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Math General Mathematics vs Applied Mathematics vs Statistics degree. Differences?

  1. Jun 2, 2012 #1
    What are the differences between these three undergraduate degrees?

    I am contemplating on being an Applied Mathematician. I have a natural desire to solve statistics and probability questions. If I were to come across a real life problem I would try to involve these concepts in solving them.

    However, at the same time I am compelled to teach math at the high school to Grade 12 level. I'm not planning to become a full time teacher, just a part-time teacher and/or private tutor. I fear that taking an Applied Mathematics degree will limit my ability to teach Pure Mathematics at the Grade 12 level.

    What are the things that an Applied Mathematician would need to be competent at? Do I really need at least a Master's degree to become qualified?
  2. jcsd
  3. Jun 2, 2012 #2


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    Hey Alshia and welcome to the forums.

    IMO the distinction is not always a clear cut thing, but the difference will often by the focus and the application.

    Pure mathematics is often concerned with a focus of abstraction, rigour, and proof of the fundamental mathematical frameworks like analysis and logic.

    Applied mathematics is an applied endeavor which uses a lot of the methods that pure mathematicians have invented and verified to be true or "close enough" for practical purposes to either completely solve real world problems, or give insights, approximations, and advice for such problems if exact solutions can not be found.

    Statistics is mainly used for understanding anything under uncertainty, and usually it translates into making sense of data for uncertain processes. Statistics has theoretical results which deal specifically about proving results in the context of uncertainty, and a good way to distinguish theoretical statistics from pure mathematics is that statistics is concerned largely with dealing with functions of random variables and samples of data to achieve statistical understandings by producing 'statistics' (which are functions of a sample).

    There are many cases where statistics and applied math coexist and personally I would consider statistics as applied mathematics anyway.

    Also one thing you should be aware of is that there is difference often in who you work for given your focus.

    Applied people will tend to work in situations where they work in an advisory capacity to people with non-technical understanding and backgrounds (or at least in a limited way). As a result you will typically have to be able to tell people your results in a way that they can not only understand, but more importantly appreciate.

    Pure mathematicians will publish in journals and can and will publish things using mathematical jargon since the focus and the nature of the audience/community is a lot different.

    If you want to be a competent applied mathematician, I would recommend you become a good communicator, know and understand mathematics of all kinds in your particular focus to a high standard, understand the limitations of models, assumptions, and methods that are used and know when they should not be used, and finally get a feeling to understand the different domains you will work in both specifically for each domain, and generally for all domains.

    If you do a degree you will get a bit of this training, but more will be acquired later on.
  4. Jun 3, 2012 #3
    Hey chiro and thank you for the excellent response (from what I can gather).

    If an undergraduate program is simply called 'Mathematics' does it necessarily imply that it is a Pure Mathematics program? I used the term 'General Mathematics' because in some Mathematics program (like MIT, from what little I've read) it says that students will be speciallizing in General, Pure or Applied Math in the final years, and if a Mathematics program is already a Pure Mathematics program from the start it makes no sense to specialize in Applied Mathematics.

    Secondly, the issue is that I need to be able to prove that a formula works (which is what much of Pure Mathematics is about) to convince myself that it's true. But at the same time I am compelled to apply the knowledge to solve tons of practical real world problems.

    Based on my past experience with how Mathematics is taught (all below college level by the way), if one takes some kind of applied course, the proof for the formulas and methods will not be studied. This will stunt my mathematical development because I need to understand the formulas and method to their very core to appreciate them.

    Thus, I am hesitant about going into an Applied Mathematics program right off the bat. I would rather apply for a General Math course before specializing.

    My default choice is to go into finance and other business-like industry, as the compensation will likely be more than satisfactory (just an intuition, correct me if I'm wrong.) However ideally I would like to involve myself in social science settings, doing research and applying the results to create innovative products for people.

    I do not mind learning how to communicate my methods in laymen's terms. I enjoy conveying what I've learned or discovered (hence why I also wish to teach part time). Your description of the criteria of a good applied mathematician is exciting to me. Noticing assumptions, understanding models, understanding limitations of each model and therefore knowing which model to apply in what situations...these are things in math that I live for.

    For example, there's this obscure investment which my parents are involved in; having no background in finance, they do not know how to calculate the gross profit and total capital invested, and thus they don't know how to evaluate the net profit. Having no financial background myself, I simply used the geometric series to solve the problem. That was a truly satisfying conundrum.
    Last edited: Jun 3, 2012
  5. Jun 3, 2012 #4
    Accidentally double posted due to technical error.
  6. Jun 3, 2012 #5


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    As far as my experience and understand goes, you will pick a specialization for mathematics that roughly corresponds to pure, applied, statistics or some combination thereof (for your information I am a math major).

    All mathematics majors will have to do the fundamental subjects of the calculus series and will have to gain at least an exposure to the other fields of non-specialization at some level.

    The major will concentrate usually on getting exposed enough to the fundamentals of the specialization.

    In statistics this means doing a year long intro university sequence in probability and statistics and then later doing sequences in inference, regression modelling, experimental design, and applied probability.

    For pure mathematics, this concerns getting a solid grounding in analysis, topology, and algebra at differing levels (pure mathematics like every field has its own unique specializations). These subjects have their own issues, but these form the core of the pure mathematics in many respects.

    Applied carries in a lot of domain dependencies, but there are non-domain subjects like optimization, numerical analysis, dynamics and chaos theory, computational issues in respect to applied mathematics and general mathematical modelling. You will also look at the 'applied results' that have been worked out which include differential equations, integral transforms, and other similar kinds of things in an applied context. Integral transforms can be thought of as in terms of decomposing things into independent characteristics and then using those for whatever purpose. Signal processing is just really a huge application of this kind of thing.

    For the second issue, this will depend on your focus. You can, even in an applied context understand enough about why something works without knowing the specifics, and a good teacher should be able to tell you this kind of thing, even an applied one. Many applied programs and sciences often have hand-wavy arguments for things that are usually good enough for the purposes of that particular focus.

    It's really up to you to establish what the focus is: this will help you decide specifically where you wish to end up eventually. Also remember that the lines between pure, applied, and statistics get blurry so don't think that you have to belong to only one or even two of these groups. Also real world problems are, by their nature, applied however lots of things that have a pure designation (like number theory) are still applied in some form. You should also note that typically what happens is that pure mathematics ends up being applied, and Terry Tao's comment was that the delay is about 50 years. With the way things are going now, I wouldn't be surprised this reducing to 10 or even 5 years in the future (possibly being nearly instantaneous at some point).

    With regards to proofs and such, I think that yes for many purposes this is fair characterization in one way or another. Again though, you have to be aware that the focus for each stream is different and there is a reason for this: each career has different goals in mind and usually what works is that if you need something complex done, you typically get people all with different focii to work together to solve something that typically one person could not do alone, even if they were competent in all areas. Think of it as a jack of all trades, master of none times n people vs n people who are really good in something distinct working together.

    There are advantages and disadvantages of the two approaches and ultimately it depends on the task at hand, but eventually this is the way it is because there is a tendency for specializations working together to do complex things being the way to go.

    Also you need to think about the kinds of problems we have nowadays: we are designing computers with god knows how many transistors, design issues, physics issues and so on as well as jets, bridges and everything else and this is why specialization is required by default because the spectrum of what is needed to know knowledge and experience wise is huge and also very specific and subtle.

    For your comment about social science, you should note that human beings are scientists by default. Most of us don't quantify our experience, place strict controls on the apparatus, or experimental conditions and write papers for journals, but we are still kind of 'naive' scientists at heart (I use the analogue of 'naive psychologist': if you haven't heard of the term, it's based on a psychological theory in the early days of psychology).

    People that have a lot of experience in a particular area are taken seriously and sometimes in some areas and industries, experience is a necessary collateral to be taken seriously and academia is actually in many cases not taken seriously.

    So you don't necessarily need to be a scientist by formal designation to get experience and use that either in an advisory or speculative capacity: people do this all the time and they make all kinds of decisions about the direction of projects or even the direction of entire companies: their collateral for doing this is mostly based on experience and this is a natural thing across the board, and it should be intuitive to understand why this is the case.

    If you want to communicate things in laymans terms, you are going to be a good applied scientist. If you end up working for a business or doing consultancy, this is definitely a requirement for the job.

    The people that want applied mathematicians want them to solve real world problems, or at least give a structured set of advice for solving them (typically decision makers end up choosing how to solve it, but because of their scarce time and importance for decision making, they get analysts and other people to do the grunt work). Definitely I think you are in the right area given this statement.
  7. Jun 3, 2012 #6
    Thank you for your response. I think I can compensate by studying the proof before attending any applied lecture, so that I do not get annoyed when the proof is not provided. Not giving the proof to me is like asking me to use the quadratic formula without knowing how to derive it; it annoys me.

    By the way, is there a path for a mathematics undergraduate degree holder to go into actuarial science?
  8. Jun 3, 2012 #7


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    There are degrees that are 'acturial science' degrees that cover all the specific coursework to prepare you for the exams but I don't recommend you do this because they are way too specialized and don't leave as many options for non-actuarial pathways as another major would.

    If you are interested in this, you should know that you can sit the exams externally with your local actuarial society (local means for your country), and if you pass the exam then that's all that matters.

    I would if I were you, do a mathematics degree with a statistics major, with a financial math course based on stochastic calculus, a few economics subjects, a few programming subjects and maybe an accounting subject.

    If you understand the statistics properly, then the actuarial stuff won't be too bad because it is going to be applications of probability and statistics to insurance and financial applications, and this is what you will be assessed on in the exams. In fact it's a lot better that you understand probability and statistics properly so that if you are thrown a curve-ball, that you won't completely be unprepared.

    By doing this, you will have the prerequisites to study for the exams, and if things don't pan out, then you can take your qualifications and apply for other non-actuarial specific roles.

    Also with actuarial, the understanding that you would 'like' to have in actuarial science is not always going to be there because you will have to remember a lot of techniques and information for this (you will also have to take in an acceptable book just for the exams: there are lots of things): think of it a little bit like engineering where you need to know so much stuff without really understanding everything completely.

    So in this sense, at least for your purposes of understanding, it would be better if you did a math degree with the above recommended extra courses, and then when you go to do actuarial study you will have the background to have the proper understanding or at least know how to obtain it using the knowledge-base you have already built.

    You can also obtain the syllabi and some past exams on your local actuarial societies website, and this is probably a good thing to do sooner rather than later so you become aware of what you will have to do in order to pass the exams, and understand the material.
  9. Jun 4, 2012 #8
    Yup. Of the actuaries & actuarial analysts I work with, the majority have a BS in math. I believe this is also true of US actuaries in general, though I don't have any hard data on it.
  10. Jun 6, 2012 #9
    Thank you, chiro(practor), for your response.

    ...do a mathematics degree with a statistics major, with a financial math course based on stochastic calculus, a few economics subjects, a few programming subjects and maybe an accounting subject.

    Can you point me to some resources that tells me exactly which financial math/economics/programming/accounting subject I should take?

    I am not fond of Accounting though...I did it in high school. Once I understood the system it became a bore. I intend to study programming by myself some time in the future. I also intend to study economics (by myself or through the undergraduate course).

    Also with actuarial, the understanding that you would 'like' to have in actuarial science is not always going to be there because you will have to remember a lot of techniques and information for this...

    Hmm. But many of these practical techniques are offsprings of the more general concepts in mathematics, are they not? So if I possess the knowledge and understanding of the core ideas and have some background in proving mathematical equivalences, then I can derive them, or at least understand future mathematical applications.

    Three more questions >.<.

    Which path (applied math vs actuary) has higher financial compensation on average, also accounting for the additional exam I need to take for actuarial positions?

    Which path offers more opportunity to learn about things that can lead to the possibility of self-employment? (investment, market trends, etc)

    Which path has less working hours on average? I prefer not to work more than 10 hours, though if I have to work 12 hours once a week I would not have too much problem with it.
  11. Jun 6, 2012 #10


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    Take a look at this if you are in the US (if not look at your local actuarial society website for similar information):


    It will probably leave you better off if you can get VEE because it means you will get credit for these kinds of exams and get them out of the way while allowing you to concentrate on the more mathematical exams.

    You will have to do some accounting and corporate finance subjects, but you won't by any means become an accountant.

    They are based on sound mathematical principles, but the point is that you are going to drown in techniques that are rather specialized applications of statistics to insurance-type applications.

    Because of this, you may not get the luxury of having the deep understanding all the time at every point in a way that you might otherwise like to. I'm not saying you won't understand it, but again given the volume of what you will have to remember, there will be points that you will just have to learn what you need to without having a chance to ponder those deeper thoughts.

    This is another reason why doing a math degree with a statistics major from a solid statistics department is a good idea because these things can settle down (concepts, knowledge etc) so that when you see actuarial stuff, you will see that it is just another application of mathematics for insurance, and you won't be scared off by the volume off stuff including the formulas, concepts and problems that you will have to deal with.

    For 1., I don't know the specifics but on average if you make it to be a fully qualified actuary, you will probably earn about 200K+ after 10 years of being qualified and this is a highly conservative figure not an optimistic one.

    The thing you have to keep in mind though is that a) lots of people hear about this and think they can get an easy job as an actuary which has its own effects and b) not many people that start the path to become an actuary ever actually finish the exams. For the accounting, look at the VEE (or similar arrangements if not SOA), or just get the exam syllabii and study materials for that subject and do it in your own time, and sit the exam externally (you will need to pay a fee for sitting the exam and getting it processed).

    For 2., if you want to learn self-employment I recommend you get a job in an area of your desired future self-employment and learn the ropes there.

    Taking courses in business subjects like economics, accounting, finance and so on is nice and provides good learning, but I don't recommend doing this for starting a business or becoming self-employed.

    If you can find work in a good business that gives you exposure to as many sides of the business as possible, then you will have a very good background for creating your own business and for self-employment.

    The accounting and finance stuff for a small business won't really be the hard parts: the hard parts will be knowing the business, industry, customers, and everything else inside out. The other thing is that you will have to build up a reputation, and this is made a lot easier if you have solid prior experience as an employee vs if you just decided to start a business without a decent background.

    If you want to become an entrepreneur, you will never ever stop learning period. You will be working absolutely flat out and it will consume you. You will not get any benefits of any kind and you will be working for minimum or even under minimum wage given the number of hours you will be working. You will most likely be working seven days a week for many many weeks in a row.

    There are so many things to say about this, but I will leave you with one piece of advice if you are considering this: get into a solid business as an employee where you can rotate around various business units/departments and get as much exposure to all facets of the business as possible. This is not an easy thing to do but if you can do it, then this will be a good training exercise for you. After years of this (At least 5 but probably closer to 10 or more), then you will be able to decide whether the self-employment is right for you.

    For 3. if you want to become an entrepreneur, then don't if you want this. Entrepreneur's don't stop working and they work these kinds of hours every single day of the work.

    For other jobs like actuarial you will be working crazy hours for every year that you are studying for the exams, which means that you will have to do this until you qualify in which time you can take a breather.

    More money usually translates into hours, experience, stress, responsibility, crappyness of the job, control or nature of regulation (medical doctor), amongst other things. Unless you are the presidents son, or will inherit the family business, then you will have to work long hours in some respect for more money.

    There will be a point though when you build up a reputation, experience, and so on where you can step back and take a breather and not have to work as hard (but still work hard), but this definitely won't come straight away.
  12. Jun 8, 2012 #11
    Maybe some of them. There are many things that appear at the job and in the exams that you cannot derive. Some examples:

    You can’t derive what’s in State and Federal regulations.

    You can’t derive the Actuarial Standards of Practice (ASOP).

    You can’t derive the code of professional conduct.

    You can’t derive the political impact that results from actuarial judgment, either internally or externally.

    You can derive methodologies, but you can’t derive which one your employer, the State government, or the Federal government actually uses.

    A strong math background does help with the preliminary exams, for sure.
  13. Jun 8, 2012 #12
    For detailed salary statistics for North American actuaries, google “Ezra Penland Salary Survey”. DW Simpson has one, too.

    Average salary is a poor measure of financial compensation. A salary distribution can have a long tail, yet almost everyone can make less than the average. The reverse is true, too.

    So are you average? Imagine two professions, A and B. The distribution of salaries in A has a very low coefficient of variance, while B has a long tail. Even if A has a higher average pay, if you happen to be above average, B may have a better expected salary for you if it’s providing opportunities for salary advancement that A does not. If you’re below average, the reverse is true: go for the tight salary distribution with the higher average pay.

    Average salary statistics also ignore the time value of money. An actuary can earn a smaller salary than a PhD and still have superior lifetime earnings due to the ability to work while learning. (Note: Math PhD’s very rarely make more than Actuaries).
  14. Jun 9, 2012 #13

    “Take a look at this if you are in the US (if not look at your local actuarial society website for similar information):”

    Thank you.

    “...there will be points that you will just have to learn what you need to without having a chance to ponder those deeper thoughts.”


    “...so that when you see actuarial stuff, you will see that it is just another application of mathematics for insurance.”

    Yes, that is what I wish to see when examining mathematical applications.

    “Entrepreneur's don't stop working and they work these kinds of hours every single day of the work. “

    I don't mind working long hours for myself, but I cannot bear doing the same for others. It's similar to how I can spend more than 10 hours studying something not in my school syllabus, but I cannot study for exams for more than 6 hours. I guess it's the perceived free will.

    Of course, as you pointed out, more hours as an employee means more compensation, but this is not necessarily true for the self-employed. I shall keep that in mind.

    Thank you for all the other information you've given me; it's pretty enriching. You seem to have an extensive knowledge on this. What is your current profession?


    Thanks for the examples and salary information.

    Yes, that's right. The salary distribution may be negatively skewed, with a long tail to the right, and it may stretch to 4-6 standard deviations away from the mean, unlike a normal distribution where the maximum stretch would be 3 standard deviations.

    Not to mention, the normally distributed average salary may be higher than the skewed average, but in return have a significantly smaller standard deviation, which limits the amount of salary increment.

    Not to mention the (probably unlikely) case where salary is bimodally distributed O.O.

    Care to refer me to some resources on calculating the time value of money? I did learn about Net Present Values (how much money gained in the future is worth currently), but nothing more than that.
    Last edited: Jun 9, 2012
  15. Jun 11, 2012 #14
    I think (but don't know for certain!) that a lot of entry level salaries are bimodal. Someone posted a graph of entry level lawyer salaries in the past and it had two clear peaks. I figure you have the Ivy League/Top 10 peeps creating a bump to the right, and everyone else creating a bump on the left. Maybe a better explanation would have to do with the choice of work, with one group working longer hours than the other.

    It would be interesting to see how pay per hour were distributed and compare.

    Just random thoughts. . .
  16. Jun 11, 2012 #15
    Just a short note: in the big picture, it may not be such a big deal if you choose one of those subjects and then change your mind later. As several posters said, the lines between these subjects are often blurry. Some examples:
    • My undergrad degree is in pure mathematics. Now I'm writing a physics thesis using math I learned from the statistics department at another university.
    • Chaos theory researchers are in the Applied Math department at some universities and the Physics or Engineering departments at others.
    • Many of the job applications I see ask for "statistics, mathematics, physics, EE, or other quantitative degree."
    • David Hilbert was a mathematician until about the age of 50, when he hired a physics tutor. A few years later, he was publishing important papers on general relativity and quantum mechanics.
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