 amic headers that we talked about last time. I won't go into that in detail here. But this last one, the middle bit, is going to be the norm dot dist of the top part, which is going to be the higher x, the 133, and then the mean standard deviation. It needs to be cumulative minus the norm dot dist of the lower x, the 109, the same mean standard deviation. It also needs to be cumulative. And then if we converted that to, or I can also think about it this way, 1 minus the two results that we got up top. In other words, if I look at my data up top, we did p of x is greater than 133. p of x is less than or equal to 109. So if I look at those two, and I think about this, I guess, I'm going to say, OK, p of x was greater than the 109, and less or I'm sorry, let me do that again. What were my two? p of x is greater than equal to 133 and less than or equal to 109. So those are the two tails, right? Those are the two ends. Greater than the 133, less than the 109. Those are the two blue sides. So if this whole thing adds up to 100% and then I have these two blue sides so I can subtract that out, I can say 100% minus those two. In other words, I'm going to say this is going to be the 30.58 plus the 6.05 minus 100%. And I should get that middle part of the 63.37. So that's another way to kind of envision that. And then I could say, OK, the z scores. So what if I had the z scores? I can do this with the z calculations, getting to the z's in the same way. It's going to be the 133 minus the mean divided by the standard deviation. We'll give us the z's. And then you can do the z in the same way, norm.s.dist with the z scores cumulative minus norm.s.dist with the z scores of the lower z score. And you'll get to the same thing. You can also do the same thing with the z scores that we got to get to that same 63.37. So this is the probability of the lower. So before I do that, just note that when I look at this graph, then this last bit, then is p of x is less than or equal to 133. And over the 109, we have our dynamic header that we'll show how to do in Excel. And one way I can do this now is instead of me redoing this whole graph based on the information to the left, I can say, well, look, I'm going to sum these two up because this is the upper and lower ends. And I'm going to say if. So I want you to do an if. And then I'm going to sum up these two. If the sum of that is greater than 0, then what do we want you to do? We want you to put nothing over here, double quotes. If not, then we want you to put the number, or then we want you to put the p of x. So by doing that, I can then get this middle part that's being graphed. So now we're graphing that middle part. So the point of this is that now we have this one graph that might be able to help us and be dynamic to answer any of these questions. We can say, OK, the high end is over here. And I can use this graph to envision if I want to look at the tail on the high end, I can use the same graph if I wanted to ask a question on the low end. And then if I want to ask a question about the middle, I can use the same graph to basically plot the middle point. Because by the way, I'm entering it into the system here. I can enter either. I'm going to say that this is always the top and the low point, and this is going to be the middle. So we'll do that in Excel if that's something of interest. So to try to get kind of one graph that will be a little bit dynamic once you put it together so that you can visualize multiple of these kind of questions. Now another question that we haven't really looked at as much is the probability of the lower end. What if we know the probability of the lower end is 45% and I'm trying to then find the x value or the z value? So in other words, I know that this probability of this end is I know that area under the curve. And I'm trying to find the point then of the x value or the z value, this value on the x. So if we ask a question like that, then we can use a formula, which would be norm.inverse, i, n, v. And then the data input will be picking up the probability instead of the x value and then the same mean and standard deviation. And that will give us then the x value, same with the z value. So if I know the 45%, I'm now backing into the z value. So I kind of reversing the algebra but doing it with a formula. So now we're going to solve for the inverse, norm.s.inverse. And now all I need is the probability, which is the 45%, because the mean and the standard deviation are kind of included in that z value calculation. And that gives us our z value of the 0.13. The 0.13 and 125, you'll note, if I look at the graph, so the 0.13 and the 125 would be somewhere around here. They're kind of the same, they're at the same point. Or if I look at 125 over here and I look at the z value, 125 and 0.18 about, because there's rounding involved. So there is that. And then we're going to say, OK, what's the probability of the upper end? So notice that these two are kind of the inverse. So 45 and 55 add up to 100. What if we're looking, what if I know then like the upper end, like this blue area. This isn't the exact number, but you know the upper end. Then how do I back into like the related x or z? So we can say, OK, that's going to be another inverse, but now you'd have to take the norm.n verse, same thing, but the probability now is going to be 100% or 1 minus the probability here. So 1 minus the 55 and then comma the mean and standard deviation. And that'll give you the x. And then if you wanted the z value, same concept, we would take the norm.s.inverse, 1 minus the 55. And then you get to this value. And they're the same numbers because we chose the 45% and the 55%. So obviously, if you're saying the probability of something that is the lower end, 45% means that the upper end is the 55%, which means you end up on the same x value, which is going to be the 125 and the 61.