in this video we are going to talk more about the central limit theorem and how it kind of looks like how it works and some of the different variables that go along with it so for for this video I'm using this application through stat Crunch and so we're going to start off with a population so assuming we know something about the population um somewhere hidden deep in the abyss of the Universe we understand that the population looks like this and so to us actually as real humans we don't really know this but let's assume this is the underlying truth and so we want to take a sample because that's something that we realistically can do so if we want to take a sample let's say we take a sample of size two so we go out and we grab two things one time so we grab two things and we get a sample mean and we can see we find a sample mean of 24 and we move forward um the sample mean here is 24 our standard deviation between these two samples is25 and the mean of our sample means so this would be our mux bar is 24.24% now we have a standard deviation of our sample mean so this is going to be our Sigma xbar we can see this exists now and then we can also see that our sample mean has our mean of sample means has started to look closer to our population mean so the more samples that we have the closer our data starts to look like our population now note that our actual sample data not really that just kind of does whatever it wants to do each time so if I do a thousand of these so I just went out and I took a thousand samples of sample size two now I can see that my mean is very close my mean of sample means is very close to my population mean my actual individual sample mean not really it doesn't tell me a whole bunch this is why it's super important to take as many samples as you can and then my standard deviation down here my sample mean standard deviation this is my Sigma xar so if you remember on your equation sheet you have that Sigma s xar is equal to Sigma s/ n so my Sigma squar or Sigma is five so if I take that 5 * 5 so I'm working with a calculator you might want to grab a calculator or write this down and follow along test it out later but if I take 5 * 5 to get my Sigma squar and then divide that by n so n is my sample size so that's going to be this number here so divide by two I get that my Sigma squared xar should be 12.5 this down here is my standard deviation so I need to take the square root of that value so the square root of 12.5 is 35355 so relatively close I add another thousand and another thousand and there we are now that we have 3,000 samples which is crazy getting 3,000 samples is insane um we can see that our standard deviation gets closer and closer and closer and now we are at almost exactly what the theory says our standard deviation of our sample means should be and then we can also see that our sample means have started to look normal so this is what our Central limit theorem says the more samples that we take the more that our sample means will look normal and so that's why we are able to use our sample means um and the normal distribution and the standard normal distribution in order to draw infes so getting a population to end up look that is a normal bell-shaped curve to look like a sample mean bell shaped curve that makes a lot of sense right so now what if we start off with with a population that actually has a uniform distribution so let's say we do something like this that's not what I wanted so let's say we have a normal distribution mean of 24 standard deviation 7.5 so let's say this ranges from what 11 to 35 38 I don't know what the math is there but roughly somewhere in there gives us a let's say it goes from 11 to 35 something along those lines gives us a mean of 24 standard deviation of 7.5 so now if I take a thousand samples it's starting to look kind of normal and so we're getting this bell-shaped curve again this is what our Central limit theorem says the more samples we take ultimately regardless of what the initial population looks like our sample means are going to take on a normal distribution so now let me reset that again now if I take samples of 30 so that magical number of 30 if I take a sample size of 30 one time so I just go out one time this is almost even starting to look normal two so if I do it again and again they're not taking on a uniform distribution they're almost forcing themselves into a normal distribution you can also see down here our sample means are also going towards that normal distribution so if I take a whole bunch you can see that normal distribution at the bottom filling in one by one by one and then if I just add a thousand to speed things up you can see the normal distribution really coming in strong here um and so again those same calculations for the samp the mean of the sample means and the standard deviation of the sample means are going to work out so our mean of sample means should equal our population mean and it's very very close and then our population standard deviation so if we take 7.5 squared / n which is 30 here and then the square root of that we get that our standard deviation of our sample means by Theory should 1369 and we're really really close 1. 382 so all of this to say that the central limit theorem the more samples we have the bigger our sample size the bigger the number of samples we take the more our distribution of what we are actually physically doing will start to mirror a standard normal distribution and that's the theory behind a lot of the work that we do and everything that we will work on moving forward if you have questions please bring them to class um I know that this is one of the topics that's kind of really out there because it's a lot more of the theory behind what we do and not so much of the practicality so please please feel free to bring questions um I can pull this up and play around with it again whatever you guys need um to get through this but that is Central limit theorem