One of my Statistics professors made great use of one of these (he called it a Quincunx) our my Six Sigma class. It was about management, and how reacting to processes you can’t control just makes things worse.
He’d pick someone from the class, tell them he was going to check their performance (like they were a Sales manager), a run a ball through the Quincunx. If the ball landed on the left, that meant they’d underperform, and they got a tongue lashing. If it fell on the right, they got praise. People got angry about the senselessness of it all.
But that was the point. The lesson: if you mandate targets on something that is essentially random and can’t be controlled, you’re going to have a bad time. (And if you react to those random results by changing the process, results get even worse — but that was another class for another day.)
I bought one of these to teach a one-shot class on experimental design and statistics.
To say it served the purpose would be an understatement. We blew through the CLT and derivation of statistical power in 10 minutes, leaving the other 110 minutes for the students to present research papers. One of the best $35 I’ve ever spent (don’t have the Amazon link handy but there are some great versions there). Highly recommended if you teach.
I really wish I had teachers who used some visual tools to teach these concepts. More than the concept itself, it is the feeling of awe that one gets when one watches these things. It is easy to forget that a lot of math comes to describe phenomenon in nature! The right teacher at a young age can have a radical impact on kids picking up STEM.
If you can post a link to a good one I'd appreciate it. I've generally found expensive ones and bad ones. I've been looking for one that's cheap and good.
As others said, if you can post a link that would be really great. When I've taught probability in years past, I always showed students Galton boards on YouTube, but a real one that doesn't break and doesn't break the budget would be much better.
You can just search Amazon. There's nothing special about a quincunx, they're simple. Any search query on Amazon, whether 'Galton quincunx' or 'Galton bean machine' or 'Galton board' will pull up a bunch, I just checked.
One of my favorite photos in the world is of the large (wall-sized) Galton board at the old Princeton Engineering Anomalies Research Lab. There are two guys, visitors, sitting in front of it and they have just used "psychic powers" to affect a run and the balls are ridiculously skewed to one side, just ridiculously, obviously skewed.
I like the photo because it's a bifurcation point for the viewer: there are two options to resolve what you're seeing:
1. It's fake.
2. It's not fake and "there's something there".
The whole PEAR Lab itself suffers from the same ambiguity: they got consistent positive results, but never so positive that skeptics could be decisively satisfied. (Not including one-off things like the photo of the visiting guys who did produce a dramatic undeniable effect.)
I managed to remember the book in which the photo is published:
"The Second Coming of Science: An Intimate Report on the New Science"
by Brian O'Leary
1992
My copy is in storage so I can't post a scan. IIRC the visitors are O'Leary and his son.
It turns out he has a wikipedia entry:
> Brian Todd O'Leary (January 27, 1940 – July 28, 2011)[1] was an American scientist, author, and former NASA astronaut. He was part of NASA Astronaut Group 6, a group of scientist-astronauts chosen with the intention of training for the Apollo Applications Program.
> A remote viewing experience in 1979 and a near-death experience in 1982 initiated O'Leary's departure from orthodox science. After Princeton, O'Leary worked Science Applications International Corporation. He refused to work on military space applications, for which reason he lost his position there in 1987. Beginning in 1987, O'Leary increasingly explored unorthodox ideas, particularly the relationship between consciousness and science, and became widely known for his writings on "the frontiers of science, space, energy and culture".
This was discussed in a really interesting way in the Pearl/MacKenzie book "The Book of Why" which I heavily recommend for people interested in cause & effect. Really opened my eyes to a lot of things I had been doing statistically but never known formally what was going on. http://bayes.cs.ucla.edu/WHY/
I fondly remember when I wrote a Galton board simulation as a computer science school project. It would show the ball falling down and simulate the coin flip at each stage, summing up the number of balls that fell into each slot. The hard thing was it had to run on a 80286. The final version had some nifty background graphics and advanced drawing routines, written in Pascal and some inline assembly ;)
If you vary the size of the opening, am I correct that that just changes the parameters on the curve?
I've known about the central limit theorem for a long time and was probably taught about it in first year, but I have never managed to sit down and understand how to prove it properly. One side effect of the theorem should be to explain least squares—if I am not mistaken then least squares was invented largely due to the central limit theorem by Gauss.
We can always do least cubes, but that does not provide us (usually) with better results.
> if I am not mistaken then least squares was invented largely due to the central limit theorem by Gauss
I'm not really speaking from expertise here, but I thought least-squares error measurement was based on the fact that the metric is easy to minimize, because taking the derivative of x^2 is easy, whereas taking the derivative of |x| is complicated.
Least cubes doesn't really work conceptually, as it would imply that if an outlier above the fitted curve is bad, then an outlier below the fitted curve is good. That's not what you want.
As someone else mentioned, the BLUE is what we are usually looking for the Gauss-Markov theorem linked tells us why it is least squares.
EDIT: I am a bit suspicious though, since the definition of variance implicitly uses least squares. Maybe someone else can explain this better, but my expectation is that variance and the reason why it also uses squares (and not something else) should follow from some set of first principles.
Yes, I think so. Intuitively, how I would like to think of this is that we are getting to this from the Euclidean distance metric. But skimming the proof seems like we use variance (which is defined using squares) as a presupposition.
Recently a good friend gave me a small Galton board for my birthday and it stands now on my desk. It is so cool to do a little "simulation" and see the magic of the central limit theorem. Highly recommended as a gift for any statistically interested person!
Is it really 50% though since more than one ball is falling through at a time? Wouldn't a steel ball hitting another steel ball affect it's possible path?
Notice how the drawn Gaussian doesn't follow the bottom row of binomial coefficients of the Pascal's triangle[1]. The model that assumes that each ball independently falls left or right exactly one unit is limited and doesn't actually describe many real world Galton boards.
I built several of these from scratch after realizing that pegboard is already set up in the right pattern and dowel pins fit into peg holes perfectly.
I demo'd it at a STEM fair and everybody has a great time. It makes a ton of noise and a great visual demo. I even ended up learning a bunch about hopper theory because I had to 3d print a hopper to feed it and it kept jamming.
I forget where I first read about hopper theory but I was fascinated to discover that so much analysis has been done on the dimensions of such a simple device. It makes perfect sense in retrospect given all the industrial and agricultural applications but it's one of those little corners of the world where you would never guess the pains engineers go to understand and optimize it.
He’d pick someone from the class, tell them he was going to check their performance (like they were a Sales manager), a run a ball through the Quincunx. If the ball landed on the left, that meant they’d underperform, and they got a tongue lashing. If it fell on the right, they got praise. People got angry about the senselessness of it all.
But that was the point. The lesson: if you mandate targets on something that is essentially random and can’t be controlled, you’re going to have a bad time. (And if you react to those random results by changing the process, results get even worse — but that was another class for another day.)