 This one is taking the 167 divided by that same total divided by the 1034, that gives us about the 16.15%. So if we do that all the way down and sum it up, then it should add up to 100% because we're taking the percent of the total. Now we can compare, say, the likelihood based on the bell curve of 13.2% to be 72 inches versus the actual data, which was 14.7. We can look at the difference then between here and here and see how close we were at each of those inch points, those inches. And that could give us more indication as to whether the bell curve would be appropriate to use or not. Then we have the Z score. Now the Z score is now we want to represent instead of with an X with a Z. So we can say how close to that middle point are we? So when we look at this in terms of a graph, the middle point on our bell curve is going to be the mean. The Z score is this down here and if it was at zero, that would be the mean. So we're counting the higher the Z score, both in negative or positive, the farther away it is from the mean zero is at that middle point. So how do you calculate it? 64 in this case is the X, the number of inches minus the average number of inches, which is 73.7, divided by the spread standard deviation 2.3, and that gives us our 4.21 in this case. So if we were to take an individual, we can say this is their actual height in inches, the X, or we can say this is how far away they are from the middle point of the population, the mean, which is the Z score in terms of standard deviations. So you can see it gets closer and closer to the mean, and then right here we're at that middle point. So remember when we're talking about bell curves, usually classically the middle point is kind of good because that's normal. Usually you kind of want to be normal most of the time. Although when you're talking about heights, obviously being taller for certain things is better. So if you're above the norm, that might be an advantage up to a certain degree. And when you're talking about pitching, then you would think there would be some kind of optimal body type or optimal height where possibly most pitchers would fall into in order to have the body mechanics to be maximized, you would think. That would be one assumption, although different pitchers might do different things. But then you can see this is going to be above the mean on this side. And then if I go into, so that's the Z score. So if I was to graph this then, we can graph the actual, this is the actual data on top of the percent. So we graphed the actual kind of bell curve on top of the data so that we can see they're pretty closely lined up. Now this is going to be a bell curve that's in the format of an area using not the actual data, but the bell curve. And if we just look at it without the orange bit here, you can see that we have the bell curve that's centered around this center point. And then it's in that bell shaped curve. And we can then look at it in terms of the X's, the inches here, and we can also look at it in terms of the Z score. We added the two X axes down below, which is really neat and we do that in Excel so you can kind of check that out. Now when we ask different questions, the common questions we might ask is what's going to be the area above a certain point? What's going to be the area below a certain point? And we might ask what's going to be the area in between a certain point? Now last time we looked at the idea that you can kind of use one graph to ask both the questions of what's the area above and the area below. Because if this is the area above the inverse, because the whole thing is 100%, the inverse 1 or 100 minus that point will be the area below. So the blue represents this point and below, the orange this point and above. So that's a great tool that you can kind of use to be somewhat flexible when you're trying to visualize this stuff, especially for people such as me that were never good at really plotting these things out on a piece of grid paper or something like that and making these nice evenly spaced out Xs and so on down below. So if we do that in Excel, we might ask these questions, right? How would we graph this? Well, we could do these three here. These are the three graphs that we made according to these questions. So we might say, is P of X greater than or equal to 79? Note that you can make a dynamic header in Excel this way by making this into a formula. The quotes, this looks complicated, but it's actually, once you do it a few times, not too bad. Quotes are what you have to do when you want plain text in a formula. And I think of it as like a knot because it's tying together this information, which is text to the next bit that I want, which is actually coming from a cell, which is that 79, making this dynamic. Because if I change that 79, this will change to 80 and so on. And then I tie that together to that end bit, which is just text with a quote with the quotes again, making this dynamic. So there we have that. And if I was to calculate it, then it would be one minus norm dot dist. So remember that if I'm looking for something above, that would be that would be the graph over here. If I'm looking at the orange side. So if I want it to be above all I have is the cumulative function. So I have to I have to take something that adds up to here. So if I want the side on the right side, I can't take a function that adds to the bottom down like this. I have to basically say I'm going to take the entire thing 100% minus the blue area. So that's what we're doing. That's what this is doing 100% or one minus the blue area. If I was to graph that over here, I can say that's going to be this bit, which is has a formula of if logical function. If the logical function.