 Statistics and Excel. Uniform distributions with dice. Get ready, taking a deep breath, holding it in for 10 seconds, looking forward to a smooth soothing Excel. Here we are in Excel. If you don't have access to this workbook, that's okay because we'll basically build this from a blank worksheet. But if you do have access, three tabs down below, we've got the example, practice blank, example in essence, answer key, practice tab, having pre-formatted cells so you can get right to the heart of the practice problem. Blank tab, blank worksheet, so we can practice formatting the cells within Excel as we work through the practice problem. Let's look at the example tab to get an idea of where we will be going as we think about a uniform type distribution using an example of rolling a dice. Now note that the die has six sides, six number on the die. If we thought about rolling the dice an infinite amount of times, which we can imagine to be the entire population, an infinite amount of rules, we would expect basically a uniform type of distribution. Then we can imagine, well what if we rolled it a finite amount of times and plotted out the distributions of a finite amount of times of rules, and we can compare that then to the uniform distribution. So that's what we will do. Let's go to the blank tab and let's first, so we're imagining the dice rule. So remember there's six sides on the die. So what would be the expected rules of any number if we were to roll it, let's say a thousand times. So we're going to say rules, let's say a thousand times, and let's format our cells. Hold on a sec, let's select the entire worksheet, right click on the cells and then format them. I'm going to go into the currency, negative numbers bracketed, make them no dollar sign and no decimals and okay. I'm also going to make it bold. You may not need to, but I'm going to work it bold here. Home tab, font group, bold. So there we have it. I'm going to hold down control and scroll in a bit. So I'm currently at the 265% on the scroll in. So I'm going to imagine that we roll it, let's roll the die a finite amount of times, which is going to be a thousand times as opposed to the population, which would be an infinite amount of times that we can imagine rolling it. And then the outcomes, we're going to imagine any one number. The odds on each rule of any one number coming up is going to be equal to one out of six. So there's six numbers on the dice. Let's go ahead and format that number. Let's make it a percent and add some decimals. So six numbers on the dice. We would expect then a one in six chance on each rule for any one number on the dice. So the expected, let's then say, well the expected rules of any number, expected rules of any one number, then if I roll it one thousand times and I have any one number, whether that be a one, two, three, four, five, or six, I would expect how many times for that number to come up. Well, if each rule has a one six chance, it would be one thousand times that percent. So we'd get one sixty seven about, if I add a couple decimals, home tab, number group, a couple decimals, one sixty six point six six on forever. So that would be what the expected results would be in essence for one number if we rolled it a finite amount of times, one thousand times. We would expect to see one hundred and sixty six point six six, number ones, two, three, and four. Now note that this expected result is actually impossible to do, because I can't get an outcome that's not going to be a whole number. So note that we're basically making a model here, a prediction based on what we know can't actually happen in real life, because I can't get a rule that's going to be not a whole number. But the model is still of course useful, because we can get the expected outcome with the model. So if I then, let's make a skinny C here, and then the headers of our table, I'm going to say these are the dice numbers, and then I'll tab and say these are going to be the expected number of rules. I'll just say expected rules and enter. I'm going to format these now by selecting these two up top. We're going to go to the home tab, alignment group, wrap the text, and then alignment group and center the text. I'm going to go to the font group, bucket drop down, make it black, and then the drop down, make it white. All right. So there we have it. And I'm going to say the number of rules is going to be one, two, three, four, five, six. And then we're going to say, what are the expected outcomes for each of them? Each of them, if we roll them a thousand times, is going to be 166.67, is going to be the expected outcome for each of them. So I'll just say this equals this outcome. And let's say F4, that's going to put an absolute value or dollar sign before the B and the three. And then I'll take that and just copy it down and put my cursor in the fill handle, copy that down. There we have it. So the total then that we would expect to have in kind of our perfect world would be equal to sum. And by the way, I'm going to start using some more keystrokes sometimes. You could hit alt enter here. Let's do that again. Alt enter. And then it'll try to sum up what's above it. So notice that keystroke could be a lot faster oftentimes whenever you're using some function, alt enter. And so there we have it. And so there's the thousand rules. That's what we would kind of expect to happen. Let's put an underline here, home tab, font group and underline it. Let's make this into our kind of format that we've been using. I'm going to select these items, font group, drop down on the bucket. And if you don't have that blue, it's in this. I'm going to make it that blue. That's the blue I like. And then font group, drop down and all borders. We can also make this one a little bit more skinny so that we can just trim this up. Let's do the same over here as well, making that home tab, font group, blue and bordered. Okay, so there we have that. Now, if we were to plot this, then I can plot this out and just say, okay, well, if I select these items, these two, and I was going to the insert and I'm going to use a bar chart to plot this one. So charts, drop down. We're going to make a bar chart, which is going to look kind of like a histogram. I'm going to pull this over, right? Because I'm going to say the numbers I want on the bottom. And I'm going to close this up a little bit. And so then I need to adjust the data. So I'm going to go in here and I'm going to say, let's go to the data up top. So I'm in the chart design and then the data. And what I want on this side, which is going to be on the y-axis, is just the expected roles. So I don't want this one, or I could basically delete this one, delete and I want the expected roles and then over here, I'd like to make sure that I have my actual labels, not the ones that they're going to make up. So I'm going to copy these items and OK. So there we have it. We're going to say OK. And it's picking up the name of expected roles. I don't really need the legend over here. I could add the names on the side. I could say these are going to be the data labels. I could add data labels and say there's the data labels and then we could also add access titles if we wanted to. So on the access titles, this is going to be the results. So I could say this equals the expected roles and on this side, I'm selecting it. I'm just going to say this equals the dice numbers. So you can format your chart thusly. And notice sometimes now that I have actually my chart information, I might not even need...