 that data, and I was to graph it or plot it out, if I look at the dice numbers, there's six numbers on the die, one through six. And if I rolled the die a thousand times, each one of those numbers, we would expect to come up to around 166.67. So our kind of perfect model that we have in our head, which is too perfect, because it doesn't take into effect, I count the randomness, because this is basically a sample, instead of the entire population of dice rules, which we imagine to be like an infinite number of dice rules, it would look like this. Now, if I was to graph that in a histogram, then we've got the dice, one through six, and the expected roll, it would be just a straight line. We would expect all of them to be 167 across the board. And obviously we have now a straight line. And notice that the straight line, you might say that the uniform, that's what the uniform distribution will be. You might say, well, look, there's only one of those, not really a family of curves. But obviously if we rolled the die for some other number, other than a thousand times, if we rolled the die 200 times, we would expect the outcome to be 200 times the point 166.66 on and on, so it'd be 33. So it's actually kind of a family of curves, because the straight line is up here. If we rolled the 200 times, it would have a straight line at the 33. So these are family of curves, which are basically straight lines, which are just straight lines, which are the uniform distribution that we would have. That would be our expected outcome, formula for it, f of x equals c, we're gonna have the same outcome because it's uniform. Nice, easy equation for us. Our predictions are nice and easy, although they're not gonna be perfect, because in real life, there's gonna be the randomness involved. Now, if we were to approximate what actually would happen, if I rolled the dice a thousand times, you could do this in Excel, and you could do it by using the random number generator, which would look like this, random between, and then the bottom number would be one, top number would be six. My voice cracked it. I'm just gonna copy, and if we copied that down a thousand times, I don't think I added all thousand, I only went down to here, but if you do this in the Excel worksheet that we will have as well, you would have a thousand numbers that are approximating, that are randomly generated as a dice rule would be random in theory, right? Out of one through six, so the likelihood of this one coming out a two was, you know, one out of six, right? So we rolled a two, then we rolled a five, then we rolled a three, then we rolled a six, then we rolled a one, a one, a four, and so on and so forth. So if we take then that data, we could say, let's do it this way, we can say, okay, now we've got the dice one through six, we've got the expected rules were even, this is what we expected to happen, but this is what actually happened. Now this actual data we're pulling in from our data set over here by basically counting the numbers that are coming up and the formula in Excel would look something like this, we're gonna say equals count if brackets, we're gonna be picking up our entire range, you could see it goes down to a thousand in Excel and then we want the criteria, so we want you to count every number in this range if it has Q2, which represents this number one. So if you find this number one in the range, count it and it says that that happened 182.