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But analysis and linear algebra are at the bedrock of statistics. Otherwise, how do you even do a line of best fit?

eesmith 5y ago
I don't know why you think analysis is the bedrock of statistics. I'll explain my understanding; perhaps that can help you clear up my confusion? "Analysis" has different meanings in mathematics than in general use. When you wrote "or Calculus / Analysis", I assumed you meant "the branch of mathematics dealing with limits and related theories" ( https://en.wikipedia.org/wiki/Mathematical_analysis https://en.wikipedia.org/wiki/Mathematical_analysis ). The real analysis course I took as an upper-level undergrad focused on the fundamentals of calculus. I did a lot of ε/δ proofs that semester. Statistics does not require that sort of analysis, so it cannot the "analysis" you refer to. "Analysis", less mathematically speaking, means "the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it" (quoting https://en.wikipedia.org/wiki/Analysis https://en.wikipedia.org/wiki/Analysis ). That goes back to at least Euclid ("Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth") (Ibid.) Historically, mathematics education taught Euclid's Elements - geometry - as a way to learn analysis. Lincoln, for example, "kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight"." (quoting https://en.wikipedia.org/wiki/Euclid%27s_Elements https://en.wikipedia.org/wiki/Euclid%27s_Elements ). Clearly, "analysis" doesn't require calculus, and surely can be taught independent of calculus. Indeed, Fermat and Pascal laid the foundation for statistics before Newton was a teenager. "Linear algebra" is also a very broad topic. When I learned to solve a problems like {given x + 2y = 5; 3x - y = 12; solve for x and y} in 9th grade, I was doing linear algebra, though it wasn't specifically told it was linear algebra. When I learned about eigenvalues and eigenvectors in college course actually named "linear algebra", it of course covered things in much more depth. But why does an introduction to statistics course requires knowing about eigenvectors, null spaces, and orthonormal basis? I mean, I vaguely remember my college statistic course, and it didn't require most of the mathematics I learned from linear algebra, real- and complex- analysis, or group theory? What I remember most strongly about that course was Bayes' theorem. That's such an important concept yet it wasn't taught in my math courses until taking a required class mostly for stats and math majors. And yet Bayes' theorem doesn't require analysis or linear algebra, and neither does it entail eye-balling lines SPSS memorization. At https://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum/Mathematics-and-statistics/Achievement-objectives#collapsible8 https://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum/M... you see some of the goals for teaching statistics to secondary school students in New Zealand: > Carry out investigations of phenomena, using the statistical enquiry cycle: > determining estimates and confidence intervals for means, proportions, and differences, recognising the relevance of the central limit theorem using methods such as resampling or randomisation to assess the strength of evidence. > critiquing causal-relationship claims / interpreting margins of error. > calculating probabilities of independent, combined, and conditional events > applying distributions such as the Poisson, binomial, and normal. Clearly it is possible to have meaningful statistics education without calculus and linear algebra as prerequisites, so I don't understand your objections at all.
- threatofrain 5y ago
  It appears you may be from NZ? I meant to summarize Calculus and Analysis as basically the same subject matter. Mathematical Statistics definitely requires Calculus / Analysis as pre-reqs (many textbooks make zero attempt for review) and Linear Algebra is a well-defined course at the undergraduate level, even if every other university has its own 20% difference on what it chooses to emphasize. Linear Algebra at the undergraduate level can be considered a replacement for Euclidian geometry; it is the formalization of that space. As for the actual course of Geometry in the US, it is an isolated anomaly in the HS to undergrad curriculum in that it is never really touched again. Its proof-heavy style and construction from geometric primitives will not be continued upon. If you mean Statistics as in the introductory undergraduate stats course, that would murder most children's interest because it involves memorizing distributions as collections of facts, and it most definitely doesn't involve understanding cause vs effect. And Stats AP is generally not trusted by universities. I talked about linear fitting as an example for data science, and I pointed out that without understanding Linear Algebra, linear fitting means to use your eyes and draw a line, and the "linear" part will literally mean a line. And what is the difference between a correlation and a linear fitting?
  - eesmith 5y ago
    By "Mathematical Statistics" you mean a subset of statistics, yes? Because I pointed out several areas of statistics which can be taught without calculus or linear algebra, like Bayes' Theorem. "If you mean Statistics as in the introductory undergraduate stats course ..." While I mentioned my being in an undergraduate stats course, the LA Times piece and most of the references I gave refer to secondary education, not tertiary. So, no I don't mean that. The AP Statistics test in the US doesn't require calculus. "Recommended Prerequisites: A second-year course in algebra" at https://apstudents.collegeboard.org/courses/ap-statistics https://apstudents.collegeboard.org/courses/ap-statistics , suggesting it's "College Course Equivalent [is] A one-semester, introductory, non-calculus-based college course in statistics". To double check, https://tb2cdn.schoolwebmasters.com/accnt_42975/site_42976/Math-Course-Catalog-Harrison.pdf https://tb2cdn.schoolwebmasters.com/accnt_42975/site_42976/M... shows Statistical Reasoning with Algebra 2 as a prerequisite, and AP Stats require only pre-calculus. https://www.pike.k12.in.us/userfiles/15934/my%20files/2018-2019%20academic%20planner%20and%20course%20description%20book%20-%20updated%20december%202017.pdf?id=6960 https://www.pike.k12.in.us/userfiles/15934/my%20files/2018-2... requires Algebra 2. Which means statistics is already being taught in high schools, without calculus or linear algebra. Do you really think those courses are 'murdering' the interest of most students any more than any of the other classes they take? The argument, quite clearly made by the math education advocates in California, is that the traditional math education progression is just that - tradition. The pieces can be put together in different ways and still give a good education, only with a balance which is more appropriate for more people. I find knowing about Bayes' Theorem to be much more useful than integration by parts, but the former was never taught in my high school. And I haven't used the latter in over 20 years. No, a high school stats course isn't going to be rigorous Mathematical Statistics. But then again, a high school calculus course also isn't rigorous Analysis. In high school I used cosine and log functions on my calculator, without knowing how they were computed. I certainly couldn't have done it by hand outside of a few well-known special cases. From what I infer looking at things like https://www.northernhighlands.org/cms/lib/NJ01000179/Centricity/Domain/107/Linear%20Regression.pdf https://www.northernhighlands.org/cms/lib/NJ01000179/Centric... , the goal of stats education is to understand how to use and apply a tool like linear fitting, and what concepts like "correlation" and "residual" mean, not the full underlying mathematical treatment. Entirely appropriate for high school, yes? Otherwise, when and how should students learn that sales prediction, height prediction, etc. are not esoteric concepts but things they can do themselves?

I don't know why you think analysis is the bedrock of statistics. I'll explain my understanding; perhaps that can help you clear up my confusion? "Analysis" has different meanings in mathematics than in general use. When you wrote "or Calculus / Analysis", I assumed you meant "the branch of mathematics dealing with limits and related theories" ( https://en.wikipedia.org/wiki/Mathematical_analysis https://en.wikipedia.org/wiki/Mathematical_analysis ). The real analysis course I took as an upper-level undergrad focused on the fundamentals of calculus. I did a lot of ε/δ proofs that semester. Statistics does not require that sort of analysis, so it cannot the "analysis" you refer to. "Analysis", less mathematically speaking, means "the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it" (quoting https://en.wikipedia.org/wiki/Analysis https://en.wikipedia.org/wiki/Analysis ). That goes back to at least Euclid ("Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth") (Ibid.) Historically, mathematics education taught Euclid's Elements - geometry - as a way to learn analysis. Lincoln, for example, "kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight"." (quoting https://en.wikipedia.org/wiki/Euclid%27s_Elements https://en.wikipedia.org/wiki/Euclid%27s_Elements ). Clearly, "analysis" doesn't require calculus, and surely can be taught independent of calculus. Indeed, Fermat and Pascal laid the foundation for statistics before Newton was a teenager. "Linear algebra" is also a very broad topic. When I learned to solve a problems like {given x + 2y = 5; 3x - y = 12; solve for x and y} in 9th grade, I was doing linear algebra, though it wasn't specifically told it was linear algebra. When I learned about eigenvalues and eigenvectors in college course actually named "linear algebra", it of course covered things in much more depth. But why does an introduction to statistics course requires knowing about eigenvectors, null spaces, and orthonormal basis? I mean, I vaguely remember my college statistic course, and it didn't require most of the mathematics I learned from linear algebra, real- and complex- analysis, or group theory? What I remember most strongly about that course was Bayes' theorem. That's such an important concept yet it wasn't taught in my math courses until taking a required class mostly for stats and math majors. And yet Bayes' theorem doesn't require analysis or linear algebra, and neither does it entail eye-balling lines SPSS memorization. At https://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum/Mathematics-and-statistics/Achievement-objectives#collapsible8 https://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum/M... you see some of the goals for teaching statistics to secondary school students in New Zealand: > Carry out investigations of phenomena, using the statistical enquiry cycle: > determining estimates and confidence intervals for means, proportions, and differences, recognising the relevance of the central limit theorem using methods such as resampling or randomisation to assess the strength of evidence. > critiquing causal-relationship claims / interpreting margins of error. > calculating probabilities of independent, combined, and conditional events > applying distributions such as the Poisson, binomial, and normal. Clearly it is possible to have meaningful statistics education without calculus and linear algebra as prerequisites, so I don't understand your objections at all.

threatofrain 5y ago
It appears you may be from NZ? I meant to summarize Calculus and Analysis as basically the same subject matter. Mathematical Statistics definitely requires Calculus / Analysis as pre-reqs (many textbooks make zero attempt for review) and Linear Algebra is a well-defined course at the undergraduate level, even if every other university has its own 20% difference on what it chooses to emphasize. Linear Algebra at the undergraduate level can be considered a replacement for Euclidian geometry; it is the formalization of that space. As for the actual course of Geometry in the US, it is an isolated anomaly in the HS to undergrad curriculum in that it is never really touched again. Its proof-heavy style and construction from geometric primitives will not be continued upon. If you mean Statistics as in the introductory undergraduate stats course, that would murder most children's interest because it involves memorizing distributions as collections of facts, and it most definitely doesn't involve understanding cause vs effect. And Stats AP is generally not trusted by universities. I talked about linear fitting as an example for data science, and I pointed out that without understanding Linear Algebra, linear fitting means to use your eyes and draw a line, and the "linear" part will literally mean a line. And what is the difference between a correlation and a linear fitting?
- eesmith 5y ago
  By "Mathematical Statistics" you mean a subset of statistics, yes? Because I pointed out several areas of statistics which can be taught without calculus or linear algebra, like Bayes' Theorem. "If you mean Statistics as in the introductory undergraduate stats course ..." While I mentioned my being in an undergraduate stats course, the LA Times piece and most of the references I gave refer to secondary education, not tertiary. So, no I don't mean that. The AP Statistics test in the US doesn't require calculus. "Recommended Prerequisites: A second-year course in algebra" at https://apstudents.collegeboard.org/courses/ap-statistics https://apstudents.collegeboard.org/courses/ap-statistics , suggesting it's "College Course Equivalent [is] A one-semester, introductory, non-calculus-based college course in statistics". To double check, https://tb2cdn.schoolwebmasters.com/accnt_42975/site_42976/Math-Course-Catalog-Harrison.pdf https://tb2cdn.schoolwebmasters.com/accnt_42975/site_42976/M... shows Statistical Reasoning with Algebra 2 as a prerequisite, and AP Stats require only pre-calculus. https://www.pike.k12.in.us/userfiles/15934/my%20files/2018-2019%20academic%20planner%20and%20course%20description%20book%20-%20updated%20december%202017.pdf?id=6960 https://www.pike.k12.in.us/userfiles/15934/my%20files/2018-2... requires Algebra 2. Which means statistics is already being taught in high schools, without calculus or linear algebra. Do you really think those courses are 'murdering' the interest of most students any more than any of the other classes they take? The argument, quite clearly made by the math education advocates in California, is that the traditional math education progression is just that - tradition. The pieces can be put together in different ways and still give a good education, only with a balance which is more appropriate for more people. I find knowing about Bayes' Theorem to be much more useful than integration by parts, but the former was never taught in my high school. And I haven't used the latter in over 20 years. No, a high school stats course isn't going to be rigorous Mathematical Statistics. But then again, a high school calculus course also isn't rigorous Analysis. In high school I used cosine and log functions on my calculator, without knowing how they were computed. I certainly couldn't have done it by hand outside of a few well-known special cases. From what I infer looking at things like https://www.northernhighlands.org/cms/lib/NJ01000179/Centricity/Domain/107/Linear%20Regression.pdf https://www.northernhighlands.org/cms/lib/NJ01000179/Centric... , the goal of stats education is to understand how to use and apply a tool like linear fitting, and what concepts like "correlation" and "residual" mean, not the full underlying mathematical treatment. Entirely appropriate for high school, yes? Otherwise, when and how should students learn that sales prediction, height prediction, etc. are not esoteric concepts but things they can do themselves?