 to make better predictions about certain types of data sets. So we have the empirical rule. That's the 689599.7 rule. Now you probably wanna memorize these numbers. They seem kind of like random, but they're so very useful because of the shape and the unique characteristics of the normal distribution. So what does that mean? It means about 68% of the data within one standard deviation. So remember the middle point is here. We're just about there, about 74 in this case. And the characteristics of the bell-shaped curve is that it's defined by the middle point as well as the standard deviation. So if we change the middle point, the mean, it would shift to the left or the right. If we change the standard deviation, it's gonna get squatter and wider or taller and thinner. So because those are the two defining characteristics, it works out that we can say that within one standard deviation, so if we took just one standard deviation away from the mean, and I'm just making up the one standard deviation here, then we would find that the area of this actually contains 68% of the data, which is a great thing to know. That's quite precise. That's very nice. Two standard deviations, then I'm just making up where that would be over here, then has 95%. So the vast majority of the data then is gonna be within the two standard deviation if we have data that conforms to a bell-shaped type of curve and three standard deviations. I'm just making up where that would be again, would be the vast, vast majority of the data, 99.7% of the data within the three standard deviations. Now note that in theory, the bell-shaped curve is one of those things that go on forever, so it goes on this way and it goes on all forever to the left and to the right, but if you go three standard deviations and if you go one, if you go four standard deviations, then again you have the vast, vast majority of the data are gonna fall into those and the fact that that's so precise is quite nice. Now remember that we're not talking about the actual data set, like if we're talking about actual data, we might plot it out in the format of a histogram and it might approximate a bell-shaped curve enough then for us to say let's use the bell-shaped curve, which is the perfect, beautiful, symmetrical curve in order to give us that nice predictive power of having the formula as well as this nice property of having 68% of the data within one standard deviation, 95% within two and 99.7% of the data in the three standard deviations. Applications beyond basic statistics. So we've got quality control, Six Sigma and other methodologies rely heavily on the normal distribution to assess product quality. So many products that are created, there's a volume component. So when you're talking about a product where all the products need to be the same and you're trying to crank out a whole lot of them like potato chips or candy like Tootsie rolls and that kind of stuff, then all of the products that are being output should be the same, but clearly there's still gonna be a little bit different. So if you're thinking about how many chips are in a particular bag, you might weigh the bag and one bag is gonna weigh a little bit different than the other bag, but you're trying to reduce the differences down to as small a reduction as possible in order to maximize efficiency. And if you think about that, the concept of us thinking of why something conforms to a bell curve in that we're saying, well, most of the results are in the middle because we're aiming for the bag to be weighing this amount, but then there's gonna be some variant of error that will be in there because of the packaging process. And so then the question is those errors will have a normal distribution typically and we can start to think about those errors and how we can measure that to then tone down or hone down our packaging process. Clearly in finance, stock returns, portfolio theories and risk assessments will often be using statistical kind of concepts like the standard deviation, the variance and so on and the distribution, normal distribution, social sciences. So psychology, sociology and other fields use the Gaussian curve for various assessments. So in science and especially in social science type of fields, then it might be quite common as we're looking at things in nature that we might be applying and using statistical concepts such as the normal distribution and scientific research testing hypotheses, analyzing sample means and drawing inferences. So whenever we're doing sampling and that kind of thing which is quite clearly quite common in sciences as well as like in political type of areas we need to know the statistics behind it including the concept of the normal distribution type of concepts. And this is something that is highly, highly important especially in fields like social sciences and those are fields where people might not be as mathematically inclined and these concepts are not just intuitive concepts. I mean, it takes some kind of thinking, it takes working with the stuff for a while to basically wrap your mind around these areas. So if we're in the social sciences you can oftentimes maybe able to create a lot of value for yourself by being someone that understands these concepts so that you can work with the data well and efficiently.