In high school statistics, why does it seem like equations come out of the sky

This is a serious observation from my experience taking AP Statistics. We are given a bunch of unexplained formulas and are expected to make sense of them in real world applications without knowing why they hold. In the algebra classes, I can at least bridge my number sense intuition with the materials to construct an equation, but in statistics we don't use such reasoning to arrive at equations, rather the equations are given and we have to think of them in an applications based context. When I ask my teacher he doesn't seem very receptive to the idea of explaining the equations from pure common sense, instead he talks about what the symbols mean.

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I guess you have a bad teacher then. –  Arthur Oct 18 '13 at 18:39
Could you give some examples of equations that seemingly have no context/theory to support them? –  The Chaz 2.0 Oct 18 '13 at 18:40
There's quite a difference in required math skill level between (a) applying the $\chi^2$ method to a given data set and (b) deriving the theory behind $\chi^2$ tests including insights why we require that each observation should be expected to occur a least about 5 times in order to make the test valid and where that degrees-of-freedom stuff comes from. –  Hagen von Eitzen Oct 18 '13 at 18:45
As an analogy- first/second graders are taught addition with carryover as a black-box algorithm but are not really told 'why' it works as they may not have the mathematical maturity to understand the 'why' part. Sure, there are exceptional students who may but the teacher usually has to accommodate the majority of the students who won't. –  response Oct 18 '13 at 18:49
Truthfully, in many cases the equations do come out of the sky. Why do we choose $p$-values of 0.05 (or 0.01)? Why never $p=0.0374$? It's not too hard to look through medical research journals and find correspondence debating whether to analyze the statistics in this way vs. that way, all while eliding any actual fundamental mathematical discussion (it's usually because one finding contradict's someone else's finding...). –  Arkamis Oct 18 '13 at 18:58

Once upon a time an illiterate was recruited as a student by Prestigious University because he was a talented football player and they badly wanted him on the team. On a midterm exam they asked him "When was the War of 1812?" He said "I don't know", and that was undeniably correct, so he passed instead of flunking out, so he could stay on the team, and everyone lived happily ever after.

Math is a subject that everyone is required to take, including those who don't want to learn math. That is a crime. A consequence is that math courses are taught the way that football player's course was taught. That way they can get everybody through. Let's say students are told that variance is $$\frac1n\sum_{i=1}^n (x_i-\bar x)^2,\text{ where }\bar x = \frac{x_1+\cdots+x_n}{n}. \tag1$$ (The version with $n-1$ in the denominator instead of $n$ is used ONLY when estimating a population standard deviation based on a sample standard deviation, and the conventional argument for doing that is immensely weaker than most who teach statistics seem to suspect, and at any rate I have a reason for avoiding it here.)

An intelligent student who wants to understand will wonder why the square root of the quantity in $(1)$ is used instead of the mean absolute deviation $$\frac1n\sum_{i=1}^n |x_i - \bar x|. \tag2$$ So what happens if the teacher tries to explain that? Students say "Why do we have to learn this? That other instructor doesn't make his students learn this. Learning that is too high a price to pay to get an "A" in this course!" They regard learning the material as a price they pay to get a grade rather than as the thing they showed up for. The reasonable solution is to expel them and teach a course for honest students. But politicians say "That other country over there is full of students who have surpassed ours on standardized tests showing that they've memorized this formula without understanding anything!!! They will slaughter us in a nuclear war next week if we don't make our students do the same!!!!". Everyone including professors believes every word of this and harshly condemns all who disagree and wants to pay higher taxes to make students memorize formula $(1)$ above. And one can't cover everything that will be on the standardized test and still have time to accomodate intelligent students who ask questions about things like this. They must be told to realize that students who want to become illiterate blue collar workers are who the course in statistics is for and curious and intelligent students should learn their place and not misbehave in class while their dumber classmates do what neither they nor anyone else wants to do.

I'll come back later and maybe add something about why $(1)$ rather than $(2)$. The short version: because the variance of a sum of independent random variables is the sum of the variances.

PS: There is also something less nocent involved: People in many fields may want to use the mathematical results derived in statistics without necessarily being able to understand the theory.

PPS: Alright, why is the square root of $(1)$ used as a measure of dispersion, instead of $(2)$? Both have the property that if you multiply $x_i$ by $c$ for $i=1,\ldots,n$, you multiply the measure of dispersion by $|c|$. They also have the property that adding $c$ to each $x_i$ does not alter the measure of dispersion. Those two properties together make them both measures of dispersion.

So find the variance of this set of numbers: $1,2,3$, and also this one: $1,2,3,4$. What's the sum of the two independent random variables? Look at this addition table: $$\begin{array}{|r|ccc|} \hline & 1 & 2 & 3 \\ \hline 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \\ \hline \end{array}$$ The sums are: $2,3,3,4,4,4,5,5,5,6,6,7$. Find the variance of the numbers in that list. It will be the sum of the variances found above.

This can be shown to work generally, and that's not hard.

A consequence is that one can apply the central limit theorem to sums like $X_1+\cdots+X_n$ and one knows what the variance of the sum of the many random variables will be.

PPPS: That the sum of the variances equals the variance of the sum, when independent random variables are added, doesn't work when you use $n-1$ rather than $n$. That's what I had in mind when I said I have a reason to avoid that here.

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I very nearly spit my coffee all over my keyboard at the opening anecdote. –  Arkamis Oct 18 '13 at 19:09
My friend, this was a glorious rant. –  Chris Dugale Oct 19 '13 at 2:00
" They will slaughter us in a nuclear war next week if we don't make our students [memorize a formula] " -- Awesome! :) –  Happy Green Kid Naps Oct 19 '13 at 5:01
"Math is a subject that everyone is required to take, including those who don't want to learn math. That is a crime." Not really. Army also takes everybody especially in the countries with mandatory service. The crime is to deprive teachers of persuasion methods available to drill sergeants. "Everyone including professors believes every word of this and harshly condemns..." As a professor, I say that this claim is very far from the truth. Every decent professional knows that to make and deploy the modern weapons we need a few clever and knowledgeable people, not a throng of morons. –  fedja Oct 19 '13 at 5:20
@fedja : There are some hyperbolic exaggerations in what I wrote. That every professor without exceptions believes that algebra should be taught to large numbers of people who did not seek to learn mathematics is exaggerated only in the word "every". –  Michael Hardy Oct 19 '13 at 20:42
1. statistics is one of the parts of mathematics that would be useful to the larger majority of citizens. So it is useful to teach it even in non specialized classes.

2. concepts about probability are very difficult to define. They are easy to explain in a superficial way, but very difficult to formalize.

3. theorems are quite difficult to prove, they require a very high mathematical knowledge.

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It's the same in college.

In my opinion, the real culprit are all of the student who want to pretend that they are scientists but can't handle the rigor of mathematics. They have to take stat courses, but they couldn't handle the material if it was taught correctly, so the courses become "memorize, apply, and pray".

I wouldn't say the "pure math" courses are completely innocent either. It's not uncommon to see attempts at trying to get around having to prove things by just having extremely (unnecessarily) complex definitions.

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Probability, and by extension, Statistics, is a difficult subject. Urn problems and the like are rather simple, but the difficulty increases sharply from there. Many ideas from Probability are difficult to formalize without use much more sophisticated mathematics, for instance, measure. Ideas seem to be thrown out of nowhere, mostly because, to go about doing them in a "proper" way would require more resources than most high schools have. Not to mention, a higher mathematical maturity than most High-schoolers posses.

If you're interested, though, you should be able to find reasonably good explanations for your questions with a few searches along the internet. In college, the problem is still there, unless you take much higher level Mathematics and Statistics courses.

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As someone who has taught "Math for Business Majors", teaching stats to anybody that doesn't have a least a couple years of college undergraduate math under their belt is very difficult. It many ways it's just as unpleasant for the instructor as the students.

Each of those formulas could take a full day of class or more to derive from basic principals. Deriving all those formulas is the topic of an upper division college math course. You simply don't have the tools needed to understand the proofs.

Mathematics is a language. What you're doing in your stats class is the equivalent of being given a phrase book in a foreign language. You can find the bathroom with it, but there's no way you can express complex ideas.

Think of it as a shop class with formulas instead of power tools. The intent is that you learn some of the uses and limitations of the tools, not that you learn how to engineer new tools.

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I think teaching high-school students interested in math could be easier than teaching undergraduates with a couple of years of college math behind them. The interest level in math has more to do with it than the amount of experience. –  Michael Hardy Oct 20 '13 at 1:15

Micahel Hardy's answer is perfect, but I can't help adding a story of my own. I had a student come to me for help with her first-year stats homework. Here is the problem she gave me:

"You're going to a two-day conference and can't decide what shoes to pack. You own 5 pairs of flats, 7 pairs of heels, and 8 pairs of boots. To save time you decide to randomly pick your shoes. If you sample two pairs of shoes, one at a time, with replacement, what is the probability you will get a pair of heels and a pair of boots in that order"

Can you see what is horrifyingly wrong with this problem? In a small way, this problem exemplifies everything that is wrong with the way statistics (and math in general) is taught. It's not about understanding what you're doing, it's about following instructions and applying rules.

I explain more specificially whats wrong with this problem in a couple of blogposts starting here.

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