Do the math: too much calculus? (2012)

The problem with statistics education is that it is generally heavily watered down, particularly the classes at the university level for business, psychology, and other majors that need the knowledge but aren't math or engineering majors. People heavily misunderstand the usage, difficulty, and nuances of stats education if they are only exposed on this level. It's better than nothing, but leads to serious issues that remind me a bit of the novice arrogant programmer vs. the grizzled veteran smart senior programmer. Generally, most people I see with limited stats education come out of it thinking mean, median, normal distributions, and at best linear regression are the whole of stats.

Another problem that also related to the parent with regard to stats vs. calculus is that calculus is actually a pre-requisite for most useful statistics work. It is true there are plenty of topics in stats you can cover without knowing calculus, but you are at a severe disadvantage and won't really understand the "why" and just be regurgitating the memorized method. I feel a lot of statistics is a combination of the "why" and "how" because so much of it is applying math at a more discretionary and opinionated level, like is done in science. Really learning stats requires a lot of math, from advanced algebra to diff eq possibly to matrices to calculus. To do good work in stats, you need a lot of tools to draw from, otherwise you're stuck with tools that often don't work (common example is people who only are familiar with normal distributions and nothing else).

Yet another problem with stats is that you can really make up whatever you want when you conduct statistical applications such as an analysis. The rule is that as long as you provide justification and proof for what you are doing - i.e. how and why, it is acceptable along with some text to explain. This is why good statisticians do things like describe and reveal their sampling methods, explain what did and did not work, and why they selected various tools. Obviously most dubious work can be quickly disputed and identified by someone who knows what they are doing, but the reality is that so many people never follow-up, fact check, or even read anything but the conclusion.

Stats is in so many ways about showing the process and justifying your answers than the quantitative answers. This is similar to financial forecasts as well for example which also use a lot of applied math, and people just believe what they read even though the evidence to disprove the report is right there. The science comparison applies here too. Just like with research studies, a lot of people don't check, verify, and re-run the results so people just believe what they read or rubber stamp it. I recall I had a very mean professor that once gave a final exam question that actually could not be solved at all using the methods we learned because the types of analysis and models we were asked to try to apply were invalid for the data. The simplest version of this familiar to many people is trying to create a best fit linear equation to non-linear data - simply draw it on paper against a data plot and you'll see it makes no sense.

If there's one important thing I learned in stats, it's to be skeptical and prove things with evidence. Nearly every time I watch the news, read an article, etc., I go nuts because they are making huge conclusions without giving you a way to understand who, what, where, why, and how about the study. The presidential elections in the US are an obvious example I see a lot with things like "Survey Monkey" passed off as scientific data in some contexts. Even represented as a casual, unscientific process, such things can be dangerous because people still accept statistics at face-value usually.

In summary, yes, people should learn statistics but it requires a lot of hard work to develop the tools and analytical skills to do it right. Like calculus, it can require many courses which can equate to years of study. Like anything, the skills involved also tend to get better with time and increased knowledge. At least an extra course or two beyond what is taught now would go a long way, but people really need to emphasize more than stats as repeating some processes of functions, formulas, proofs, and such, and more as a way of thinking and set of tools. Stats should teach you to be very paranoid and skeptical, and verify what you see in the world around you as best you can before you believe it. Once you do that, like the parent says, you will see so much clearer what is true, false, dubious, could be better, etc. in the world around you.

I TA'ed physics at one of the top Ivy League universities in the US, and what I found amazing was that we didn't require calculus for our introductory mechanics course. You don't get into these universities without taking all the highest level courses in high-school, which means that almost everyone in the class must have taken at least one semester of calculus. But we still avoided the simple v = dx/dt calculations, the logic being that physics was too difficult to combine with the math they'd all learned years before.

My friends in economics told similar stories: "Don't worry, we don't have to use calculus to calculate marginal cost, we can use 'the midpoint method'", followed by 20 minutes of explaining an arcane nondeterministic procedure to approximate a derivative.

Math should be motivated by science and modeling, otherwise it's just an exercise in diddling numbers.

The huge problem as I mentioned elsewhere is that stats is really applied math. You do learn a lot of regurgitation-style stuff - formulas, proofs, blah blah, but it's all useless without applying it. Moreover, stats draws from so many different areas in math and requires analytical skills that are closer to many of the things scientists emphasize in learning to conduct studies and experiments. This makes it almost unsuitable to teach the more valuable parts of stats at lower levels because the students can't possibly be prepared.

At best, I think you can teach some general things about stats and the mentality I described of being analytical and skeptical. Unfortunately, stats is just really hard for anyone to properly and comprehensively learn who isn't going to be able to invest a lot of time both learning pre-requisites and then all the different areas of stats. It's like learning to be a carpenter and only understanding how to work a hammer, but not a saw, measuring tape, or anything else.