[SOUND] I'm going to use this animation to show you how central limit theorem works in practice. If I have an underlying population with a normal distribution it is not a surprise that if I take samples from here over and over again, those sample means will also if plotted will look normal. So, what I'm going to do is I'm going to change my underlying population to something other than normal by just randomly clicking observations and this is what my population looks like and as you can see it's far from being normal. As a matter of fact if you look here it says that for the population distribution that I have created, the mean is 14.80, the median is 11, so far from being the same. So it's a bell shaped curve where the require needs to be almost the same. And it has a standard deviation of 9.45. So now what I'm going to do is that I'm going to animate it. And, first time that I will click on this, what you will see is that I have two plots being plotted here. Distribution for means when sample size is 5, and distribution for means then sample size is 25. So the first time I click on this it's going to draw five random observations from my population, drop them down here, and then plot the average for those five down here. Then it will go and take 25 random observations from my population, drop them here and then take the averages of those 25 and plot them here. So I'm going to animate this one so we can see one, two, three, four, five, and their sample mean came here. As you can see up here now it's dropping 25 from that and once it has 25 observation it will drop the mean of those 25 right here which just happened. going to show you one more time what this will look like. So this is the second sample that it will be drawn from that population. The sample of five is being taken. Their mean came down and now the second sample of 25 is being taken. And the second one came out. So now you see that I have two repetitions, which means I have taken two samples of 5, and 2 samples of 25 and these are the respective means, so what you're seeing here is that this is a distribution of my population. What is going to be developed here is the distribution of the sample means for 5 and 25. And central limit theorem will tell me that if I do this often enough this will look normal even though this is not normal. So, animated 5 times and you can see what happens. This is 7 repetitions and 7 repetitions. So and when I say automatically do it, it's not going to show you how it's dropping it, but it's the same principles happen. As you can see it's right now nothing near normal distribution. But it is about repeating this a lot so let me repeat it 10,000 times and see what happens to the distribution of the sample means. Remember, these are sample means and we are looking at the distribution of those. So what happens at 10,000? It is looking a lot more normal! Look at it. It's almost normal here. And of course, it's tighter. As the sample size goes up. What happens if I list 100,000 times. Again, you will see this. This is what the Central Limit Theorem shows us. The beauty of the Central Limit Theorem is that if you have any population, given a large enough sample size, and this isn't even large enough, we want it to be at least 30 or more, we will start getting a sampling distribution that is normal. And this is why then I can say what is the variability I expect, if I take just a sample out of here. So if I took a sample and all of the sudden the mean is here, it's quite an outlier. It doesn't happen that often. So, what I'm going to do now is show you how these values are calculated. So this is the standard deviation of the population and the shape of this is what we call standard error. And the standard error is calculated by taking the value of the population standard deviation and dividing it by square root of the sample size. So, if I take 9.45 and divide it by square root of 5, I should get 4.22 likewise if I take 9.45 and divide it by square root of 25, I should get standard error of 1.89. so what it says is that the distribution of these means is lot more cohesive, lot less varied compared to the population. Furthermore, the larger your sample size gets, the tighter these samples will become, these sample means will become. They're clustered around one central point. Now look at what we get for our estimation of the mean. Remember, the mean of our population is 14.8. What happened to the mean of the sampling distribution 14.78? Very close. When we did it for N=25, even closer, 14.79. And this is the beauty and the strength of Central Limit theorem, that shows that if you do sampling, if you do your sampling correctly, then you should get a sample that is a good and fair representation of your population, therefore the statistics that you come up with using your sample information would be a good point estimator of your population parameter.