wait, but is this really a demonstration of the central limit theorem? isn't it just a demonstration of the fact that the final location of a random walk is normally distributed? i mean you're basically just applying the law of large numbers to a random walk, not SUMMING the random walks. knowwhatimean?
It is a wonderfull mnemonic to remember the central limit theorem, I'll never forget it after watching this. But this should be accellarated. at least 5 times. One minute video would be optimal.
In the video, the balls are scattering because the simulation makes them :-) Basically at every downward step, each ball makes a random decision about whether to go right or left. Over time, the accumulated differences in random decisions result in very different positions in space! This same effect occurs in real life in various physical devices people have created to demonstrate this mathematical truth. In those devices, the falling balls tend to impact small obstacles like pins.
The PEAR laboratory (now closed) at Princeton University had a physical ball drop machine to demonstrate probabilistic principles such as the Central Limit Theorem. I wonder what happened to it.
How is this meaningful in anyway whatsoever, all the balls are being droped in from the center, it seems logical that there is a higher probability of the balls landing in the center, no?
Thanks for the comment! I agree it is obvious that the balls will tend to land in the center. The mathematics predicts more, however... namely that the resulting distribution will have a particular shape (namely that of a gaussian).
@Booredyeah Yes, that's why the center-of-mass of the distribution is directly under the point where the particles enter the frame. But the details of the shape of the curve require some deeper reasoning.
The CLT doesn't say that the sum of enough iid rv's looks like a normal dist. It says that with a sequence of iid rv's from any distribution, as n approaches infinity, the distribution of the sample means will approach a N(μ, σ²/n). The sum of the random variables is only used to define a new set of rv's so that the means will approach a standard normal.
@palui But the sum of iid rv's and sample mean of idd rv's only differ by a factor of 1/n. Thus, they should look the same, only in a different scale. So I think the CLT applies to this experiment very nicely.
It's interesting how the middle two values--predicted to be greatest--do not follow this prediction. Is this due to modeling error because the particles must move +dx or -dx at the first time interval? This means there should be two events, +dx then -dx, to realign to the middle. Am I simply overthinking this?
I think you are over-thinking it. The issue is likely just that N = 18000. If N = 1800000, they would probably be closer to prediction. Probably. Random is often more random than you think.
Indeed this arrangement is a probabilistic anomaly, the fact that the beads are forced to move on the first iteration should not matter, they have a fifty percent chance of skewing left and fifty of skewing right. The law of large numbers says that as the number of falling beads approaches infinity this first iteration jump should not matter and the left skewdness and right skewdness will cancel out giving a true normal distribution.
Actually, DrNate may be close to correct. This resembles a modeling error based on the ratio of the dx value and band width. A simple example would be a dx of 5 and a band width of 14 -- two out of every five bands would record two impact events, boosting its volume (try it). The pattern here is less dramatic but is noticable: from center, to the left and right, bands 2,4-5,7-8,10-11,13-14 are disproportionate to 1,3,6,9,12. The dx/band ratio is the cause. Maybe Mr Schaeffer can provide details.
I love the CTL, it's my favorite in all of mathematical statistics. One can give a "physics" argument, feathers, sand, etc. But you must keep in mind that the Browian charastic (randomness) in a particles motion is important here. Sand is fall in a much more predictable fasion due to it density and shape. Feathers "flutter" therefore the drag generated by the surface in free fall simulates random fluctuations in the feathers motion, while a sand grains path is much more predictable.
if you sample until infinity (limit n tends to inf) why doesn't the distribution degenerate to a point mass since we have sampled the entire population so the sample mean should equal the population mean parameter with no variance?
Yes, there is convergence as n goes to infinity. But instead of the distribution degenerating to a point, it converges to a normal distribution as n increases (which is the content of the Central Limit Theorem). For instance, the proportion of paths falling into one of the buckets gets closer and closer (in the sense of probability, i.e. that large deviations from the expected value are unlikely) to that predicted by the normal distribution (shown in the video by the blue line).
did this help you in everyday life? did you ever calculate the probability of a type II error for the confidence interval at a 97% level of the chances of getting 4 stars in the clt video? ...kidding. but does really help later in life..stats?
that doesnt answer the question: most businesses hire statisticians that only dicate themselves to doing statistics....by computer; so...i mean i guess scientists could use stuff like this....but still whats the probability of the average guy using this?...lol get it? "the probability"....lol...ok
Next time you take any medicine you better be damned grateful that somebody knew how to calculate confidence intervals. Also, every advertising company knows fine well that their businesses depend on collecting data and analysing them. Every engineering or manufacturing process is optimized by using statistics.
So the answer you seek is: YES stats does really help in later life. How do you think government runs? How do you think transportation is organised? Jesus you have a small mind.
@bettiethebadger The CLT is defined to be used with a population N that isn't fixed, such as obtaining rv's from a die roll. The rv's, in this case, can be generated without n approaching N.
that is not amazing at all what is it trying to prove? Of course more of the balls will be in the middle if they are originating from the middle at the top. Pretty boring
yet not all of the particles land in the exact downward position. they are centered around a certain point. go try it with physical materials (sand e.g.) and it will yield the very same results.
You guys are misunderstanding it. This will NOT happend with sand. But try with something lighter like feathers.
The point is not that most land in the middle. The point is that not only do they land in the middle as anyone would expect, but they form a specific curve as you see of probability, not just a random shape with most in the middle.
You´re wrong about the sand buddy. It doesn´t matter if you do it with sand, feathers or something else. What matters is the "n" number of times you repeat the same experiment. If "n" is large enough, you will get an aproximation to a normal distribution. By other words... as "n" tends to infite, you aproximate to a normal distribution. Imagine a little boy at a beach throughing sand behind his back. The more times he troughs, the more the "mount" will look normally distributed.
The question is: did it approximate well enough? What about the gap on the left side? Is it within "chance" or a mysterious flaw in the random number generator?
The Seattle Science Center had a display of this for YEARS 70's to 90's....it used real balls, and was 15-20 feet high. The balls were taken via toothed scoops on chain drive on the left side to the top, and dumped in the center funnel, then went through a field of pegs, then the tall catch boxes, as above.
It was mostly glass, so you could see everything. After it was full (to the curve) a bell would ring, and metal plates would swivel down in each compartment, dumping the balls into the base. It was called the probability machine!!
Yes, I got the idea from one of my favorite exhibits at the Chicago Museum of Science and Industry, which was exactly as you describe! Guess that many science museums had similar displays. It's amazing that such a deep theorem of abstract mathematics has a physical demonstration.
it's basically distribution.. it'll apply to statistics
lovelplants 1 month ago
Y ESAS MAMADAS QUE PUTO GRINGOO!!!!
machincuisi 3 months ago
Let's see if I can get this comment seen in my statistics class.
boxedbocchan 4 months ago
The only reason I came here is because Roland Good III told me too!
Suckittrebek5252 4 months ago
Nothing random about the results
RayWillJohnson 5 months ago
Thought I saw purple Jaws with blue teeth... never-mind... carry on.
bosshoss502 7 months ago
wait, but is this really a demonstration of the central limit theorem? isn't it just a demonstration of the fact that the final location of a random walk is normally distributed? i mean you're basically just applying the law of large numbers to a random walk, not SUMMING the random walks. knowwhatimean?
slonopochron3000 1 year ago
this is amazing.
slonopochron3000 1 year ago
It is a wonderfull mnemonic to remember the central limit theorem, I'll never forget it after watching this. But this should be accellarated. at least 5 times. One minute video would be optimal.
2002budokan 1 year ago
In the video, the balls are scattering because the simulation makes them :-) Basically at every downward step, each ball makes a random decision about whether to go right or left. Over time, the accumulated differences in random decisions result in very different positions in space! This same effect occurs in real life in various physical devices people have created to demonstrate this mathematical truth. In those devices, the falling balls tend to impact small obstacles like pins.
benschaeffer 1 year ago
what causes them to scatter though?
123qwerty 1 year ago
The PEAR laboratory (now closed) at Princeton University had a physical ball drop machine to demonstrate probabilistic principles such as the Central Limit Theorem. I wonder what happened to it.
deskset24 1 year ago
How would you 'add together' two normally distributed datasets
dvaidr 1 year ago
How is this meaningful in anyway whatsoever, all the balls are being droped in from the center, it seems logical that there is a higher probability of the balls landing in the center, no?
Booredyeah 1 year ago
Thanks for the comment! I agree it is obvious that the balls will tend to land in the center. The mathematics predicts more, however... namely that the resulting distribution will have a particular shape (namely that of a gaussian).
benschaeffer 1 year ago
@Booredyeah Yes, that's why the center-of-mass of the distribution is directly under the point where the particles enter the frame. But the details of the shape of the curve require some deeper reasoning.
snuchia 1 year ago
that looks fake.....
LazyBillyDog 1 year ago
Very nice! Thanks for sharing!
racastilho 1 year ago
The CLT doesn't say that the sum of enough iid rv's looks like a normal dist. It says that with a sequence of iid rv's from any distribution, as n approaches infinity, the distribution of the sample means will approach a N(μ, σ²/n). The sum of the random variables is only used to define a new set of rv's so that the means will approach a standard normal.
palui 1 year ago
@palui But the sum of iid rv's and sample mean of idd rv's only differ by a factor of 1/n. Thus, they should look the same, only in a different scale. So I think the CLT applies to this experiment very nicely.
racastilho 1 year ago
It's interesting how the middle two values--predicted to be greatest--do not follow this prediction. Is this due to modeling error because the particles must move +dx or -dx at the first time interval? This means there should be two events, +dx then -dx, to realign to the middle. Am I simply overthinking this?
DrNate89 2 years ago 2
I think you are over-thinking it. The issue is likely just that N = 18000. If N = 1800000, they would probably be closer to prediction. Probably. Random is often more random than you think.
richdiesal 2 years ago
Indeed this arrangement is a probabilistic anomaly, the fact that the beads are forced to move on the first iteration should not matter, they have a fifty percent chance of skewing left and fifty of skewing right. The law of large numbers says that as the number of falling beads approaches infinity this first iteration jump should not matter and the left skewdness and right skewdness will cancel out giving a true normal distribution.
Chrisisnotyourmom 2 years ago
Actually, DrNate may be close to correct. This resembles a modeling error based on the ratio of the dx value and band width. A simple example would be a dx of 5 and a band width of 14 -- two out of every five bands would record two impact events, boosting its volume (try it). The pattern here is less dramatic but is noticable: from center, to the left and right, bands 2,4-5,7-8,10-11,13-14 are disproportionate to 1,3,6,9,12. The dx/band ratio is the cause. Maybe Mr Schaeffer can provide details.
coulie27 2 years ago
@Chrisisnotyourmom You are referring to the CLT, not the LLN. They are connected, yet also are different things. Cheers.
caviper1 1 year ago
Comment removed
Theoneandonlysparky 2 years ago
This video's interesting if you have quantum mechanics in mind.
GuerrillaForce 2 years ago
Very nice, i have also seen this with mario Kart on the wii - i have a video as a response its cool!
natacus2007 2 years ago
Twist ending! The CLT is disproven.
geoffgurak 3 years ago
I love the CTL, it's my favorite in all of mathematical statistics. One can give a "physics" argument, feathers, sand, etc. But you must keep in mind that the Browian charastic (randomness) in a particles motion is important here. Sand is fall in a much more predictable fasion due to it density and shape. Feathers "flutter" therefore the drag generated by the surface in free fall simulates random fluctuations in the feathers motion, while a sand grains path is much more predictable.
69erthx1138 3 years ago
REMEMBER: You're unique, just like everyone else.
Amishman35 3 years ago 6
This comment has received too many negative votes show
what is this bullshiot
BigRy8 3 years ago
if you sample until infinity (limit n tends to inf) why doesn't the distribution degenerate to a point mass since we have sampled the entire population so the sample mean should equal the population mean parameter with no variance?
bettiethebadger 3 years ago
Yes, there is convergence as n goes to infinity. But instead of the distribution degenerating to a point, it converges to a normal distribution as n increases (which is the content of the Central Limit Theorem). For instance, the proportion of paths falling into one of the buckets gets closer and closer (in the sense of probability, i.e. that large deviations from the expected value are unlikely) to that predicted by the normal distribution (shown in the video by the blue line).
benschaeffer 3 years ago
did this help you in everyday life? did you ever calculate the probability of a type II error for the confidence interval at a 97% level of the chances of getting 4 stars in the clt video? ...kidding. but does really help later in life..stats?
rrbr1991 3 years ago
Its required for any business and science related job.
richi1173 3 years ago
that doesnt answer the question: most businesses hire statisticians that only dicate themselves to doing statistics....by computer; so...i mean i guess scientists could use stuff like this....but still whats the probability of the average guy using this?...lol get it? "the probability"....lol...ok
rrbr1991 3 years ago
Well, if your going to interpret data.
richi1173 3 years ago
Next time you take any medicine you better be damned grateful that somebody knew how to calculate confidence intervals. Also, every advertising company knows fine well that their businesses depend on collecting data and analysing them. Every engineering or manufacturing process is optimized by using statistics.
So the answer you seek is: YES stats does really help in later life. How do you think government runs? How do you think transportation is organised? Jesus you have a small mind.
Theoneandonlysparky 2 years ago 2
@bettiethebadger The CLT is defined to be used with a population N that isn't fixed, such as obtaining rv's from a die roll. The rv's, in this case, can be generated without n approaching N.
palui 1 year ago
Sorry about the errors ... by through i meant throwing of course. My english sucks :P
iamtchiko4 3 years ago
vry nicely done
Cagebreaker21 4 years ago
that is not amazing at all what is it trying to prove? Of course more of the balls will be in the middle if they are originating from the middle at the top. Pretty boring
SmakYourSelfRightNow 4 years ago
yet not all of the particles land in the exact downward position. they are centered around a certain point. go try it with physical materials (sand e.g.) and it will yield the very same results.
deaschowieda 4 years ago
You guys are misunderstanding it. This will NOT happend with sand. But try with something lighter like feathers.
The point is not that most land in the middle. The point is that not only do they land in the middle as anyone would expect, but they form a specific curve as you see of probability, not just a random shape with most in the middle.
jelleoelle 3 years ago 2
You´re wrong about the sand buddy. It doesn´t matter if you do it with sand, feathers or something else. What matters is the "n" number of times you repeat the same experiment. If "n" is large enough, you will get an aproximation to a normal distribution. By other words... as "n" tends to infite, you aproximate to a normal distribution. Imagine a little boy at a beach throughing sand behind his back. The more times he troughs, the more the "mount" will look normally distributed.
iamtchiko4 3 years ago
Chance is not allways "Random" in that Random is vary much a value label or personal judgement.
richardbrompton 4 years ago
The question is: did it approximate well enough? What about the gap on the left side? Is it within "chance" or a mysterious flaw in the random number generator?
clray123 4 years ago
The Seattle Science Center had a display of this for YEARS 70's to 90's....it used real balls, and was 15-20 feet high. The balls were taken via toothed scoops on chain drive on the left side to the top, and dumped in the center funnel, then went through a field of pegs, then the tall catch boxes, as above.
VideoJunkei 4 years ago
It was mostly glass, so you could see everything. After it was full (to the curve) a bell would ring, and metal plates would swivel down in each compartment, dumping the balls into the base. It was called the probability machine!!
VideoJunkei 4 years ago
Yes, I got the idea from one of my favorite exhibits at the Chicago Museum of Science and Industry, which was exactly as you describe! Guess that many science museums had similar displays. It's amazing that such a deep theorem of abstract mathematics has a physical demonstration.
benschaeffer 4 years ago
I counted 2,935,126,274,643,951 balls in this experiment. Anyone got a different count?
onionpizza 4 years ago 7
Hahah That was funny!
besnikl 4 years ago
ya everyone got a count??? :)
wwwdotcomdotmy 4 years ago
Congratulations, Rainman.
clray123 4 years ago