 Distribution represents what's normal, the average, what normal people do. So notice that the bell curve is kind of interesting, too, because before that, you've got to think that people might not have had as tuned a sense as to what normal is. Like the bell curve almost seems like it kind of defined normal. So people that are too tall used to be thought of as kind of weird. Now they're kind of idolized for being very tall. And that's always been the case, I guess, to some degree. But too tall, you'd be saying that's abnormal. Abnormal was bad. Well, how do you know it's abnormal? Because we have a bell curve, and on these types of things, it's too small is abnormal. So now it's weird. It's outside the range. So you've got to think before the bell curve, did people really think in terms of how abnormal something is? Or something like that. But in any case, the middle point is around 74. So you can see that the Z score is around zero. And then the Zs are positive going above it from that point. OK, so that's the Z score. So we have those questions. So now we can kind of ask questions, such as we can we could say, OK, well, what if a test score is equal to 90, the operator is less than or equal to. So if we had something like this, p of x is less than or equal to the 90, how would we get that calculation? Because remember, if I look over here, we're going to say, OK, well, 90, if I go down to 90, I can see that the likelihood of me getting 90 would be 1.92. But that's not usually what I want. I want usually greater than or equal to 90, or less than or equal to 90, possibly. But either way, if it was greater than, if I, what's the likelihood I get above 90, I could, you think, well, I could add these up. I can add all this up down to here. But again, you can't do that exactly. That'll give you an approximation because we're talking about the area under the curve. So you could do that. It'll give you an approximation, but it's not going to be exact to get an exact number. This is less than. So this would be the likelihood of something less than or equal to 90 would be equal to norm dot dist. The x is going to be up top. The mean 74.92 standard deviation 10.09. And then the the x is 90 and the cumulative and then the cumulative bit. Do we want it to be cumulative? We do. So that means it's going to add everything up, which gives us the likelihood of 93.24. This isn't the test score that we're looking at the likelihood that we have something at a 90 or below. The likelihood of getting a 90 or below based on the bell curve is 93.24%. In other words, we would expect 93.24% of participants to score 90 or less on the test. If I looked at the z score, I could say, okay, well, what about the z score then? Where do they line up with the z score? Well, we've got 90 minus the mean, which is 74.92, which is 15 divided by the standard deviation 10.09. And that gives us a z score of 1.49. Remember that zero is normal. Zero would be at the 74. So remember that normally you would think with tests, well, average should be 70, right? Wasn't that like, but then you got to think about, well, what are averages on this particular test? In this case, the average score, the middle point is 74, around 75. So if you get a 90, so 74, 75 would be zero in terms of z scores. If you get a 90, you're clearly well above that. And so we're at the 1.49 in terms of z scores. And then if we look at this one, we can calculate this same thing, the probability of x being below or equal to 90, not with x this time, but with z scores. So the z scores and is kind of like another is another way that we can represent things by x or the z score, right? And if I do it with a z score, the formula would just be equals norm dot s dot dis. And there's only two things we need to do the z and cumulative. We only need the z because notice that the z score itself calculated and included the x, the mean and the standard deviation. So all we need is the is the z and then one for cumulative to get to that same answer if we had the z score instead of x, and we wanted the likelihood of getting a less than or equal to the 90. So then if we if we want a question like greater than or equal to 90, then it would be a similar thing over here. We can say, OK, well, if I go over here, now I'm looking at 90, which has a 1.29 percent likelihood, and then you would sum everything up down to here. But that will not be exact. But you can do that as an approximation to be more exact. We can use a formula one. You could do it this way. You could say, well, if to get less than 90 is that. So I can say 100 percent minus 93.24 gives us 6.76. In other words, if I'm going to have less than or equal to or greater than or equal to 90 is 93 percent, and it has to add up to 100, then then the likelihood of the other side greater than or equal to 90 is only the 6.76 percent. Or you can do it with our calculation. So now we have to say it's the same thing, norm dot dist, but now we're taking one representing 100 percent minus the norm dot dist of the X 90 percent, the mean that we saw up top, same standard deviation, and we want it to be cumulative. So that's doing the same thing we did up here, one minus the answer we got before, which is doing it in one cell now, one or 100 percent minus the answer we got before. And then we can do the Z score, same thing. So we have our Z score, same Z, but now we're asking getting over the 90 or that Z score. Same concept, one minus what we got the norm dot s dot dist. So we can do the same thing just like we did up top with either X's or Z's, whatever the case may be. And then we could say, what if we're doing in between? So the next question you would ask is, well, what if I, what's my likelihood of getting somewhere between a 75 and, what did I say, a 90? I can