This is such a great story, it should be included in every intro stats class (I did, back when I taught intro stats).
Gosset didn’t have the mathematical background to derive the correct distribution theoretically, so he figured out what it was by simulating drawing samples of different sizes thousands of times and fitting curves. Simulating, in those days, meant writing numbers on thousands of cards, then shuffling and drawing a sample. Calculate the mean and standard deviation. Repeat. Thousands of times. He published the result with an apologetic shrug for not being able to prove it properly.
It's interesting how mathematically shallow most stats presentations are. In most other areas I've studied, you start from some basics like axioms and gradually build up machinery by proving theorems etc. But presentations I've seen of the t-test focus on when and how to use it, without going very deep into the derivation at all.
This leaves me skeptical of the movement to replace calculus with stats in high school. It's true that an ordinary citizen will find stats more useful. But for students who will go on to become scientists and engineers, I think they should study calculus. Calculus is a better on-ramp to the sort of rigor you need in upper-level math. And I'm concerned that a bad "cargo cult" stats class may be worse than no stats education at all. Calculus education seems harder to screw up.
Calculus is also mathematically shallow in that sense: the subject where you start with axioms and gradually build up the machinery of calculus is (Real) Analysis, which is not part of the standard calculus curriculum and which the vast majority of people taking calculus will never study [1]. A typical Calculus class expects students to memorize and use things like trig function integrals which are presented without proof; not so different from memorizing and using statistical tests presented without proof, in my opinion.
In an intro statistics class I think conceptual depth is more important than mathematical depth. It's more important that students really understand the concept of probabilistic inference, both hypothesis tests and confidence intervals, than that they understand the mathematical derivation of the t distribution [2].
Unfortunately intro stats classes often fail on this count as well. One of the (many) straws that eventually broke my desire to teach was a committee decision--a committee composed entirely of people not teaching intro stats--to disallow students from bringing formula cheatsheets to exams, effectively forcing us to make the students memorize formulas rather than focusing on conceptual understanding.
[1] When I took Real Analysis there was a calculus class that met right before us in the same room, which often ran over so that the calculus students would be packing up as we entered the room. One day as we're sitting down one of them asks us what class we're there for, and then asks what Real Analysis is all about, since he's never heard of it. One of my classmates responded with the absolutely perfect "Well, our homework last night was integrating x^2 from 0 to 1."
[2] I'd say the same goes for Calculus, for what it's worth; actually understanding what an integral means is more important than being able to set up the Reimann sum and take the limit.
> A typical Calculus class expects students to memorize and use things like trig function integrals which are presented without proof; not so different from memorizing and using statistical tests presented without proof, in my opinion.
You can see the derivatives based on their shapes, it is very intuitive that way unlike statistics. Calculus tries to show the intuitive explanations for the different parts, and it works very well it is how we get engineers and those engineers has built almost everything we see in modern society.
Intuition is much more useful than formalism unless you need to prove things, and calculus is one of the most intuitive subjects in math once it clicks. That intuition then stays with students for their entire lives, often they don't understand they got it from calculus class. Humans has very poor intuition for things like velocity and acceleration until after they take calculus, but people forget that since those things feels so extremely obvious after you have taken the class. That just shows how great the class is, that it helps solidify such important concepts that people don't even realize they used to have trouble with them.
> Unfortunately intro stats classes often fail on this count as well
Because unlike calculus statistics is very unintuitive. The only ways to teach it is either via extreme formalism or by showing people what numbers to put where, there is no intuitive way to teach statistics as far as we know. That makes statistics classes and knowledge you learn there churn much faster than the intuition built in calculus, making the class inherently less useful.
All the useful things I know about statistics I learned in middle school, stuff like confidence intervals, sampling bias etc. Everyone learns that in middle school, later stats classes just focus on churning numbers, I learned nothing in later stats classes I was forced to take in college.
Not sure why you are downvoted for this. Your points have merit. I agree that stats classes can have a "cookbook" flavor and do not generally lead to a deep understanding of probability. But I would rather fix the stats classes than abandon the topic.
Does anyone really argue to replace calculus with stats? I thought the idea was to offer both and let students choose based on their interests.
Propbabilities, combinatorics, logics and sets are the most valuable things from high school maths that benefitted me all the way through from teenager to professor.
Calculus is intellectually stimulating, but for my line of work (dealing with uncertaintly, risk, decision making, AI), other parts of mathematics are more useful. However, I would not argue calculus should be replaced. I would argue for more "proper" maths to replace "recipe-like" maths. It's more important to go deeper on a topic than what the topic is.
> (dealing with uncertaintly, risk, decision making, AI)
Rate of change, slopes, finding maximal points etc is extremely important in all of that, and that concept you learned and got intuition for in calculus. Calculus isn't worthless for you at all, you just undervalue it.
You might not have needed to study more calculus than the original courses, but that just makes that course even more valuable, it is so useful in so many domains even though it is a low level math class.
Um. Wow. That's quite a story. But, it's not real. "owever, Guinness had a policy of not publishing company data, and allowed Gosset to publish his observations on the strict understanding that he did so anonymously."
I'm 1906, Gosset was the guest of Person at UCL, and since Gosset had a First in Math, and Professor Pearson was the leading mathematician and publisher of the Bell curve..
Gosset spent a year at UCL. University College London. A year with an expert looking over his shoulder? I would think that he would publish with an extreme amount of confidence, forgoing the need for an apolocetic shrug, which I have never ever heard of. Never, and I have a degree in math with a minor in Statistics. They had playing cards. You are arguing for large sample sizes, which is not economical - precisely against the design of the test - which looks surprisingly suspicious.
> I would think that he would publish with an extreme amount of confidence, forgoing the need for an apolocetic shrug, which I have never ever heard of. Never, and I have a degree in math with a minor in Statistics.
Good for you. As you might have guessed from reading that I used to teach statistics, I have a bit more than a minor in the subject. Your attempt to appeal to authority, not to put too fine a point on it, falls flat.
Just because you haven’t heard of a thing don’t mean it isn’t true. We can, after all, just read the original paper:
> Before I had succeeded in solving my problem analytically, I had endeavoured to do so empirically. The material used was a correlation table containing the height and left middle finger measurements of 3000 criminals, from a paper by W. R. Macdonnell (Biometrika, i, p. 219). The measurements were written out on 3000 pieces of cardboard, which were then very thoroughly shuffled and drawn at random. As each card was drawn its numbers were written down in a book, which thus contains the measurements of 3000 criminals in a random order. Finally, each consecutive set of 4 was taken as a sample—750 in all—and the mean, standard deviation, and correlation5 of each sample determined. The difference between the mean of each sample and the mean of the population was then divided by the standard deviation of the sample, giving us the z of Section III.
As for the apologetic shrug, in the course of the “analytic solution” we have:
> The law of formation of these moment coefficients appears to be a simple one, but I have not seen my way to a general proof.
and then after a bit more math guessing the correct distribution based on the moments
> Consequently a curve of Prof. Pearson’s Type III may he expected to fit the distribution of s2.
My story is slightly off; Gosset only used one sample size rather than several different sample sizes. But he did use simulation with thousands of hand written cards as his approach to the problem, he did fail to prove the correct distribution (moments are not sufficient to determine the distribution), and he did publish with an apologetic shrug.
> But, it's not real. "owever, Guinness had a policy of not publishing company data, and allowed Gosset to publish his observations on the strict understanding that he did so anonymously."
Except that this part is true. Obviously, he is well-known in academic circles, but Guinness did have a policy against its employees to publish their research using a pseudonym[1].
[1] Specifically, they can publish with three conditions:
1) To not mention Guinness or its competitors,
2) To not mention anything about beer (so topics specifically about beer is forbidden), and
3) To not publish using their surname (which in practical effect is to publish using a pseudonym).
Gosset didn’t have the mathematical background to derive the correct distribution theoretically, so he figured out what it was by simulating drawing samples of different sizes thousands of times and fitting curves. Simulating, in those days, meant writing numbers on thousands of cards, then shuffling and drawing a sample. Calculate the mean and standard deviation. Repeat. Thousands of times. He published the result with an apologetic shrug for not being able to prove it properly.