[MUSIC] Hello, so far you discussed population and sample, and the distinction between those two. Just to remind you what difference the between population and sample is. So in terms of what we're looking for, we're not talking about population, we're talking about parameters. What parameters the population has? When we're talking about sample, which is the smallest size of the population, the selected size of the population can be randomly selected, most probably, we're talking about statistics. Now we're going to do some exercises with those notions. Specifically, I will try to explain how we use population in order to measure its mean. So what is mean? Mean is the notion that describes what will be the average of all the population values. If we talk about a simple population, let's take the following example where we have GDP per capita for ten EU countries. In this example you will see Bulgaria, Belgium, Portugal, France and other countries in the European Union that are represented with their GDP, corresponding GDP. So in the table that we have seen just now, the ten countries that represent the sample of the entire population of European countries. And we're interested will find out the average, which is the statistics of this specific sample. In order to find out the average, which we'll denote by X upper bar, we will be able to do that by first of all summing up all the values, And divide it by the sample size. The sample size is ten countries and the sum of all the values, Is 195,557. If we divide it by 10, we find out that the average GDP in our sample is equal to 19,555.7. Another interesting aspect that we can see, Is the VAR, What is the variance? In order to understand the variance, let's try to picture it, first of all, what it represents. Let's take a horizontal line, let's draw a horizontal line which represents GDP. And if we plot all the observation of GDP of all the ten countries, so one, two, three, four, five, six, seven, eight, nine, ten. If we plot all the observations on this line, we'll find out somewhere that the mean is somewhere between here in the middle, lets assume this is our mean or average. The question that we ask ourselves when we want to find the variance is, How far away those observations, Are from the actual mean? And we can do this with this example, but let's try to take a different example. An example that we'll consider for variance will be the following example. We'll consider the population of five firms. And those five firms will have a profit margins and they will be the biggest firms according to the Forbes' magazine in 2005. So first of all, we can do the following. In order to find the variance, as we did before, we will find the mean, Of this population, so we sum up all the values, Of margins, 12 and 5, and divide it by the sample size, in our case, it is 5. If you do the calculation properly, you get that the average margin of this population is 10.8% margin. The next step will be, as I said before, is to find out the differences between the actual observation and the mean of the sample. So the first difference will be, (10- 10.8), the second difference would be (11- 10.8). The third difference would be (16 -10.8). The third difference would be (12- 10.8). And the fifth difference will be (5- 10.8). [COUGH] Those differences represent the difference between the actual observation and the mean. Because we're interested in the overall dispersion of observation from its mean, we will notice the following thing. Some differences, Will be negative. Some differences will be positive. And again, some differences will be negative. It might be the case, that if we sum all of them together, because at the end, we want to find the overall dispersion of all the observation from their mean. What we need to do is we have to get rid of the negative observations. Because, when we sum it might be that they cancel out the positive observation, and we end up with zero of dispersion, which is not entirely true. Because we know there are some observations which are on the right hand side of the mean and some observation on the left hand side of the mean. So in order to get rid of that, what do we do, We square the dispersion around the mean, and summed them up all together. Again if you do it properly, you'll find that this is equal to 62.8. And this will be the variance, this will be the sum of dispersions, but as in the mean, what we need to do, we need to divide it, By the population size or the sample size in this case. And what you get is 12.56. So what does it mean 12.56? 12.56, represents a value that is saying, how far or how dispersed all the observation from the mean. Now remember what we did before, we squared. And the reason why we squared, I remind you, is to get rid of the negative signs here and here. But as we square it, of course, we add an additional information, which is not necessarily needed, say, for our example. So the way to get rid of this, although it's not going to be entirely mathematically correct, what we need to do is, is to square root the whole variance. And when we square root the whole variance remember, when we square and sum that, it's not the same as we square root the figure and get back to the situation where we don't have the squares. It's not mathematically correct, but we have to somehow get rid of those squares. And the way to do that is by square rooting the variance. So when we square root the variance, which is this, We actually get the standard deviation. And the standard deviation in this case, Is 3.5. In other words, the higher the number here, the higher the figure here, the higher the dispersion around the mean of the figures. So the more they are surrounding the mean, the more they're closer to the mean, the lower the variance as well as the standard deviation, will be. The further away they are from the mean, the higher the variance and standard deviation will be. [MUSIC]
