I received the following email from someone who wishes to remain anonymous:

I am a longtime reader of your blog and it, along with other factors that I will explain briefly, has motivated to pursue a second masters degree in statistics and machine learning. The problem is, my math isn’t great. I understand statistics and probability conceptually but I can’t always follow the math. I took Intro Calculus but that was 10 years ago. I’m a bit scared and overwhelmed and dont know where to start so I was hoping to get some feedback and guidance from you. A little bit about me: I am in my late 20s, have a degree in public policy from ** University, and work as an analyst. I started college as a pre-med student before realizing that I did not have any interest in the field of medicine outside of my desire to help other people. I’ve always been a bit of a policy wonk so I finished undergrad a year early and enrolled in a public policy masters program. I started focusing on quantitative analysis after I read The Black Swan and Fooled by Randomness and took courses such as Applied Econometrics, Public Finance and Stats I and II. I loved these courses but I did not have the opportunity to delve deeper into them in a 2-year program. Additionally, these courses were not very math heavy. Anyway, I’ve been following your blog for the past few years and I found a university program in Statistics and Machine learning that appeals to me. I thought about taking Calc I and II, Linear Algebra and Differential Equations (in that order) through MIT’s Open Courseware site before I apply to the program. Do you think this is a good course of action? Do you have any other recommendations for building a solid foundation that will allow me to get the most out of this program?

I don’t know the best advice to give here, but here are some thoughts.

1. If you’re not so good at math, you might not get much out of taking a bunch of college-level math courses remotely. It’s hard enough to learn this stuff if you’re taking it in college; I’d guess that a remote course would be even more difficult. And taking 4 courses in sequence is a lot.

2. If you’re interested in statistics and machine learning, then programming skills will be important, I’d guess more important than math. So I think you want to make sure you have a solid foundation there.

Perhaps the commenters have some other suggestions?

Just apply to the program and see if you get it. Like you, I’ve always been better at statistics and probability concepts than I ever was at anything calculus related. As a food microbiologist, who’s grown increasingly interested in policy, I find that I can pretty much figure out any math related issues that I need to to advance my career and my understanding.

I think if you take a bunch of math classes online, it will just wear you out and be demoralizing. Just focus on the degree program and the next step to get you there and then get you through. Don’t add extra work for yourself.

I am actually in a very similar situation although I consider myself good at math although I only took single variable calculus but I love math. I want to get into statistics or machine learning but currently work in accounting and my degree is in finance, econ. How do I stand a chance to get into a good Ivy League level data science/statistics type program like one at Stanford or YU how can I get in without having much of a previous background but feel strong in my math abilities.

By the way there are a bunch of somewhat similar type of questions including those asked by me on quora if you google it…

If you are very motivated it is possible I took my first algebra Class When I was 28. I was able to get my PhD in statistics but it took a lot of work and I had to doa great deal of catch up tofollow in my Courses where youngestudents had all the math under their belts. It takes perseveriance and the concepts don’t really make sense until much later. If you feel you are headstrong enough to keep at it even though you will always feel likyou are running behind the pack go for it, It does even out later.

good luck

Wow, the fact that you were able to get a PHD in statistics is really motivating! I turn 28 in July and even though I took calculus in college and two econometrics courses at the graduate level, I am starting from scratch to build a more solid foundation. Right now, I working through some pre-calc textbooks, then I’m going to do calc 1 and 2 independently. From there, I’ll probably audit a few math courses at my local university based on everyone else’s caution against doing everything online. I would love to chat with you about how you went from taking your first algebra course at 28 to getting a PHD in statistics. Let me know if that is ok and I will post my email.

Gilbert Strang’s lectures on MIT OpenCourseWare are a pleasure to watch, and so are Denis Auroux’s (you may especially enjoy his just-in-time blackboard clearing technique). But you can’t learn math by watching a movie or reading passively. You have to solve problems and have someone tell you where you went wrong. I would recommend doing the Graduate Certificate in Statistics, at the University of Sheffield. It’s an amazing course; I did it in 2011-12. I reviewed it here:

http://vasishth-statistics.blogspot.de/2011/12/part-1-of-2-review-of-graduate.html

This course has three modules, Math, Probability, and Statistics, and brings you to the point that you know enough to start an MSc in Statistics. You’re constantly doing problems that are graded, and that’s where most of the action is in the course. If I had watched the relevant courses online on MIT OpenCourseWare I would not have learnt as much (I know because I have done that too).

If you are one of those rare people who can sit down and solve problems on their own in a disciplined manner, and learn from your mistakes, then you can probably get away with following online courses like Strang’s and Auroux’s.

I will definitely look into that graduate certificate. I was thinking about getting a graduate certificate at my local university so I might just dive headfirst into that and see if I can get in. I agree, I need to do problems. The pre-calc books I’m using now have lots of problem sets with answers and explanations. I cant learn unless I do lots and lots of problem sets so I completely agree with you.

Major in philosophy of “statistics and machine learning”. No math required.

I was in your situation and got into a CS Machine Learning program. You can muddle through it like I did catching up on the required math on the way but it’s hard on the ego and inefficient.

In retrospect however, the math you need for machine learning is a bit of a subset of the courses you mention, and then a few others. I am thinking of putting together a “here is the math you //really// need for ML” wiki. For example, aside from cosine and some infinite series, you can get by with nearly no trig, however you need to get used to thinking in N dimensions all the time. Geometric thinking is useful. Also I found the matrix notation tough at first until I started drawing rectangles of the matrices. (Everyone uses the Matrix Cookbook anyway !)

So until I get my act together and put up that resource. Here are the things I wish I had spent a summer on. You don’t need to do the entires courses, but here are some chapters that would have been helpful to me.

- Linear Alg: If you can understand SVD and everything that leads up to it, you’re mostly there. Try Strang Chaps 1-5. Decompositions, and geometric intuition. Great videos of him on line.

- Calc: Make sure you have the basic single var derivatins and integration down, then practice partial derivatives. Don’t forget basic algebraic tricks like completing the square. Make log and exponent rules are memorized. You will spend a lot of time in log space. If you can do a Lagrangain you’re mostly there.

- Convex Optimization. Skip Chaps 1-5 of Boyd and try some of the exercises. He’s also an entertaining lecturer and recently did a new version of his videos. Just to have the convex optimization vocabulary down is a huge help. Many ML problems are fundamentally optimization problems.

- Stats: Focus on the form of exponential family distributions, the notion of a conjugate prior, and some know the stats vocabulary. Bayes Rule, conditional probability and associated algebra. Honestly from an ML standpoint I was shocked how little classical stats I used. I imagine though if I was modelling I’d need more. You can feel the division between the stats and engineering communities here.

- MATLAB: If you are in a non applied ML course you will probably be working in MATLAB. Most ML research is still using this. It’s not a hard language, and pretty easy to take a formula and move it into matlab. Maybe Numpi/Scipy is gaining ground too. Just find some code implementing a simple ML routine and try to understand it. Something < 50 lines.

- Christopher Bishop's Pattern Recognition and Machine Learning may be in your future, or Trevor Hastie Elements of Statistical Learning. Find a copy and tackle some easy bits, the appendix etc. It's worth getting used to the fact that they are dense, but also that you don't need to know it all. Most ML courses cover 1/3 of Bishop.

- Peter Orbanz' course slides are clear, concise explanations. Worth reading through: http://stat.columbia.edu/~porbanz/teaching/W4400/W4400S14_01May14.pdf

my 2c

Skip=Skim !

This is great. Thanks! I am doing pre-calc now, then going to go through calc I again and then calc II. Looks like I dont need to do a full course in linear alg or diff eq so that is good. I’ll add MATLAB to my list of programming things to learn. Right now, I’m focusing on Python and R.

The question I would ask such a student is, ‘What do you intend to do with this degree?’ and ‘To what depth and scope of this area do you intend to go?’. And when someone says ‘I’m not so good at math?’ what do you mean by this? What do you feel honestly are your strengths? Your deficiencies? Is it that the formalized maths as a language do not make sense?

Before I even began this sojourn, I’d begin with a personal ‘statement of purpose’. Be honest with yourself. Why do you want to do this? What do you wish to achieve? Do you have more of an ‘applied’ mindset, or are you a ‘novelist’ (sometimes the 2 can be intertwined, but in the sciences, to do the latter, one must be able to see the forest from the trees – this has a more abstract or theoretical quality. One frame of mind is not necessarily exclusive of the other, but the latter does require a deep and sometimes unique understanding of one’s subject.). What are your strengths that would aid you in this area? What weaknesses might hinder you? Are the weaknesses sufficient to require remediation in your career path?

I admittedly have an axe to grind in this area. Many people want to do things like this simply for a piece of paper and a title. Then they want to manage such programs in industry without ever having traversed the conceptual minefield and rigor to really have a sense of direction in quality design and development of work in the intended field or industry they choose. This in my opinion is a rather large problem in both the tech industry as well as others. Little expertise, but plenty of schills claiming it…and perhaps directing (not leading/facilitating) a group in a company that doesn’t know any better. The analogy is “teaching without having done it first”. There are a very few of the ‘self-taught’ who have the capacity to do great things with a ‘bit’ of formal direction and their own minds, even in this field – but this is an extraordinarily rare person. And in today’s speed culture, those who don’t have a command of the underlying machinery of what they build will have a hell of a time deconstructing problems when they arise.

Ok, soapbox off. Back to what is ‘required’ and ‘reasonable’ for the student’s purpose…

In the maths, for statistics, many programs require at least 2 semesters of calculus as the calculus is the fundamental language which expresses probability and therefore statistical concepts. The other is a solid understanding of matrix theory. In my opinion, this means being able to rather fluidly read/write and truly understand the abstractions presented in each concept as if they were a language to you (and they are). The key in my opinion is, *understanding the concepts in the language of the concept*.

For machine learning, the formalisms are no less required to be able to read/write/speak/understand, but I don’t believe that these require any further maths than the above than say discrete mathematics (which in computer science is a definite requirement). You will need to understand basic algorithm design, and I would imagine data abstraction, and data structures. The only maths required here may be say, conversions between different numeric bases.

I think the above would be the bare minimum. However, none of what I’ve discussed here deals with some of the various analytic dimensions, such as space, time, and frequency. For example, time-domain applications (i.e. time-series) from the statistical nomenclature break down into ordinary and oftentimes 1st order partial differential equations. Those 2 subjects (ODE and PDE) give one the ‘under the hood’ understanding of the mathematics of time/rate, etc. In the space dimension, various algebras are quite useful as well as topology, geometries, etc.

If decisioning becomes interesting (where in cognitive computing it is a mainstay in design), then things like game theory and combinatorics then become useful for understanding.

These additional things I mention truly go into the depth and scope you are willing to go to understand the craft, which is why I think it critically important to go back to the beginning and truly understand your aims for doing this work, what you wish to contribute to it, and what you’re willing to do to get there.

Very good points, Philip. So, right now I’m working with a health technology start-up that is building a mobile app. The algorithm’s ability to both predict user behavior and develop targeted recommendations based on various inputs is one of the apps most marketable features. I work on the business development side of things but the back-end of the app is really fascinating and I’d like to work more with that. Even if it fails, I’d like to have statistical and programming skills to build predictive algorithms. I’ve been able to provide some conceptual direction on what the algorithm should do based on my philosophical understanding of statistics and probability but that is it. In that same realm, I’d like to start a political consulting firm that does robust election modeling and forecasting for conservative (note that I did not say Republican) and libertarian candidates at the local, state, and federal levels. The right has a serious data problem and I’d like to be part of its solution. This would combine my love of politics and statistics in a really neat way. Now that you’ve read this, what would you suggest?

You’re dilemma isn’t a unique one, though very important for what you consider fulfilling for your career. You, like others, will often ask

“Can I develop and lead, too?”.I can go a couple of ways with this. One in the early startup sector, and the other in the more mature company.

Taking the 2nd on first as I’m more familiar with this arena, I can genuinely say that in a mature corporation, leading and contribution are in sometimes both possible within scientific tracks for roles that resemble that of a ‘principal/lead/etc’…something. In some companies, those with formal ‘manager roles in quantitative disciplines have opportunities to contribute in development as well. Though generally, managerial tracks will take you on just that – more project management, facilitating, etc, etc. than development contributions.However, those same companies do not generally hire those without some sort of advanced training/experience that can be demonstrated in an interview (that may or may not include a requisite degree in a requisite field). Of course that assumes that said company knows how to interview this sort of candidate in the first place. At any rate, the advanced degree and demonstrable depth and scope in it as a requirement is commonplace (and I believe moreso in the past few years).

In the startup world, I feel this is strictly a forward looking vision question.The answer to that lies in where you see your startup. Will you be the CEO, or a technical scientific director as well as a founder? Some hybrid chief evangelical scientific operational financial executive rockyroad smorgasbord? To what degree and perhaps for how long? Just from mere observation, I get the feeling that startup founders may develop something early on with the chief motivation to stay small, patent, and sell a technology for someone else to continue development. That’s perfectly legit. Others still intend to develop something with the intent to build the business around the development. That’s equally legit, but a different animal.In my opinion, once a company begins to grow and expand, the founder has to begin making some choices on whether they want to be a

contributoror theleaderof the company.And it also depends on whether you, as a lead of any sort within your own company intend to dictate what gets developed and how

(this is a slippery slope for any company). This is generally the part in the play where I notice that founders in many of the cases I watch have begun co-founding, and/or stratifying existing roles.Some founders, if they do not intend to take the CEO position, will look to hire that person (usually a down the road event). Many that I’ve noticed tend to hire scientific talent (either as contract or perm) that, at least in resume, tends to precede their own so they can do the other needed work of a growing business. In these cases, they still want to stay in the idea game, but the scope of development is simply not something for which they have time. That’s kinda where the rubber meets the road in my mind. One cannot do

everything.So here, you have a choice to make, depending on what your longer term vision is

(for startups which I often read about that succeed, having a pretty solid and even detailed longer term vision appears to be fairly common advice).The short of the previous…..I think you might best choose your academic needs/wants based on some basics:

Your

intendedrole(be clear about it both short and long term).Your

longerterm startup visionYour intended

depth and scopeof your role(what will you be spending the most of your time doing?)The

resources and timeyour role will take(if it is more of an executive position)Your

Plan B – risk contingencies….i.e. what happens if you decide startup is not ‘it’ for you or something goes kablooshJust simple honesty– what DO you wanna do vs what do you have the personal capital/resources to do? What motivates you the most? How long is that likely to motivate you?I also am interested in pursuing graduate studies in statistics but lack an extensive math background, and I used the question feature on mathbabe and found it helpful. Here is what Aunt Pythia told me.

http://mathbabe.org/2013/08/24/ask-aunt-pythia-5/

One thing, I do not think you can get grades from MIT open course ware, and feedback is useful and might be important to an admissions committee.

It also seems worthwhile to do data analyses with R and have some stuff online via a public account.

I *still* do not get O’Neill’s angle is on the ‘alter ego’ thing she does. Andrew maybe you should start one of these yourself. What would your name be? :)

On the one hand, I’d never want to discourage anyone from learning more math, statistics and programming …

But as with others, before someone spends a lot of time doing it, they really have to be sure about what they want to do.

In particular, they may competing with people who studied all that math in high school and college by the time they were 20, and had been programming for years before that.

The real difficulty is the elapsed time it takes for math to sink in, and the long prerequisite chains. It is possible to catch up, but very very hard.

I wouldn’t limit ‘when’ someone begins the process, though I agree it certainly helps to have a good understanding of the road ahead. And as far as competing against a fresh graduate, never underestimate one’s veteran experience out of school. It very well may be that the person who had difficulty in maths at an earlier age may, having experienced and observed life outside of the classroom, find them less daunting later in life, and for a variety of reasons.

I often think that people’s lack of confidence in the maths, stats, and disciplines that explicitly integrate them, are ground in negative school experiences (whose sources are varied), and thus their resulting viewpoint from them. I don’t believe that only the chosen few have the aptitude and capacity to learn them – but certain behaviors and fears that have developed in others simply need a bit of sublimation to something else to ‘unlearn’ past experiences.

“And as far as competing against a fresh graduate, never underestimate one’s veteran experience out of school.”

Yes, I agree, so let me say a little more on what I meant, via an example.

(For general context, see Languages, Levels, Libraries, and Longevity, in ACM Queue.

Back in mid-1990s, I was back at Penn State for other things, and they’d asked me to give a short lecture to a senior computer science class and then answer questions on computing and industry.

1) I talked about algorithms+data structures (taught in a course they’d all have taken), but from the viewpoint of the surprises that could happen when running on real machines with cache memory hierarchies, rather than the idealized machines found in Don Knuth’s books. The point was that they didn’t need to be come computer architects, but they did need to know where there might be troubles that would cause surprise performance losses.

2) Questions: the first question was from a very eager guy in the back of the room:

Q: We’ve learned C, C++, Java, etc. What’s the most valuable language in business out there?

A: (sigh) You might not like the answer….English.

Q: (What?) (Young woman next to him nudges him, says: “See, I told you so.”)

A: Look, it’s good to learn programming skills and you’ll need them to start. But unless you want to sit in a cubicle and write code for the rest of your life, and unless you’re the best, compete with a lot of others, you need to do higher value-add. You need to be able to talk to customers/users/funders of software, turn them into requirements, express ideas, make presentations, work in teams, etc. The programming skills are necessary, but not sufficient for a good career, for most of you.

Q: (light dawning) Is that why we had that software project management class?

A: Yes … and I actually started some of that when I was teaching CS411 about 25 years ago. I’m glad to see more of it.

Q: Ahhh. (woman pokes him again, and says “see, I told you so.”)

Anyway, one needs to distinguish between:

a) Doing enough of something that one can do it often and fast, with skill

and

b) Studying it enough to understand it, its applicability, limits and surprises … and then if at all possible, using some expert-created software to do higher level and higher-value work.

After all, once upon a time, doing the work of a few R statements meant writing some Fortran code, which is what scientists and engineers at Bell Labs were doing, an impetus for creating S in the first place.

Anyway, understanding the goals (as Philip M suggested earlier) is crucial … my only advice is to make sure you’re targeting something where (decent math/programming+experience) is competitive with (very good math/programming skill alone.)

I feel as though I’m in a similar situation: I don’t develop intuitive understanding of the complex mathematics very quickly (I’d need a month if you wanted me to derive the SVM dual, or backpropagation in neural networks). It took me an entire summer of working with my current Ph.D. advisor to fully grasp spectral clustering! But that’s where my advice would come in: don’t wear yourself down right now with online courses. Get yourself an advisor who understands your situation and can play to your strengths; I find the chalk-talks absolutely critical to intuitively understanding how the theory works. If I have someone I can bounce ideas off of and talk one-on-one with, I grasp the concepts much more quickly than in a lecture setting.

tl;dr my advice would be find an advisor or mentor who can help you along. Hit the MIT Open Courseware only once you have an idea of how things work.

What Nicholas said was spot on, if you’re doing the master’s at an engineering/CS school. And I would also add that Linear Algebra and Matrix Calculus is probably more important for you, going in, than other subjects (although what Nicholas said applies there too). If I had to rank the subjects in order of “importance” (where important means it would be absolutely required to be fluent in the subject and “not have to think about it” when starting the master’s) it would be Lin Alg, MATLAB(or Python with Numpy, or even R, it’s easier to program from scratch ML algos in MATLAB and Python than R, and the opposite applies when you just want to use something off-the-shelf to do some analysis), Optim, Calc, Stats. And from these subjects, focus on the parts that Nicholas spoke about (eg you’ll probably never need classical stats stuff on an ML MSc, and it’s possible you could get away with only knowing about conjugate priors, multivariate models, and Bayes rule). Also Bishop’s Pattern Recognition and Machine Learning is essentially a tutorial, and pretty easy for self-studying and using as a guide (much more so than Tibshirani I think) so you could skim through it and see what you think challenges you.

Finally, contrary to the most comments here, in my opinion the right classes from MIT Open Courseware would be very helpful. I really don’t think you’ll need a whole course on Differential Equations though. Better make that the first lectures from Boyd’s Optim class.

This is really helpful! Thank you ranking the classes in order of importance! My plan was to brush up on my calculus then do lin alg.I am going to teach myself Python and R, I didnt think MATLAB would be useful if I knew R but everyone recommends that so I’m going to add MATLAB to my list of programming languages. I have a lot of friends who are proficient in MATLAB and R so they will help me with that. For optimization, is that Calc III or is it just listed as “Optimization” on a list of courses offered at the university level. I’ll try calc on open courseware and see how I like it. From there I’ll either continue doing my math courses online or start taking classes at the university level. Also, I’m VERY familiar with Bayes rule/theorem at the philosophical level. That is how I found Prof. Gelman’s blog actually – through looking up something on Bayes theorem haha.

Hi Everyone,

First, thank you for your advice,Professor Gelman!I’m glad I decided to wait until you posted your advice before enrolling in any self-directed online courses. Second, thank you to everyone who has responded so far. The advice has been really helpful and I will reply to each comment individually. I’ll also post an update with a new plan of attack based on everyone’s suggestions. Thank you all again. This means a lot to me!

I think it’s a good idea to take classes online if you don’t need the formal certification. In my experience online courses can be far superior than what’s available locally and they are of course free. I can recommend Jim Fowler’s courses for Calculus I and Calculus II. He also has a multivariate calculus course under development that is quite good. You can view these on Coursera or at http://ximera.osu.edu/. In my experience the MIT videos are great but if you can get a platform that engages you to do hundreds of problems you will be much better off.

I’ll second this comment. I’m in my mid-30s, and recently refreshed my calculus with the Ohio St classes on coursera and just enrolled in a Stats MS program part time while I work (I was also influenced by this blog to do so, by the way). They gave me the confidence I needed to enter the program since it had been over a decade since taking those classes formally. I did their multivariate calc class too, but it’s probably overkill for what you need for most masters stats programs. For matrix algebra, I’d recommend using something like Khan Academy to learn the pieces you’ll actually need (assuming you don’t need college credits for it, that is).

I’ll agree with anyone who said Linear Algebra is probably the most important, especially if you’re not completely new to calculus.

As for programming, learning loads of languages at the same time seems like a great plan, but it quickly will become a mess. Would advise not learning more than two, especially if you have no prior experience. R is great, since you’ll be able to do lots of statistics with it very quickly. Python is a great language too. I’d have to advise holding off on learning MATLAB, it’s clearly adding the least “marginal benefit” of the three.

I recall Bill Curtis of the old MCC publishing research in the 1980s, I think, that indicated roughly that people were more effective at programming the more languages they knew. Knowing those other languages presumably gave people additional “models” for approaching algorithms.

I did this thing, basically — a stats MS after a social science PhD, when I’d had 7 years away from math and going on 15 years away from calculus. It was hard as hell, but worthwhile, because of two things: 1) I had enough practical experience doing applied stats during and after my PhD that I was pretty sure I wanted to make a career in the field, and 2) I had the support of my partner, without which the whole thing would have been impossible (especially given that we have a small child). Figuring out your goals and your support team are two important preliminary steps, I think.

Things that helped while I was in the program:

1) This book: Healy, Matrices for Statistics, because I took undergrad linear algebra and it was really not sufficient for understanding my grad regression text

2) Finding some peers of whom I could ask all the dumbass math questions my teachers would’ve kicked me out for asking in lecture

3) Making up extra homework to drill with when the given homework was too hard to be useful (peers were good for this too)

4) In my case, keeping my job while in school helped, because it gave me a place to feel like I knew what I was doing. Knowing how to connect applied questions to methods and code, and knowing how to connect those things to the underlying math, aren’t the same skill, and it’s possible to be great at some but not all of those. Also, getting paid like a grownup is a balm for the soul.

I was a math major and I’m not uniformly good at all areas of math. No one is.

For review, and to learn more about MOOCs, I sampled several online calculus classes. My absolute favorite is Calculus 1 from Coursera taught by a team of excellent professors at Ohio State. I wish I learned calculus this way the first time.

Seriously, if you teach math, you need to watch these master teachers at work. They also produced an online free textbook and an online practice site where you attempt calculus problems. If you get stuck, you can get hints, one step at a time, with links back to the lectures and texts that are relevant for solving that problem. There are also optional programming lessons that give an intro to numerical analysis. Seriously great MOOC, the best math MOOC I have ever seen. (No, they did not pay me to write this.)

I did not sample their Calculus 2. It is by the same team, so I would expect it to be the same caliber.

My math prof used Anton’s Linear Algebra book, but I referred frequently to my roommate’s copy of Strang’s book. When you get to higher math, you may find it helpful to get more than one book because different authors present material with their own twists. Some will resonate more strongly with you than others. You can find older versions of linear algebra books cheaply on used book websites. edX also offered Linear Algebra and you may find those lectures helpful. The class is over, but you can still access all the courseware.

https://www.edx.org/course/utaustinx/utaustinx-ut-5-01x-linear-algebra-1162#.U53CRpRdU7E

If you want to learn Differential Equations, try OpenCouseWare.

http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/

I think this is a really tough question to answer without knowing what is meant by “not great with math”, as others have said. But I am happy to share my experience. I was generally “good at math”, but entered a Stats program with a limited background.

I was always good with math in grade/high school but started slacking a bit while in Calculus I and my AP score didn’t count for college credit. I began as a math major in college, and went through Calc 1 and 2 (again), getting mediocre grades in both. I went on to start the basic foundations/theory courses. But, the 19 year-old in me got the best of me (*groan*), I withdrew, and I switched to Psychology. The main reason being that I didn’t have the attention span to do weekly homework assignments on theory and I thought Psychology was more interesting. I would have switched to Statistics since it was a more applied discipline, but my small liberal arts college didn’t have that option.

I went to graduate school in a traditionally non-quantitative program but wanted to show off some quantitative chops for when I went on the job market. My university/program had relatively liberal guidelines on credits for PhD courses and I applied to the Statistics program as a dual-degree student without having more than Calc 2 (7 years prior) and an intro grad sequence in Stats meant for non-stats-majors. I never took linear algebra formally, and had no real introduction to differential equations.

My first advice is to learn the very basics of something like R before you go into the program (or whatever language they use most for the courses). If you are comfortable with basic functions in R (even just loading data, packages, taking an average, snooping through your data) you will be able to focus on the statistical procedures you are learning, rather than trying to learn both R and the statistics/math at the same time. I had taken a basic introduction to statistics for graduate students sequence that used R throughout, and I would NOT have been able to get through the Statistics program without having that R introduction first. I was able to spend my time on the methods, rather than the programming.

I learned the very basic matrix stuff on the fly–the use of this was limited to the very basics–and Diff Eq wasn’t even mentioned during my courses, really. I had to dig pretty deep and practice a lot with my integration, etc. from Calculus when I got to my theory courses. It was clear throughout that I was well behind my peers in the math background, but I was at or above the class average in most of my courses. And this was in a Top 15 graduate statistics program.

The exceptions were the two required theory courses. I struggled through those (Casella and Berger based). It was really easy for me to follow along with the proofs in class knowing just basic matrix notation and calculus. But without a strong calculus background, there was a lot of staring at the page trying to know where to start when I attempted the HW assignments.

I think this is ultimately a totally different math skill from taking the derivative or integrating a straight forward function like 2x^2 + 5. In a book like Casella and Berger, for example, you’re given a function and have to understand the steps to take to get through the problem. They’ll throw in twists and turns that you haven’t seen before, and the key is recognizing where those come in and how to address them with the algebra and calculus based tools and tricks you learn. But the algebra and calculus itself is relatively basic stuff that is taught in high school. In the theory courses, rather than learning how to do them, you’re trying to identify when to use them.

The program I was in has changed the theory requirements since I was there–they used to require the first 2 courses that the BioStats PhDs took, but now have dedicated MA/MS courses for them. I’m guessing this is to alleviate the types of issues that I had when I was in the program for those not interested in pursuing a PhD in the field. I suspect that if you did a Policy graduate degree and got through their statistics and econometrics courses to the point that you know the basics without serious struggle, you would be in OK shape to get through an MA/MS program in Statistics.

However, I can also say that if I had gone through the full Linear Algebra, Differential Equations, etc. background, I feel like I would have gotten a good bit more out of the program. The part where I think they would have helped is in programming/modifying code for methods I need in my work now, but there aren’t necessarily pre-made packages for. Having that background allows you to think more about the underlying mechanisms and how to address them, rather than the applications alone.

i have never had a satisfactory interaction with a math or stats person: either I can’t explain myself to them, or they can’t explain themselves to me

So, way, way more important then any technical skills – can you talk to the people who need statistics ?

For instance, I do lab biology – mostly test tubes where I add 5 or 6 different liquids to each tube, do some mixing and heating, and measure the color in the tube

Stats peple will say, randomize the order of liquid addition; lab people say, I just can’t do that as a practical matter

result: no communication that is useful

avoid nonsense. About a year or so ago, there was a blog fest about a paper that looked at p values in abstracts, and concluded that most of the p values were truth.

That has to be wrong; so, don’t do stuff that non stats people find really, really, really wierd – or at least, write the paper in language that they don’t understand

Back in 2004 I was applying for a second Masters as well, this time for AI/Informatics at Edinburgh. I also was weak in Math (failed Linear Algebra the first time I took it in college along with some ‘D’s in calc) and had been out of school for a decade. What I did was contact the Math graduate programs in my city (lucky for me it was NYC) to connect with PhD students in the dissertation phase looking for some extra pocket money (Thanks CUNY!). For two hours almost every week over a few months, I was tutored on all of the topics I felt I never really understood – Trig, Logarithmic functions, Calculus, Linear Algebra, Logic (FOL etc). Most of the more advanced topics (to me at least) I took from Russell and Norvig’s AI text. What was great, was that it was at my pace and I could ask all of the deeper questions about how things worked, rather than just focusing in on syntanx/empty string manipulations. Not exactly cheap, but way less than half the cost of just one audited class at Columbia. I also have a couple blog posts that I think cover a lot of the topics you will cover in a machine learning course – some of it is more advanced than you would get at a masters level, but should be a good overview of types of math you will need.

http://conductrics.com/data-science-resources/

http://conductrics.com/data-science-resources-2

Good luck!

Matt

If something mathematical becomes critical in you work – it’s very very hard to delegate – if you don’t _get it_ you are at real risk of being importantly wrong. On the other hand, many things one might learn in mathematics likely will never be important (Andrew’s favourite, I think, is measure theory). I also believe being good in math is not very predictive of being good in statistics – especially applying statistics.

What I found helped the most for me, was to do a two term course serious math course with upper level mathematics majors and interact and compete with them. Try to do one piece of math really well. It happened to be linear algebra and that likely was as good a choice as any. Before and after that, I had audited courses in the mathematics department (preferentially to a statistics department), but after that it was more useful.

With modern computing, I believe mathematics is mostly needed just for learning and communicating to other statisticians (e.g. journal publication) rather than applying statistics thoughtfully and pragmatically. Or maybe I should say the mechanics and manipulations of math as abstract thinking is really important and that may need some mathematical experience to obtain.

These ideas (avoiding math) were related here http://magazine.amstat.org/blog/2013/10/01/algebra-and-statistics/ The same ideas might be helpful to someone trying hard to get the math when they get stuck or want to see more concrete explanations of what the math is actually making possible.

K?: I largely agree with what you wrote. It’s strange what math can do for you, though. The J language Wiki has an example doing word wrap of a text document; they frame it as a transitive closure problem and come up with a simple solution.

So, yes, start with these bits of advice, but, if you do math-related things, learn continually in a range of areas that support what you want or need to do.

BTW, with all the recommendations about linear algebra and programming languages, one might consider simply learning and using a language such as J. You get to do both at the same time, and you see new programming constructs that simplify problems. For example, Hadley Wickham’s split-apply-combine approach in plyr is simply J’s very concise “under” conjunction (&.).

Bill: > It’s strange what math can do for you, though

It can do lot but mainly I was suggesting do one math topic really well and then audit specific topics rather than trying to cover all the math topics that are used in statistics as most people cannot spare the time and energy.

The “you don’t _need_ math” to do statistics any more claim was tempered with “I should say the mechanics and manipulations of math” but it suggests I am dismissing the value of math. What I mean to dismiss is that math is the only way one can do and understand statistics. Even then I have to temper that with maybe math is the only way learn how to think abstractly enough to understand statistics.

Interestingly I did attend a J course by Ken Iverson about using J to teach calculus 1980/90s. It was interesting but likely had little impact. What he said was interesting “I am telling things in this course I needed pointing out to me when I took my first calculus course. That may not apply to you. I don’t know what you need to have pointed out.”

[…] as important to know some programming as it is to know some math. (And that’s relevant to our discussion from the other […]

I don’t see any unambiguous “yes” votes in this thread, so I’ll add one: this sounds like a good plan. Your public policy coursework probably gave you a better conceptual foundation than you think, especially in terms of thinking about the “why” of statistical questions.

I went into a quantitative masters’ program (probably about halfway between the one you already took and the kind you want to do next) after being out of school for ten years, and while I had always been somewhat mathematically minded, it took some adjustment to get back in the groove. I did similar online preparation to what you are describing, although I would echo Andrew’s suggestion that you incorporate some statistical programming into your preparation as well- there are some good online courses on programming in R, for example.

Also…there are really two sides to this question– getting yourself on the level where you will succeed in the coursework, and convincing others that you will succeed. I slipped in the side door by moving from a less quantitative track to a more quantitative track after already starting. You should put some thought into showing your quantitative abilities in a “3rd party validated way,” apart from your individual study on online courses.

Have a look at the mathematics courses from the “Teaching Company”. They are not free, but more “professional” than a MOOC and easier to understand.

There have already been so many great posts, I worry about just contributing general noise. However, the story is so eerily similar to my own I felt compelled to, if nothing else, give a “you can do it too!” In high school I had a math professor suggest that I was math learning disabled and that I would never, and I repeat that he emphasized never, be able to make it past geometry. That was a pretty scarring event in my young life, so much so that I pretty much said I’m going to do something in the social sciences/qualitative space; to hell with math and analysis. I finished undergraduate early (BA Political Science – failed Calculus my freshman year further cementing the haunting words of my math teacher), worked for a year at a law firm (thankfully realizing lawyering wasn’t for me), and then made my way to graduate school. I got a MA in International Relations. During my time in graduate school I had two very formative experiences: 1) I worked for a great group of analysts in DC doing counter-insurgency modeling. They needed someone to do research (read: go to the Library of Congress and dig) and generally build spreadsheets. They had a PhD in Statistics on staff and I spent a good amount of time with her seeing what my research turned into: awesome models. 2) I spent my second year of my MA working with a former World Bank economist. Once again I was doing “research” whiel the big boys and girls did the analysis. Once again, awesome models resulted. I think these two experiences together made me realize, perhaps belatedly, that I wish I had been doing the analysis the whole time.

I graduated in 2008, in the New York area. I was basically at ground zero of the financial crisis with two holes in my head (two political science degrees for the layman). Somehow I got an internship that turned into a manager’s position at an international oil firm. I spent the next three years busting butt making it work, doing a job most MBAs would have been happy to do. I was working 90 hours a week, making good money, and had the trappings of someone important. I wasn’t happy though. I still had this siren’s call of data and analysis rattling around in my head. At the beginning of my third year of work I said… that’s it. If I don’t at least try to get a degree in statistics and analysis I’ll die with regret. So… I somehow took Calc 1, 2, and Linear Algebra while working those crazy hours and juggling a fiancée. I got into a stats program in the CUNY system and proceeded to work my work day, go to class for four hours, and then go back to work till one in the morning – rinse and repeat everyday for two years. I graduated this May with my MS in statistics. I’ve been working for USAID the past year and a half as a Empirical Analyst, creating a system that helps compare countries across USG program areas. I just finished doing a large machine learning project to try to create clusters of policy areas (still deciding if I was successful or not). I also have been working on a paper to try to suss out some of the relationships between economic and democratic reforms by using statistical modeling. Somehow I made it into statistics and doing data analysis that I think is meaningful, despite the teacher who said I would never make it past geometry.

I started my MS in Stats (or at least the journey) at 27 (close to you). I turn 30 this month (so it’s taken me close to four years with all the pre work included). I will be the first to say that I am not strong in math. I have to constantly reteach myself aspects of Calc and Linear Algebra, and re-review models and their assumptions. It doesn’t help that I do social science statistics which is bedeviling to say the least.

Unlike the previous things I’ve done in my professional life though, I am happy re-learning it though. I suppose the parable of this story is that it’s totally doable even if you aren’t strong in math. It’s hard as hell and you will question whether you have the endurance to do it. But if I can do it, I’m certain almost anyone can. You just have to knuckle in. Immerse yourself in it and live and breathe it. Ok… enough of the Tony Robbins schtick. Get out there and do it.

PS. Learn R. She is a beatiful, seductive mistress who will infuriate you to no end. But the results are undeniable.

Upon reflection I didn’t really answer the questions posed. I got much too swept up in the narrative of similarities.

Simply put, I think classroom settings are the best place to learn math. I’m a huge proponent of the MOOC culture (I’ve taken about twelve of them at this writing), but I think learning math is best done in a highly structured environment. You need a place where you can ask questions and work with other students. At least, that’s how it worked for me. As for building a good foundation: read blogs, watch youtube videos, take MOOCs on stats, and always practice.

If you feel you’re not good at maths but are drawn to technical subjects it may be that, like me, you actually do have that kind of way of thinking but missed out on following that path when you were younger. With no maths post-16 I started studying stats at age 28. Now, five years later I’m doing a PhD in statistics and I’m starting to feel happy doing analytical theoretical work on paper as opposed to programming it and hoping for the best. As Andrew said programming is a good way to feel your way into it, but you can pick that up as you go along too.

If you want to follow courses online that’s fine, but don’t think it will all go in first time. You expose yourself to these things repeatedly and then one day you have a messy-looking likelihood that needs differentiating and you decide to give it a go, and that’s when it clicks.

Hi, I have fair knowledge of SPSS and SAS. I want use MATLAB for regression analysis or multivariate analysis. I am a Bio person basically. I want to learn MATLAB, so that it helps me in my career. Is it advisable for me to learn the software being non-math person?