 that randomly push or pull the final value. So in other words, if you think, well, the bell curve has this middle point, and then these other values are kind of clustered around that middle point, and then they basically taper off to the left or the right in somewhat of a uniform fashion. Why might that happen? Well, any scenario like grades, for example, if you have a bunch of people that are of the same skill, or at least they're in the same grade, they've had the same education to that point, then you would expect that the middle point would be somewhere in the middle. Most people would be around the same area if they've had the same education, but there's going to be then a distribution based on whatever IQs, based on study time, based on how they're feeling at the time, and those nudges are going to nudge away from that center point, which kind of would give you this bell shape type of distribution type of scenario. If you think about heights or many different things in nature, you might think that the height of a human being is around whatever the height is, right, 5'10", or whatever. You could say the height of a human being around that, but obviously different things are going to nudge that height. One of those things, well, genetics can, of course, nudge the height higher or lower, and then people's diets and whatnot, and their circumstances throughout life could nudge the heights higher or lower around that middle point, which tends towards possibly like a normal type of distribution. So these influences often average out making the resulting distribution bell shaped. So let's take a look at some practical examples of the Gaussian distribution bell shape curve or normal distribution. So physical traits. So whenever we're thinking about things in nature oftentimes, and we're trying to say, well, how tall is something? How long is something? Often we think about, of course, human beings, but anything else, you know, animals, plants and whatnot, you might see similar types of things where you'd say the heights of men and women, for example, differ in means, but both show bell shaped distributions. In other words, if you looked at men and women, then the men are typically going to be taller on average. That, of course, doesn't mean that there's not going to be some women taller than men. It means that on average, the men are going to be taller than the women. So if you think about a bell shaped type of distribution, you would think that the spread of the bell shape, the standard deviation, would be somewhat similar, but it would be shifted on the man side a little bit to the right because the mean, the middle point would be a little bit to the right. So you'd have similar kind of curves you would expect bell shaped type of curves that would have a similar distribution you would expect in terms of standard deviation, but the men's curve would be a little bit to the right. And so then we have sports statistics, of course. Now remember that sports statistics, and we'll take a look at batting averages and compare a couple of years as one of our examples, the sports statistics is very closely related to performance statistics at a job because that's what sports are. Sports are us judging the performance of players playing the game being their job. So sports conforms itself quite nicely to statistics, but many other things and other jobs can be quantified. And anytime we can quantify them, we want to because that allows us to measure performance well and fairly based on based on standards that are that are uniform and therefore fair across the board, right? So baseball batting averages, for example, differences over the time, but the overall shape remains Gaussian. So in other words, if you had batting averages for the current year versus way back to like the 1920s or something like that, you would expect that because you're talking about something similar to like greats performances, because all of the players in the league should have similar type of skill levels because they're all the best of the best at that particular time, you would expect their performances like grades to be kind of in the middle and then taper off and resulting in somewhat of a bell shaped type of curve, whether you're talking about the 1920s or the or you know, somewhere in 2020 or whatever, right? So but you will also might expect that there would be differences in terms of possibly the mean how the middle point of the distribution and you can get into questions as to why that would be. Are there differences in the way they pitch? Did they change the bats in some way or the ball? Did they tighten the winding of the ball or something? All that kind of stuff. And you would expect that there might be a difference in the spread of the data as well, which again could lead to questions as to why that might be. And that also leads us to possibly be using the Z scores, possibly to help us to compare who's the best not just in one season, but comparing way back when to today or something like that. Games of chance poker hands distributions of average values of card hands from many games. So clearly when we look at games of chance, then statistics and the normal distribution is going to play a vital role in areas there. We have the binomial as a special case. We talked about a binomial distribution in prior presentations and in certain circumstances, the binomial distribution has a bell type of shape. So when in the number of trials is large, binomial distributions approach a normal distribution due to the central limit theorem, the CLT. So probabilities and intervals. Now here's the formula up top. We're not going to get into a lot of detail about the formula, because this is a statistics and Excel course. We're not really diving into the math that deeply, because the Excel is going to help us to apply the math. And that's kind of the point here. We've been given a gift of some having gals, you know, figure out this curve, which is so very, very useful. And our point the point of that now isn't oftentimes when we're thinking about application, at least, to be able to plot this thing out or do the math related to it, it's to be able to recognize that you're looking at a dataset that might conform to a bell shape or normal distribution. And if it does, then you can use the Excel to help you out to to create the normal distribution, which has those beautiful symmetrical characteristics, allowing us to