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Galton Board

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Re: Galton Board

#11

Post by flopska »

lordnacho wrote: post #6
mikorym wrote: post #3

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…

This might be what you're looking for: https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem

Ah, the good old BLUE (Best linear unbiased estimator). Had this in statistics and really really liked the proof.

Re: Galton Board

#12

Post by whatshisface »

thaumasiotes wrote: post #4
mikorym wrote: post #3

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…

> 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 th…

Least squares is also popular because it is equivalent to projection in a vector space.

Re: Galton Board

#13

Post by vmurthy »

apathy wrote: post #7

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 recomme…

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.

Re: Galton Board

#14

Post by mtrimpe »

thaumasiotes wrote: post #4
mikorym wrote: post #3

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…

> 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 th…

https://www.jessicayung.com/mse-as-maximum-likelihood/

Re: Galton Board

#16

Post by dekhn »

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.

Re: Galton Board

#17

Post by mikorym »

thaumasiotes wrote: post #4
mikorym wrote: post #3

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…

> 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 th…

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.

Re: Galton Board

#18

Post by mikorym »

lordnacho wrote: post #6
mikorym wrote: post #3

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…

This might be what you're looking for: https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem

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.

Re: Galton Board

#19

Post by eldavojohn »

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/
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