On the next page, we are then going to come up with three more words to answer this second question: How many mounds are there in my distribution? Now, I need to first emphasize, what does the word 'mound' mean? I need to emphasize a mound is not a single bar, but rather a mound is like a peak. It's like a peak in the mountains. And so, if we are to trace the outside of each of these histograms, you are wanting to ask yourselves: How many peaks did I have in my mountain? Did I go up the mountain once, did I go up it once, hit a valley, and then go up it a second time? Did I have one peak and then a valley, and then a second peak and then a valley, and then a third peak and then a valley? When it comes to this concept of modes, you're not looking at the bars, but rather you're looking at the outline, the shadow of the histogram. And you're asking, how many peaks did this histogram have? Because in particular, if your graph has exactly one mound, we say that this is a unimodal description. When there's only one clear point where you're going up and then down, we call that a unimodal graph. In situations where there are two distinct mounds, two distinct peaks, we call that a bimodal graph. 'Uni' and 'bi' are making a lot of sense; they're the prefixes for one and two. Lastly, when you have three or more mounds, three or more peaks, we call it multimodal. And I want you to see that we have three distinct mounds in this last graph. And yes, yes, it is intentional that I put two bars with that second mound because I'm trying to emphasize that even though two bars made that second mount, it only made one peak. One, two, three peaks. Now, when it comes to this particular statistics class, our particular elementary statistics class, majority of the graphs that we are going to look at are going to be unimodal. In Math 10, most of the graphs are going to be unimodal. And really, it's because this is an elementary basic starting statistics class. And so, because of that, most of the graphs in this class are going to be unimodal. But we are going to talk about bimodal just a little bit, just a little bit, just so we understand why these unimodal graphs are going to help us make these bimodal graphs. Alright, so let's talk a little bit about the bimodal and multimodal graphs. Ultimately, when it comes to looking at graphs with multiple mounds—when I say multiple mounds, I mean two, I mean three, I mean four, two or more mounds—when you're looking at graphs with two or more mounds, generally, the reason why you will have multiple mounds is because you're looking at two very different groups, two very different groups that have been combined into a single graph. What do I mean by that? Well, the graph on the left is representing the height of all statistics students. So when I say all, I mean I am looking at both females and males. Alright, and if we're talking about height, we all understand that generally when it comes to height, females are going to be ever so shorter than males. And so, what we can do is think about each group, each gender separately. We can see here when looking at the female's height, we can see that the most common height is 65 inches. When looking at the men's height, we can see the most common height is somewhere between 70 and 72 inches. I want you guys to see that when we are looking at a single group, notice when we are looking at a single group in both situations, we were having a unimodal, one-mound graph. And yet, when we then combined all the females and males into one big group and graphed it, I want you guys to see here we ended up with a bimodal graph. Why? Because the first mound is representing where the majority of the females had their height, at around 65 inches, and that the second mound is representing where the majority of the males have their height. And so, because of that, I wanted to just show you guys very visually why do bimodal graphs occur, why do multimodal graphs occur? They occur because when you look at the individual groups—just the females, just the males—you can see in these unimodal graphs where those main mounds occur and how they will create the various peaks. Why did I not include that mound on the far right with the bimodal graph in the top? It's because the peaks are representing distinct and high peaks. It's just like if you were to go climbing Mount Everest. You're going to climb Mount Everest and you hit various peaks. You would count those as your successes, those as your distinguished peaks, versus if you climbed over a little rock, you would not count that as a victory of climbing a peak. And so, when it comes to thinking of these peaks, peaks are emphasizing even what we think about with mountains. It's the point, the top of those mountains. And so, again, when it comes to thinking about the top of the mountains, that's the reason why when looking at these unimodal graphs of one group, it is going to be those singular tallest peaks that then form the two tallest peaks in the bimodal graph. The whole point of graphs is just to make observations, because what we'll find with graphs is, while they are helpful, they don't give us the details we need. It's kind of the equivalent of when you use Google Maps, right? When you use Google Maps and you type in your point A to B, notice that originally it gives you that aerial view of the road map between San Francisco to LA. And while that's helpful, it helps us to understand we're moving southward, it helps us to understand we're going through the middle of California, it doesn't give us the details of where do we turn left, right? What exits do we take? And so, in the same way, the graph is that aerial view of us just trying to get a sense of what is happening in this graph. And so, in particular, what we're seeing when these bimodal or multimodal graphs occur is that there are multiple values that are going to occur most often. So, there are two more distributions I want to define for you guys. First one is the word 'uniform'. When it comes to the word 'uniform' and we think about kids who go to private schools and they wear uniforms to school, what is that emphasizing? What is the uniform emphasizing? The same. The same clothes. In the same way, when you see the word 'uniform' in statistics, it's emphasizing the idea of 'same'. Same, except what's the same this time? It's that we have the same frequency for all the observations. And so, how does same frequency translate into the graph? It means then that every single data value is the same, all the same value, creating a horizontal line, meaning there's no mounds. Uniform distributions will have no mounds because every observation has the same frequency. We're going to very, very, very lightly talk about uniform distributions one more time in Chapter Six, and that'll pretty much be it. And it's because more often than not, your observations are not going to all be the same, but I wanted to emphasize this definition so that you guys, if you ever come across it, understand uniform distribution means same frequency. So, instead of having a mound, it's just a horizontal line. Every single value has the same frequency. Alright, let's go back to the top of 2.2, and let's remember that when it comes to shape, you look at its symmetry—symmetric or skewed—and you look at its mounds. Does it have one, two, or three mounds? Now, when it comes to statistics, you will find majority of the data will be symmetric. A lot of data is symmetric, and more so, as I already said earlier, a lot of the data you look at will be unimodal—one value that will occur the most. And so, drum roll, there's a very important word we need to introduce in statistics, and it's a distribution that is both symmetric and unimodal. When you have a distribution that is both symmetric and unimodal, the distribution is called normal. You 100% need to know what this word is. We are going to discuss normal distributions in Chapter Three, in Chapter Six, in Chapter Seven, Chapter Eight, and Chapter Nine. Yes, half of the material in this class is going to discuss normal distributions. And so, you should tattoo this word on your arm. Normal means symmetric and unimodal. Normal means that your graph is symmetric—left and right-hand side or mirror images of each other, or at least as close to mirror images of each other—and second, it will be a unimodal graph, only one distinct peak. The final and third question we ask for the shape of numerical data is outliers. Outliers are those values that are extremely large or extremely small that don't fit what the rest of the data looks like. And just like what was asked in the chat earlier, the question is, you know, it seems like we're eyeballing it. Including outliers, is there a mathematical way for us to identify these outliers? And in particular, this question is, yes, in Section 3.5, I'm actually going to teach you guys an explicit formula to be able to identify outliers. But until then, until then, we're just going to eyeball it just so that we can get a sense of even generally if we think there are outliers.