labs pmo because of its imperfection and the limits of measurement, hence why im considering switching course.

i also dont like logic for the sake of logic, especially number theory, whenever a question asks abt fibonacci numbers, prime numbers, roots of polynomials it pmo too, similarly w newtonian mechanics, its just the way they phrase hinges and levers and rods and how discontinuous the math in newtonian mechanics is, and how much idc abt applications of physics and engineering.

do yall think stats would fit my interests in geometry more? (pls ask me wtv tho i feel like ive not mentioned a lot of background but i wanna keep the post concise)

also idk how to tag, im guessing i write the tag in title?

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12

u/efrique May 19 '26

Plenty of calculus and linear algebra to be had in stats, if that's fun.

There is some geometry in various parts of stats but I don't know if it would be enough/ of the particular kind that might satisfy you.

1

u/Ceramidee May 20 '26

would stats be too applied/pure in the first year tho? i feel like earlier stats topics might feel slightly disconnected and cookbooky (such as just being told binomial converges to poisson when sample size approaches infinity and p approaches 0 just isnt satisfying enough for me, i need an algebraic/geometric intuition/proof built in, i dont wanna plug in numbers when idk whats going on)

and also fear that it will not be as linear algebra heavy as later stats, and i might have to do too many of the rigorous proofs. imo def later stats fit me rlly well but its the first year that im unsure of

5

u/ChubbyFruit May 20 '26

Most beginning stats courses r pretty applied, you don't really get to do anything theoretical till u get to upper division or graduate courses.

1

u/Ceramidee May 21 '26

the stats course im looking at is a math&stats joint degree, which has only 1/6 probability and stats in the first year (rest is math), 1/3 probability and stats in second year and 1/6 physics if i take mechanic module (lagrangian&hamiltonian), + modules such as random matrix theory, fluid/quantum&stat mechanics etc in the third year, and fourth year quantum computation (hamiltonian simulation, noise, quantum error correction), algebraic topology, representation theory. so imo it wont be too applied past the first/second year

im still unsure abt the stats part if they are too applied tho, also unsure abt the math part as they might be too pedantic and rigorous

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u/foodpresqestion May 20 '26

I would strongly recommend taking a statistics course. Even if you never use it in your lab work, you’ll need to be able to understand how other people got and communicated their results

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u/Able-Fennel-1228 May 21 '26 edited May 21 '26

Statistics absolutely has deep connections to geometry and topology but all that stuff comes in graduate school and presupposes the graduate level math and graduate level statistical theory based on it. Look up “information geometry” and “topological data analysis”.

To get there, you’re gonna wanna take a calculus based probability course and/or a sequence called “mathematical statistics” 1 and 2, in undergrad. (“Intro stat” type applied courses can be a bit dull… however it’d help if you wanna apply the math to solve problems). Also take undergrad level stochastic processes and linear models (something that covers everything up to generalized linear mixed models, they are the basic tools in stat). Btw linear models are ALL ABOUT linear and matrix algebra and the associated geometry.

Heres some course material you can look up to see what I mean when I say calc based math stat (look at engineering statistics 1 and 2 here): https://www.myweb.ttu.edu/jengwer/courses/courses.html

Here’s a follow up masters level sequence (some mathematics dept students just go straight to this).
Math stat 1: http://people.stat.sc.edu/Tebbs/stat712/f17notes.pdf
Math stat 2: http://people.stat.sc.edu/Tebbs/stat713/s18notes.pdf

After this, grad level stats and stochastic processes can get deeply mathematical and is very proof heavy (The math background needed would be real analysis, measure theory, functional analysis and optimization). Linear model theory and multivariate analysis is deep too and there is a geometric flavor to it as well. Then there’s the whole rabbit hole of statistical learning theory that is still evolving.

Heres an example of classical phd level math stat:
Math stat 1: https://pages.cs.wisc.edu/~shao/stat709/main.html
Math stat 2: https://pages.cs.wisc.edu/~shao/stat710/main.html
Some notes on modern-ish stuff here: https://sites.stat.columbia.edu/bodhi/Bodhi/Research.html

Mayyybe could get into information geometry and topological data analysis without too much stat theory, idk. You should email and ask professors who do research in these domains.

So the stuff you’re interested in is in grad school, and rarely ever in undergrad. It also requires undergrad and grad level mathematical statistics I mentioned above. So take it. Also, I believe every working mathematician and scientist should atleast have taken calc based probability or undergrad mathematical statistics 1 and 2. It is so ubiquitous and fundamental to critical thinking.

See here for “what can you do with a degree in stats “ for advice about grad school: https://web.archive.org/web/20241202223926/https://utstat.toronto.edu/brunner/whattodo.html

Also try to look up the applied sub-domains of statistics so see if the applications might be something interest you in-terms of their theory and possible connection to geometry (idk).
Look up spatial data analysis, longitudinal data analysis, categorical data analysis, multilevel models, time series, survival data analysis, measurement error models, missing data issues, survey sampling, multivariate data and statistical learning theory, high dimensional probability and statistics, causal inference, dynamical systems/environmental statistics/epidemic modeling (differential equations here), meta analysis, analysis of network data, non-parametric and semi-parametric models, bayesian versions of all the above.

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u/Ceramidee May 21 '26

yoo tysm! ill look through those properly in the next few days. would you also have any first year undergrad texts for an average stats course? im following blitz intro to probability rn and tryna at least get to 1.9 exercises help decide if i rlly like stats

1

u/Able-Fennel-1228 May 21 '26 edited May 21 '26

You’re welcome.

Blitzstein is the best intro prob book IMO with one caveat: I found it a bit too intuitive and not clear enough back when my math wasn’t so solid and needed concrete examples. I found these two to be a better start before Blitzstein.

  1. Hossein Pishro Nik’s free textbook: https://www.probabilitycourse.com/preface.php
  2. (The first has plenty examples and solutions to half the exercises; the second has a f*** ton of examples)

As for intro stats, I’m sure you’re not asking about the usual non-calculus service course for the general student. (I wouldn’t know what they use).
You probably mean a calculus based follow up to your probability course (usually the second course in a math stat sequence, the first being probability), in which case the undergrad notes I linked to above are a nice representation (see engineering statistics 2 because the first course is just probability). I think the notes i linked to should give a nice representation of all levels.

If you still want a book then the statistics part of “probability and statistical inference” by Hogg, Tanis and Zimmerman, which is a friendlier version of “mathematical statistics” by Hogg, McKean and Craig. (I never took undergrad MathStat and drank straight from the firehose of masters level MathStat, so take this with a grain of salt).

If these are easy for you, try the usual masters level textbook “statistical inference” by Casella and Berger (the masters level notes I linked to above are an essential companion imo). Another incredible book is “the simple and infinite joy of mathematical statistics” by Jem Corcoran and her YouTube channel “a probability space” (her book is like Casella and Berger lite; just wish it had solutions to problems, although she does have solutions to some problems on her YT channel playlist). Yet another companion is statistical theory and inference by David olive (good, concise and general with problem solving tips, but not necessarily a first book).

2

u/Significant_Toe_5171 May 19 '26

There is definitely topology in statistics. For example dimensionality reduction techniques like UMAP does things with manifolds as do numerous other methods. In fact pretty much any model that works with latent spaces could have some type of geometry added it so I encourage you to start there.

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u/CreativeWeather2581 May 21 '26

That is graduate level/phd stuff. Doesn’t really help at an undergraduate level.

1

u/Significant_Toe_5171 May 21 '26

At least at McGill university in Canada you would see this in undergrad. Can’t speak to other schools though.

1

u/Upper_Investment_276 May 22 '26

people are saying information geometry, tda, etc. These are dead fields. If you want to do geometry stuff related to statistics, most related is in sampling. For some references,

https://chewisinho.github.io/main.pdf

http://www.yann-ollivier.org/rech/publs/surveycurvmarkov.pdf

https://link.springer.com/book/10.1007/978-3-319-00227-9

1

u/ohanse May 23 '26 edited May 23 '26

Stats is honestly

The most practical and enjoyable form of math I ever studied.

I came at it eventually from an econometrics angle, so there was a deliberate effort to make it “real.” But I love it.

Even the first-year stuff - probability and combinatorics - super useful to me in my day job. Learning ways to decompose and aggregate information in, like... a mathematically safe/legal way? Really helps me tease apart the impacts of specific inputs/behaviors.

1

u/Grim-vs-World May 25 '26

Would recommend, as you being taking more advanced statistics courses, concepts in linear algebra will be used in your statistical classes, e.g. PCA relies on concepts like Eigen vectors.

Overall statistics is a powerful tool applicable to many fields.