The Law of Small Numbers

[@paulg.9596]

Vsauce, MinuteEarth, The Great War, and Conjecture all in one day? What're the odds

[@samramdebest]

let's do the math...

[@conjecturemm]

Vsauce??? Gotta check my inbox.

[@conjecturemm]

We could do the math by calculating how frequently they've been uploading over the past, say, 6 months. And then get a rate for how frequently they upload and combine them all for the odds. If anyone wants to do this (and show math) I will feature it in a video!

[@Ruminations09]

I'll get back to you in a bit with my results.

[@Ruminations09]

Okay, I might not be the first on, but I got *0.0085%* - I took the videos that each channel has made in the past 6 months, between the dates of December 13, 2015 and June 13, 2016 (so I included the videos that these channels released today). This is a 184 day span of videos that I'm taking into account. Here are the results of that: Conjecture: *19 videos released* MinuteEarth: *19 videos released* Vsauce: *3 videos released* The Great War: *90 videos released* - The number of videos released over 184 is the likelyhood of the channel releasing a video on any given day. For example, Conjecture has about a *19/184* chance of releasing a video on any given day. - To determine the odds of them releasing together, I multplied their odds together. So: *(19 x 19 x 3 x 90) / (184^4)* This gives us 0.000085035 OR *0.0085%*

[@julianpaullopez3801]

Amazing explanation, I complement it with the reading of the book and I could understand it perfectly.

[@conjecturemm]

Commenter "Paul 'Turtles' G." pointed out that today Vsauce, MinuteEarth, The Great War, and I uploaded a video today. Then asked what are the odds. We could do the math by calculating how frequently they've been uploading over the past, say, 6 months. And then get a rate for how frequently they upload and combine them all for the odds. If anyone wants to do this (and show math) I will feature it in a video!

[@DaviddeKloet]

Every day you can find 4 channels that uploaded that day. Today it happens to be these 4 and tomorrow it's 4 different ones.

[@conjecturemm]

agree

[@magic16136]

Slightly unrelated: it was a bit difficult to decide which of the three videos to watch first when I logged onto KZbin today but in the end I chose yours.

[@magic16136]

I know you mentioned four videos but I'm only subscribed to three of the channels.

[@Ruminations09]

I'm working on it right now actually, I saw the other comment.

[@Keronin]

Great video! I really enjoyed the bit at the very end where you were talking about how your samples for the "counties" were showing your whole point of the video. Keep up the good work! Brother, I'll see you on weekday.

[@conjecturemm]

Thanks! I haven't heard that said in a while haha, I like it :)

[@markncl100]

Awesome video mate. Reminds me of a video you posted couple of years ago on misleading statistics (which I quote often) although this one goes farther into the maths and our interpretation of the results. Truly thought-provoking.

[@conjecturemm]

Thanks, Mark! Yeah I guess I like interesting statistics :)

[@DrawCuriosity]

I also made a video (though covering different aspects of randomness) about how crap we are at understanding randomness! In a way, we are generally primed to look for relationships and spot outliers, which explains why we jump to those conclusions - and definitely proves that we could all benefit from better education in statistics at school. Great video as always!! :D

[@conjecturemm]

Thanks! I will check out yours, too. Yeah you know I took a stat class but it focused on the formulaic aspects of it, not really applying it to the real world.

[@magic16136]

This was a fascinating video! It was really interesting to see you doing the sampling stuff: it made the topic a whole lot easier to understand. Statistics are crazy. They're so cool but can also be very misleading if not considered in context as you point out! Also I think I now know what my next book on Audible will be when I finish the one I'm on!

[@conjecturemm]

Thanks, Lorenzo! Kahneman originally did the thought experiment with 4 and 7 marbles (population pools of 8 and 14 respectively) but I didn't have marbles, haha. And yeah, sometimes it's hard to even remember there is a greater context, because we suck at that. I think you'll enjoy Thinking Fast and Slow!

[@ElchiKing]

Fun fact: Not only statistics have the problem of the law of small numbers: In mathematics there is a theorem which says, that all norms (i.e. possible natural ways to measure lengths in space) for finite dimensional spaces are equivalent. Thus, in theory, if you have a method to approximate a solution, you only need to consider your favourite norm. And now comes the "but": If you want to implement your method you usually want to be sure that your method converges quickly (which usually means far less steps than the dimension (number of variables)). However, in this case, the convergence rate depends on your chosen norm (one can even show that you can have arbitrarily good or bad results by simply choosing another norm)

[@conjecturemm]

So you can choose whichever norm you'd like, but the convergence rate will differ based on the norm. If it doesn't matter which norm you choose does it matter what convergence rate you have?

[@ElchiKing]

***** Well, if a sequence converges according to one norm, it does so according to the others, but only eventually, not in the first few steps. In praxis, the convergence rate is important, since otherwise you could solve (at least some) problems directly in less time and approximations would be pointless...

[@conjecturemm]

How many people in your field subscribe to the theorem that norms are arbitrary? That seems problematic haha

[@ElchiKing]

***** For us mathematicians, what I said is usually not a problem, since we assume everything to be infinite (and thus, it is not important, how fast convergence is). It's just smething, you need to keep in mind

[@BitterspitBand]

I have to thank you. I watched this video when you uploaded it. Now two weeks later, I realized during a conversation with a friend that this might explain mass shooting statistics in couple of smaller countries.

[@conjecturemm]

Glad this helps. I would be curious to see those statistics, if you have them.

[@BitterspitBand]

I don't remember specifically what statistics I was watching. But it's quite normal to see Norway and Finland ranking as high as The US in the mass shooting statistics (annual mass shooting deaths/100,000 people or something like that). Those statistics always felt weird for me, because murder rates usually correlates with the number of guns in the country. But mass shooting statistics doesn't always do that. And those two countries are one of the safest in the world otherwise. I know that the mass shooting statistics are complicated and there's no way that this is some sort of final answer to those things. But Finland and Norway both have only 5 million people and mass shootings are so rare that this seems to be the case. If you only look back 15 years for these kind of statistics (they only looked like years 2000-2015 in the statistics I was watching) wouldn't that mean that it's more about the bad luck in smaller countries where the sample size is small? Finland doesn't have usually that many mass shootings, they just had like three in a row and then nothing for many years. And there's been most likely just one mass shooting in Norway but it killed over 70 people. Could this be a partial explanation to those statistics in Norway and Finland? It sure looks to me like it is but I might understood something wrongly from your video. Sorry if it's too long or hard to understand. English is not my first language and I suck at explaining things shortly.

[@ZacCoventry]

Excellent explanation. Thank you.

[@TheImpe98]

Great video! I've always found it facsinating how statistics often are so counterintuitive. You did a great job explaining it, though!

[@conjecturemm]

Thanks!

[@aasmund_ks6369]

Brilliant video. This really suits your style. Keep it up!

[@conjecturemm]

Thanks :)

[@cinkisses]

this was super awesome !!!!!! i really needed help with understanding this for my class thank you very much :)

[@conjecturemm]

Of course!! Glad it helped :)

[@themonitorlizard3019]

I'm new to this channel, and I love it! Keep up the good work

[@conjecturemm]

Thanks!! Hope you enjoy it

[@Macieks300]

As to small schools: the Law of Small Numbers is one of the explainations, the other is that those schools are better, but to prove either one you probably would have to get deeper into the statistics and performances of these schools

[@conjecturemm]

Yeah, though we'd have to ask if they're better because they're small or because of something else. It could be that they're actually better and it wasn't an error of sampling, but we can't say with 100% certainty that it was their size, either, if they are actually better. But according to the fact that a similar proportion of small school were in the worst category as the best, Kahneman says we can rule out size as a primary causal factor.

[@photon_shines]

How does one get into reading nonfiction books like the one you are talking about? I love novels, but I don't know how I can read a nonfiction book (even in topics I'm kind of interested in) without it feeling like a chore or feeling frustrated that nothing is being retained.

[@lemake72]

For me, it's about finding nonfiction books that are by engaging authors, and my interest will follow. It's probably a safe bet to try some popular nonfiction books, like Freakonomics or anything by Malcolm Gladwell. I began with and have generally stuck to these. One word titles are also a good clue. :) Concerning retention, I can't remember most details, but there are some that may stick with you, and you can always take away the "thesis"of any book and whether or not you were convinced. Happy reading! (Disclaimer: I don't read enough, and these are just underdeveloped opinions.)

[@conjecturemm]

Hey, like +lemake72 said I would start with really popular ones like Gladwell or maybe Ariely. I actually got into non-fiction through Gladwell's Outliers and Ariely's Predictably Irrational. As for more intense ones that are still interesting? Just force yourself to sit down and do it. It doesn't sound fun, but it's what you have to do. That's what I do researching KZbin videos sometimes, and though I know it will be fun, I often have to convince myself to do it.

[@skellingtonmeteoryballoon]

5:02 nice break down. your chart was easier to read with B G as opposed to xx xy charts. muchas gracias. nothing wrong with including causality in ex plainin

[@anakimluke]

Very good video! :) Though, the music credit at the end was a bit unnecessary. Maybe put it in the vid description next time? About the county paper thing, I like to think like as if you would take a very small number, like 1, the extremes would really always be THAT extreme.

[@conjecturemm]

I don't mind the music at the end, though I could put something like my twitter or "subscribe!" And yeah, seriously. All your counties would be 100 or 0

[@mobiletuner]

This blew my mind a little. I guess, you can't make any correct judgements without knowing the statistical laws. Another great example of statistical fallacy is when head injuries skyrocketed after British military started to issue helmets during WW1. At first, they thought that it was a disaster and considered ditching helmets, until they've realized that helmets just moved a massive amount of people from "killed in action" category to a mere "head injury" category.

[@conjecturemm]

Ooh, that's pretty cool! Yeah stuff like that is always fascinating to me. There's a part in Thinking Fast and Slow about regression to the mean, which is basically that after people do surprisingly well/poorly, they'll usually do worse/better in their next performance. Someone thought it was because they were giving praise or criticism, but it had no relation to what was actually going on.

[@janichirag123]

Amazing video. Probably will share with my colleagues at work. More such statistics based videos please if you could.

[@conjecturemm]

I like them too, so hopefully so :) Thanks!

[@Huntracony]

I didn't guess it from the title, I guessed it because I know you're interested in statistics and how they can be manipulated. None of the less, I felt very smug when you confirmed I was right.

[@conjecturemm]

Lol glad to hear it

[@KevinVandyTech]

I'm actually in the middle of Thinking Fast and Slow right now. My philosophy professor recommended it last semester.

[@conjecturemm]

It's really good. However I will say that I think the first part (it's in something like 4 or 5 parts) gets old pretty quickly. I put it down for a while for that reason, and am glad I picked it up again (in the second part with more examples). First part was a lot of theory and second is more examples. Though the theory is important to understand what's going on in the latter parts, but I still think you'd be able to get the gist of it all without as much theory TL;DR: push through (or skip) the first part because it gets better

[@KarstenOkk]

Did...Did we do something after 6:58?

[@conjecturemm]

What do you mean?

[@User-to7nb]

Oh my god. This is great! Thank you for this; you need more subscribers ASAP

[@conjecturemm]

Aw thanks :)

[@deviledarts]

Honestly, I'm so happy KZbin recommended the Vsauce/other people parody video. You make very good quality content that I wish would get more recognition! Also these laws are very annoying while taking multiple choice questions on a test because I always want the answers to be somewhat equal, but it's possible to have 3 times more C than A. This messes up my brain and causes me to change the answer to make it more equal; therefore, my test scores are lower DAMN IT STATISTICS! Why do you do this to me!

[@conjecturemm]

Aw thanks, Alex :) Yeah, that's a great example I wish I had thought of. It doesn't matter how many you get in a row, even though it seems super weird to us to get C C C C C B. The sequence is equally likely as any other, it's only the total end result that will differ

[@deviledarts]

yeah :D

[@grobacz]

Good one. And real life examples. Gets you to think. I like it.

[@conjecturemm]

Yay! All I'm trying to do ;)

[@KatyCunningham]

The outro omg 😂😂😂 small sample size inception

[@conjecturemm]

Yeah the original experiment was going to include 10 examples haha nope

[@anandrewbond]

Very interesting... love the new ending!

[@conjecturemm]

Haha thanks. Playing around with it because I have no idea what I want to do with it !

[@eliosgreek8028]

Great video ❤

[@thevisiblegardener5663]

Wow. I made a video on the exact topic from Thinking Fast and Slow just a couple of weeks before this one. Nice video!

[@heyitsalex99]

under 10k subs, mad, i was genuinely here before he becomes mainstream

[@conjecturemm]

Aw :) Thanks!

[@sunn7045]

Amazing video as always!

[@conjecturemm]

Thanks!

[@afterparty3695]

thank you

[@smit9779]

actually i didnt get the chapter properly thanks for explaining

[@habibhassan7770]

This was great. Thanks

[@Alexs2000w]

i love your vids and id like 2 give u a tipp.Try to play some music in the backround and see if u like it or not

[@conjecturemm]

Yeah I've tried it and I feel like it doesn't fit my style. I will probably make a video explaining this more on my other channel, "everything else"

[@MrSaber777]

Nice explanation pro

[@Narradorenprimerapersona]

Wow, quite an interesting video. You just gained a new subscriber! Lots of luck for next videos, you look quite confident and I hope you do well :D

[@conjecturemm]

Thanks so much! I hope to be making a lot of more fun videos!

[@Soliloquy084]

You might not be sponsored by Audible (yet?), but you could sign up for Amazon's affiliate program if you wanted to.

[@conjecturemm]

Interesting. I'll keep this in mind if I ever want to plug it again, haha

[@Soliloquy084]

Matt Gray and Tom Scott did a hilarious video on what people bought after clicking their amazon links, so it was in my mind.

[@KatyCunningham]

Matt I miss your main channel videos man!!!!

[@conjecturemm]

So do I. I am literally editing video right now and trying to get them to come out soon. I can't say exactly when but I can promise you that I am working on stuff right now :)

[@HodanHussein]

I Initially ended up here because of school work. But this was really interesting!

[@conjecturemm]

Glad you enjoyed it!

[@OrchidAlloy]

Great video!

[@conjecturemm]

Thanks!

[@elliottmoore4405]

Could have just as easy been a video on correlation vs causation.

[@conjecturemm]

It is slightly different, though.

[@ahmh1000]

I totally love this video. At the minute and a half mark i was like ooooh and had a dumb smile on my face ^^'

[@conjecturemm]

:D

[@KevinVandyTech]

Statistics can't limit Stephen Curry!

[@abcdefg9213]

Completely unrelated comment: Matt, are you a statistician?

[@conjecturemm]

Nope, I am a college student (studying psychology and education) and a KZbinr.

[@AnstonMusic]

Good one.

[@conjecturemm]

Thank you, inspector.

[@AnstonMusic]

***** Oh, wow. xD

[@davidclarke7865]

Great video :)

[@conjecturemm]

Thanks!

[@aniseedus]

Big league or not, this is definitely good.

[@conjecturemm]

Thanks! I'm a little slow on uploading videos right now but jumping back into it soon, haha

[@MrDan710]

You are awesome, you really are! This is so incredible important to know, why is our human brain so bad at times...

[@conjecturemm]

Haha thanks. It's just not something we're good at!

[@ScottWorthington]

You may want to reconsider your boy/girl example. I'm sure you know that the sex of a child is determined by the father. Some men are more likely to produce male children, others are more likely to produce female. So, with each successive male child, the probability of the next one being male increases. Same with females. Statistically speaking, these are not independent events. Gender is not random. As for your final statements, that's just bad science. You designed and executed a testing protocol and when you didn't get the results you wanted, you added more trials until the results supported your predetermined conclusion. How do you know that 20 is the appropriate number if trials? Because the results confirmed your hypothesis? I'm not trying to bust your chops. You're a smart young man and I enjoy your videos. I have a problem with "sciencey stuff" masquerading as science. You can do better.

[@conjecturemm]

Hey, Scott! So as I’ve learned from other commenters, the boy/girl thing isn’t exactly 50% haha. I didn’t know that. But if you substitute this boy/girl example with any other binomial distribution (like flipping a coin with B on one side and G on the other) then this is how it would work. As for your second comment, I get what you mean. But I still think it’s a little unfair to call it science masquerading as science. I wasn’t aiming to perform a legitimate, statistically significant experiment. I was just trying to recreate the results that have been proven in statistically significant experiments/mathematics. My increasing the sample size doesn’t strike me as a bad thing because all it did was *increase* the experiment’s validity-it’s not like I cut out certain numbers or didn’t use certain data. I just increased the amount of trials because science tells us that with enough trials we will see the result Kahneman and I described. Still, I see your point. It seems like I said “well I didn’t get what I was hoping for so I did a second experiment and that looked nicer so I’m sharing that one with you instead of the other one!” But (as I see it) all I did was make the experiment more valid without manipulating or changing data. The original data is visible too, btw. It’s the top row of the city and county numbers. The bottom row was the second set I ran. I will keep this in mind. Thanks for the feedback :)

[@TheImpe98]

Great video! I've always found it facsinating how statistics often are so counterintuitive. You did a great job explaining it, though!

[@conjecturemm]

Thanks!

[@pilapim3491]

Great video!