6 ms·

Seeing Theory: A Visual Introduction to Probability and Statistics

519 points by max10541 6y ago

xabush 6y ago
Very interesting and helpful !
chews 6y ago
What a beautiful study resource.
pwaivers 6y ago
This is a really cool website!
suyash 6y ago
nice use of interactive data visualization to explain concepts, would love to see more teachers use this approach.
non-entity 6y ago
Hdjgkgogpj
blululu 6y ago
On a different thread this morning someone bemoaned the lack of statistical education - a sentiment that is widespread among people who have studied and worked with statistics and probability. It is really exciting to see pedagogical tools that help explain basic but important concepts like distributions and sampling. Great work.
- qsort 6y ago
  Agreed, this is extremely well-done. Even worse than the general lack of statistical education, I feel the teaching of statistics and probability suffers of the same problems as calculus/real analysis. Introductory statistics classes ramble at length about how random variables are functions from a probability space to a measurable space, but everyone who actually 'gets' the concept behind it eventually thinks in terms of realizations (i.e. much more similarly to what this tutorial does). Intuition without theory is shallow, but theory without intuition just leads to you eventually forgetting the theory.
  - Eugeleo 6y ago
    I’ll copy my comment from other place in this thread, because I think it might be relevant here. I feel that most math subjects are treated either as full-on “fluff” (e.g. calculus, all computing, no theory building) or full-on theory (real analysis). A combination of intuition AND rigor is hard to come by. With that said... What textbook(s?) would you recommend for a thorough self-learning of statistics? I’m looking for both intuition _and_ mathematical rigor — not all proofs, but not all fluff either. I’m a bioinformatics student and I will have a semester of combined probability/stats some time this year, but I think that won’t be enough to support me given my preference for DS-based bioinformatics jobs. I’m reading Feller right now for the probability stuff, but I’m unsure about statistics. I don’t even know what the relation between probability and statistics is — most similar questions I found online (i.e. “How to learn stats?”) are answered with a “Read this probability book and you’re good”.
    - qsort 6y ago
      > I feel that most math subjects are treated either as full-on “fluff” (e.g. calculus, all computing, no theory building) or full-on theory (real analysis) My background is computer science and I had a similar experience. Just a caveat: I'm not arguing that we should stop teaching theory, quite the contrary: most of the times we err on the side of the fluff. In particular, the fact that many reputable institutions are cutting formal logic, computability theory, etc. from their CS curriculums is an absolute disgrace. Intuition is hard to teach (easy to fall into the 'monads are burritos' trap) and it's something you have to work for yourself if you want to develop. My point is just that lack of intuition/operative knowledge will lead to your theoretical knowledge of the field being less in-depth and generally less helpful to you. I honestly don't think it really matters what book you are studying as an introduction to a subject, as usually introductory courses are teaching well-established theory that everyone knows/agrees on. If you have no prior knowledge, a decent starting point is this: https://www.amazon.com/gp/product/1981369198/ https://www.amazon.com/gp/product/1981369198/ the author's website has similar content: https://www.statlect.com https://www.statlect.com
      - Eugeleo 6y ago
- plants 6y ago
  I am so glad I am not the only one that feels this way. In high school, I didn't have to take a single probability/stats class. In college, as a CS major (!!), I had to take a single intro stats class that was completely insufficient. And when a stats education is insufficient, god damn is it insufficient. No motivating examples whatsoever (what distribution would I use to measure ${real world process}? why would I need to calculate ${X} about the distribution?), just formulas that you're expected to memorize and vomit onto an exam with no understanding of why you're doing what you're doing at all. What is the deal with this? Why isn't stats commonly taught in school when it is by far one of the most prevalent disciplines? And why, on the rare occasion when it is taught, is it so abysmal? Statistics forms the basis for all of science, for god's sake. I've since developed a patchwork understanding of statistics on my own from various resources I've found the time to consume. For the record, I grew up in the US.
shekharshan 6y ago
A couple of years ago I was just learning Python and was playing around with matplotlib. Running simulation of a dice roll 100, 1000, 10,000, 100,000, and 1,000,000 times started to show how the distribution starts to catch up with the expected 1/6th probability of each face. I was thinking how good it would be to teach young students this way.
- gbrown 6y ago
  Definitely! Also, not just young students. If you can get over code-phobia, doing random experiments in a class can be really illustrative. When I teach hypothesis testing, I always teach it both from a simulation perspective and from a traditional perspective. For one, by doing the simulation part directly it's easier to see the "under repeated sampling..." logic inherent in frequentist procedures. Additionally, it's possible to do simulation-based procedures where traditional methods break down (think: permutation tests).
- nextos 6y ago
  You can get quite deep this way. Generative models map well to programming concepts. Mixtures are quite similar to composition, and hierarchical models can be understood as inheritance. Lots of classical models like HMM, LDA, etc are quite similar to those presented in the GoF book in the sense they combine composition and inheritance in some particularly interesting manner.
- ArtWomb 6y ago
  In an effort to reduce screen time, I recently tried to instigate a game of classic table-top Dungeons & Dragons. And I swear, kids were even more interested in the BigInt N-sided die function I cribbed in a python shell than any demons or demigods ;) Seeing Theory interactivity is very interesting. I think if there is one canonical example to tie it all together it would be something akin to "estimate the likelihood of an extremely rare event". Say, you're a top astrophysicist at NASA and you have to give the President a briefing on the improbability not impossibility of an extinction level asteroid event. And you must justify how those beliefs are informed by and change with data. It ties everything together: physically based world models, event spaces, conditional probabilities, monte carlo sampling and entropy estimation. And would be really fun to boot!
laddng 6y ago
"If you roll 2 six-sided dice, what are the chances you roll at least one dice above 5 (5 or 6)?" A nice trick to visually solve this in your head I heard once is: If you think of rolling two dice as a square. X and Y are each dice. You get a 36 square board. Getting 1 six is just the upper boarder. 6 on the top, 6 on the right (6 and 6 overlap). So 11 out of the 36 squares.
- borishn 6y ago
  Chances of rolling below 5 are 4x4/36 = 16/36, so above are 20/36 And here is the board: https://www.edcollins.com/backgammon/diceprob.htm https://www.edcollins.com/backgammon/diceprob.htm
  - astatine 6y ago
    There is the small matter of "equal to 5"
- will_pseudonym 6y ago
  Another way to think about it using the square board concept would be to figure out how many ways you can get not the result you're looking for, and take 36 minus that number for the number of possible squares out of 36 squares. So getting "no 6's" on either die would be the square on the board of 1 through 5, by 1 through 5, or 25 squares. So inverting that we'd arrive at the 11 squares. In studying probability, I found that accounting for the "overlap" as you described it was more tedious in more complicated problems than just always calculating the joint probabilities and inverting them.
jtsuken 6y ago
"Don't trust your intuition". This should be the basis for all teaching in statistics and probability. If all goes wrong, it should be the one thing everyone remembers from their statistics education. And yet year after year, everyone is starting with E(X)=sum(x*P(x)) and has no idea what it was about afterwards. With calculus and linear algebra your gut feel is about right no average. You can quickly get a feel for trajectories, acceleration and distances (derivatives and integrals), areas, volumes, amounts, etc. But on probability your gut-feel will always fool you. In the end, you see a handful of math bloggers bemoaning the lack of education in probability and the nonsense being discussed by journalists and politicians. And it hardly matters whether it's an election or a pandemic. The lack of understanding of uncertainty and the false belief that one can reason about these without looking at the numbers too closely is dangerous. Sorry about the rant. But... Dear creator of seeing-theory.brown.edu, if there is one thing you could change about the project to make it different and infinitely more useful: Please start the first chapter with the goat problem[1], then go through a couple of examples from chapter 10 in Thinking Fast and Slow[2], the discuss information (maybe with a simplified version of Mendel's pea experiment[3]), discuss distributions and leave expectations and variances for much-much later. [1]: https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem [2]: https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow [3]: https://www.sciencelearn.org.nz/resources/1999-mendel-s-experiments https://www.sciencelearn.org.nz/resources/1999-mendel-s-expe...
- whatever1 6y ago
  We are very good at finding correlations. It is still very hard to prove causality in natural phenomena from experiments, specially when we cannot control them. This became blatantly obvious in the covid outbreak where nobody had a clue for months about whether masks would help or not. Edit to clarify: It is very hard to prove to causality and be sure that you did not mess up.
  - jtsuken 6y ago
    You are right about people confusing causality and correlation. Otherwise this site wouldn't be so funny: https://www.tylervigen.com/spurious-correlations https://www.tylervigen.com/spurious-correlations You are wrong about people being good at finding correlations. I rarely met people who can process a sufficiently large sample size in their memory to calculate any significant correlation results. Whereas guessing correlations from charts exposes you to a number of optical illusions, which will fool the brain into seeing things that don't exist. There may be a propensity to make more type 2 errors and see correlations between any random things such as 5G and COVID, but I haven't seen any research on that.
nicetryguy 6y ago
Gorgeous web design
Layvier 6y ago
So so cool ! And it goes to show how poorly probabilities and statistics are usually taught, it's such a waste. I'm working on a non profit project aiming in parts to aggregate this kind of pedagogical tools into a collaborative learning map and serving it in a personalised way: https://sci-map.org https://sci-map.org. Early phases still, but if people are interested to contribute please hit me up!
- ivan_ah 6y ago
  sci-map sounds very interesting. Have you looked at metacademy.org before? They did a lot of good work on the data model (concepts, resources, learning pathways, etc), and also collected a lot of content, mostly on computer topics. https://metacademy.org/graphs/concepts/bayesian_logistic_regression#focus=gluv53rd&mode=explore https://metacademy.org/graphs/concepts/bayesian_logistic_reg... Sadly the project is no longer actively developed but if you haven't seen it yet, you should definitely check out for inspiration: https://github.com/metacademy https://github.com/metacademy
  - Layvier 6y ago
    I didn't know about, seems very similar indeed! I will check it out in details, thanks a lot for sharing !
Eugeleo 6y ago
What textbook(s?) would you recommend for a thorough self-learning of statistics? I’m looking for both intuition _and_ mathematical rigor — not all proofs, but not all fluff either. I’m a bioinformatics student and I will have a semester of combined probability/stats some time this year, but I think that won’t be enough to support me given my preference for DS-based bioinformatics jobs. I’m reading Feller right now for the probability stuff, but I’m unsure about statistics. I don’t even know what the relation between probability and statistics is — most similar questions I found online (i.e. “How to learn stats?”) are answered with a “Read this probability book and you’re good”.
- mtzet 6y ago
  I kind of like M.G. Bulmer's "Principles of Statistics". It's short and to the point so there's a chance of getting through it all. I really like the discussion of distributions in terms of raw data, it makes thinking about mean, variance, higher moments etc., much easier. It also doesn't skimp on the mathematical theory, but it doesn't allow itself to get bogged down by it. That said, there's a chance I just read it late enough in my career to be more ready for its content.
- wodenokoto 6y ago
  > I don’t even know what the relation between probability and statistics is That's a great question, and I think the lines are more than a little blurry. My attempt at an answer would be: Probability: Given a set of dice and coins and an order for rolling and throwing them, what is the chance of a specific outcome? Statistics: Given a set of outcomes, what dice where rolled? So if you want to know if smoking kills, you tally up medical history, and use statistics to see if there is a relationship between smoking and dying. If you want to know the probability of smoking killing you, you look at the risc each cigarette brings to the table and tally it up using probability theory. More elegantly phrased examples can be found on Stack Overflow: https://stats.stackexchange.com/questions/665/whats-the-difference-between-probability-and-statistics https://stats.stackexchange.com/questions/665/whats-the-diff...
- marksbrown 6y ago
  Probability & statistics by de Groot is the standard text I believe. Full of examples and questions.
deleted 6y ago
[deleted]
- deleted 6y ago
  [deleted]
  - deleted 6y ago
    [deleted]
johndoe42377 6y ago
Brain does not count. It updates/alters its structures - the process which humans model with numbers and back propagation. Numbers do not exist outside of human cultures.
kyrers 6y ago
Amazing. Thanks for sharing!
Aeolun 6y ago
Got stuck at chapter 2 since there was zero indication what a valid syntax for a set was.
eithed 6y ago
I must say that this website/way of reading a book is simply gorgeous
mettamage 6y ago
How come that an undergraduate person (at the time) makes one of the most compelling statistic textbooks? Is it because there are many more amateur statistic textbooks in existence, or published attempts at one (so more chance for a runaway success to be picked up)? Or is it because people in the statistic textbook industry don't feel this frustration and/or don't dare to take any risk?
jp0d 6y ago
This looks brilliant. This can be very useful to grasp the concepts of probability and statistics in a visual way. I've been struggling to understand some of the concepts and I hope to use this as a supplement. Although, I don't believe it can replace a university course or a proper text book.

519 points by max10541 6y ago

xabush 6y ago

Very interesting and helpful !

chews 6y ago

What a beautiful study resource.

pwaivers 6y ago

This is a really cool website!

suyash 6y ago

nice use of interactive data visualization to explain concepts, would love to see more teachers use this approach.

non-entity 6y ago

Hdjgkgogpj

blululu 6y ago

On a different thread this morning someone bemoaned the lack of statistical education - a sentiment that is widespread among people who have studied and worked with statistics and probability. It is really exciting to see pedagogical tools that help explain basic but important concepts like distributions and sampling. Great work.

qsort 6y ago
Agreed, this is extremely well-done. Even worse than the general lack of statistical education, I feel the teaching of statistics and probability suffers of the same problems as calculus/real analysis. Introductory statistics classes ramble at length about how random variables are functions from a probability space to a measurable space, but everyone who actually 'gets' the concept behind it eventually thinks in terms of realizations (i.e. much more similarly to what this tutorial does). Intuition without theory is shallow, but theory without intuition just leads to you eventually forgetting the theory.
- Eugeleo 6y ago
  I’ll copy my comment from other place in this thread, because I think it might be relevant here. I feel that most math subjects are treated either as full-on “fluff” (e.g. calculus, all computing, no theory building) or full-on theory (real analysis). A combination of intuition AND rigor is hard to come by. With that said... What textbook(s?) would you recommend for a thorough self-learning of statistics? I’m looking for both intuition _and_ mathematical rigor — not all proofs, but not all fluff either. I’m a bioinformatics student and I will have a semester of combined probability/stats some time this year, but I think that won’t be enough to support me given my preference for DS-based bioinformatics jobs. I’m reading Feller right now for the probability stuff, but I’m unsure about statistics. I don’t even know what the relation between probability and statistics is — most similar questions I found online (i.e. “How to learn stats?”) are answered with a “Read this probability book and you’re good”.
  - qsort 6y ago
    > I feel that most math subjects are treated either as full-on “fluff” (e.g. calculus, all computing, no theory building) or full-on theory (real analysis) My background is computer science and I had a similar experience. Just a caveat: I'm not arguing that we should stop teaching theory, quite the contrary: most of the times we err on the side of the fluff. In particular, the fact that many reputable institutions are cutting formal logic, computability theory, etc. from their CS curriculums is an absolute disgrace. Intuition is hard to teach (easy to fall into the 'monads are burritos' trap) and it's something you have to work for yourself if you want to develop. My point is just that lack of intuition/operative knowledge will lead to your theoretical knowledge of the field being less in-depth and generally less helpful to you. I honestly don't think it really matters what book you are studying as an introduction to a subject, as usually introductory courses are teaching well-established theory that everyone knows/agrees on. If you have no prior knowledge, a decent starting point is this: https://www.amazon.com/gp/product/1981369198/ https://www.amazon.com/gp/product/1981369198/ the author's website has similar content: https://www.statlect.com https://www.statlect.com
    - Eugeleo 6y ago
plants 6y ago
I am so glad I am not the only one that feels this way. In high school, I didn't have to take a single probability/stats class. In college, as a CS major (!!), I had to take a single intro stats class that was completely insufficient. And when a stats education is insufficient, god damn is it insufficient. No motivating examples whatsoever (what distribution would I use to measure ${real world process}? why would I need to calculate ${X} about the distribution?), just formulas that you're expected to memorize and vomit onto an exam with no understanding of why you're doing what you're doing at all. What is the deal with this? Why isn't stats commonly taught in school when it is by far one of the most prevalent disciplines? And why, on the rare occasion when it is taught, is it so abysmal? Statistics forms the basis for all of science, for god's sake. I've since developed a patchwork understanding of statistics on my own from various resources I've found the time to consume. For the record, I grew up in the US.

shekharshan 6y ago

A couple of years ago I was just learning Python and was playing around with matplotlib. Running simulation of a dice roll 100, 1000, 10,000, 100,000, and 1,000,000 times started to show how the distribution starts to catch up with the expected 1/6th probability of each face. I was thinking how good it would be to teach young students this way.

gbrown 6y ago
Definitely! Also, not just young students. If you can get over code-phobia, doing random experiments in a class can be really illustrative. When I teach hypothesis testing, I always teach it both from a simulation perspective and from a traditional perspective. For one, by doing the simulation part directly it's easier to see the "under repeated sampling..." logic inherent in frequentist procedures. Additionally, it's possible to do simulation-based procedures where traditional methods break down (think: permutation tests).
nextos 6y ago
You can get quite deep this way. Generative models map well to programming concepts. Mixtures are quite similar to composition, and hierarchical models can be understood as inheritance. Lots of classical models like HMM, LDA, etc are quite similar to those presented in the GoF book in the sense they combine composition and inheritance in some particularly interesting manner.
ArtWomb 6y ago
In an effort to reduce screen time, I recently tried to instigate a game of classic table-top Dungeons & Dragons. And I swear, kids were even more interested in the BigInt N-sided die function I cribbed in a python shell than any demons or demigods ;) Seeing Theory interactivity is very interesting. I think if there is one canonical example to tie it all together it would be something akin to "estimate the likelihood of an extremely rare event". Say, you're a top astrophysicist at NASA and you have to give the President a briefing on the improbability not impossibility of an extinction level asteroid event. And you must justify how those beliefs are informed by and change with data. It ties everything together: physically based world models, event spaces, conditional probabilities, monte carlo sampling and entropy estimation. And would be really fun to boot!

laddng 6y ago

"If you roll 2 six-sided dice, what are the chances you roll at least one dice above 5 (5 or 6)?" A nice trick to visually solve this in your head I heard once is: If you think of rolling two dice as a square. X and Y are each dice. You get a 36 square board. Getting 1 six is just the upper boarder. 6 on the top, 6 on the right (6 and 6 overlap). So 11 out of the 36 squares.

borishn 6y ago
Chances of rolling below 5 are 4x4/36 = 16/36, so above are 20/36 And here is the board: https://www.edcollins.com/backgammon/diceprob.htm https://www.edcollins.com/backgammon/diceprob.htm
- astatine 6y ago
  There is the small matter of "equal to 5"
will_pseudonym 6y ago
Another way to think about it using the square board concept would be to figure out how many ways you can get not the result you're looking for, and take 36 minus that number for the number of possible squares out of 36 squares. So getting "no 6's" on either die would be the square on the board of 1 through 5, by 1 through 5, or 25 squares. So inverting that we'd arrive at the 11 squares. In studying probability, I found that accounting for the "overlap" as you described it was more tedious in more complicated problems than just always calculating the joint probabilities and inverting them.

jtsuken 6y ago

"Don't trust your intuition". This should be the basis for all teaching in statistics and probability. If all goes wrong, it should be the one thing everyone remembers from their statistics education. And yet year after year, everyone is starting with E(X)=sum(x*P(x)) and has no idea what it was about afterwards. With calculus and linear algebra your gut feel is about right no average. You can quickly get a feel for trajectories, acceleration and distances (derivatives and integrals), areas, volumes, amounts, etc. But on probability your gut-feel will always fool you. In the end, you see a handful of math bloggers bemoaning the lack of education in probability and the nonsense being discussed by journalists and politicians. And it hardly matters whether it's an election or a pandemic. The lack of understanding of uncertainty and the false belief that one can reason about these without looking at the numbers too closely is dangerous. Sorry about the rant. But... Dear creator of seeing-theory.brown.edu, if there is one thing you could change about the project to make it different and infinitely more useful: Please start the first chapter with the goat problem[1], then go through a couple of examples from chapter 10 in Thinking Fast and Slow[2], the discuss information (maybe with a simplified version of Mendel's pea experiment[3]), discuss distributions and leave expectations and variances for much-much later. [1]: https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem [2]: https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow [3]: https://www.sciencelearn.org.nz/resources/1999-mendel-s-experiments https://www.sciencelearn.org.nz/resources/1999-mendel-s-expe...

whatever1 6y ago
We are very good at finding correlations. It is still very hard to prove causality in natural phenomena from experiments, specially when we cannot control them. This became blatantly obvious in the covid outbreak where nobody had a clue for months about whether masks would help or not. Edit to clarify: It is very hard to prove to causality and be sure that you did not mess up.
- jtsuken 6y ago
  You are right about people confusing causality and correlation. Otherwise this site wouldn't be so funny: https://www.tylervigen.com/spurious-correlations https://www.tylervigen.com/spurious-correlations You are wrong about people being good at finding correlations. I rarely met people who can process a sufficiently large sample size in their memory to calculate any significant correlation results. Whereas guessing correlations from charts exposes you to a number of optical illusions, which will fool the brain into seeing things that don't exist. There may be a propensity to make more type 2 errors and see correlations between any random things such as 5G and COVID, but I haven't seen any research on that.

nicetryguy 6y ago

Gorgeous web design

Layvier 6y ago

So so cool ! And it goes to show how poorly probabilities and statistics are usually taught, it's such a waste. I'm working on a non profit project aiming in parts to aggregate this kind of pedagogical tools into a collaborative learning map and serving it in a personalised way: https://sci-map.org https://sci-map.org. Early phases still, but if people are interested to contribute please hit me up!

ivan_ah 6y ago
sci-map sounds very interesting. Have you looked at metacademy.org before? They did a lot of good work on the data model (concepts, resources, learning pathways, etc), and also collected a lot of content, mostly on computer topics. https://metacademy.org/graphs/concepts/bayesian_logistic_regression#focus=gluv53rd&mode=explore https://metacademy.org/graphs/concepts/bayesian_logistic_reg... Sadly the project is no longer actively developed but if you haven't seen it yet, you should definitely check out for inspiration: https://github.com/metacademy https://github.com/metacademy
- Layvier 6y ago
  I didn't know about, seems very similar indeed! I will check it out in details, thanks a lot for sharing !

Eugeleo 6y ago

What textbook(s?) would you recommend for a thorough self-learning of statistics? I’m looking for both intuition _and_ mathematical rigor — not all proofs, but not all fluff either. I’m a bioinformatics student and I will have a semester of combined probability/stats some time this year, but I think that won’t be enough to support me given my preference for DS-based bioinformatics jobs. I’m reading Feller right now for the probability stuff, but I’m unsure about statistics. I don’t even know what the relation between probability and statistics is — most similar questions I found online (i.e. “How to learn stats?”) are answered with a “Read this probability book and you’re good”.

mtzet 6y ago
I kind of like M.G. Bulmer's "Principles of Statistics". It's short and to the point so there's a chance of getting through it all. I really like the discussion of distributions in terms of raw data, it makes thinking about mean, variance, higher moments etc., much easier. It also doesn't skimp on the mathematical theory, but it doesn't allow itself to get bogged down by it. That said, there's a chance I just read it late enough in my career to be more ready for its content.
wodenokoto 6y ago
> I don’t even know what the relation between probability and statistics is That's a great question, and I think the lines are more than a little blurry. My attempt at an answer would be: Probability: Given a set of dice and coins and an order for rolling and throwing them, what is the chance of a specific outcome? Statistics: Given a set of outcomes, what dice where rolled? So if you want to know if smoking kills, you tally up medical history, and use statistics to see if there is a relationship between smoking and dying. If you want to know the probability of smoking killing you, you look at the risc each cigarette brings to the table and tally it up using probability theory. More elegantly phrased examples can be found on Stack Overflow: https://stats.stackexchange.com/questions/665/whats-the-difference-between-probability-and-statistics https://stats.stackexchange.com/questions/665/whats-the-diff...
marksbrown 6y ago
Probability & statistics by de Groot is the standard text I believe. Full of examples and questions.

deleted 6y ago

[deleted]

deleted 6y ago
[deleted]
- deleted 6y ago
  [deleted]

johndoe42377 6y ago

Brain does not count. It updates/alters its structures - the process which humans model with numbers and back propagation. Numbers do not exist outside of human cultures.

kyrers 6y ago

Amazing. Thanks for sharing!

Aeolun 6y ago

Got stuck at chapter 2 since there was zero indication what a valid syntax for a set was.

eithed 6y ago

I must say that this website/way of reading a book is simply gorgeous

mettamage 6y ago

How come that an undergraduate person (at the time) makes one of the most compelling statistic textbooks? Is it because there are many more amateur statistic textbooks in existence, or published attempts at one (so more chance for a runaway success to be picked up)? Or is it because people in the statistic textbook industry don't feel this frustration and/or don't dare to take any risk?

jp0d 6y ago

This looks brilliant. This can be very useful to grasp the concepts of probability and statistics in a visual way. I've been struggling to understand some of the concepts and I hope to use this as a supplement. Although, I don't believe it can replace a university course or a proper text book.