 our actual curve that approximates the actual data. And this is the blue representing the curve and the orange representing our actual data. So now we're plotting the data on top and you can see it kind of lines up. I'll show you how we did that in a second here. So then just to note here, we can say the frequency. Now I might want to compare my P of X information to my actual data. And I have a problem with that because my actual data over here is being represented is actually outputs, my whole numbers over here and my P of X is in the likelihood of something of a 34. So what I could do is try to convert my P of X into a number by multiplying times the number of samples that we have, or we can try to take our actual data and make it into a percent, which is probably the more common thing. So what I want to do to compare my actual data is I want to look at my actual data over here. So there's my actual data. I want to count how many times each of these data fall into the buckets. Now note that these numbers aren't exact because I'm looking at, for example, on this 35, how many data sets in our actual data are within or above set 34 up to and including 35. And then how many items in the data set count them if they're above 35 up to and including 36. That's gonna be our frequency calculation, which looks like this, frequency, the data array would just be highlighting all of our data saying that's what I want you to count basically. And then the bin array being all these information, which are the bins that we want you to put those counts into. So if we said this, how many times did you count 34? How many times did you count above 34 up to and including 35, zero scrolling down? We're gonna say how many times did you count up to and including or over 40 up to and including 41, two, two times that that happened in our actual data. How many times did you go above 50 up to and including 51? How many of those were in our data set, three of them? And so on and so forth. And if we go all the way down, we can see the data. If I add up all of that data, it adds up to 1,000. That should be the actual number of grades in our sample data. That's how many sample data points that we generated. That's a nice check number that we have to make sure that we picked all of them up and put them in an appropriate bucket. Now, what the problem here is that I can't compare that 10 to what I got in my norm.dist because the norm.dist is 0.79. I could multiply the 0.79 times the 1,000 sample, right? I could say, well, I had a sample of 1,000. I would predict based on the norm.dist times 0.79 that it would be 1,000 times 0.0079 would be 7.9, right? I could do it that way. I can multiply all these times 1,000 to compare it to these actual counts or I can take my actual counts and divide it by the total, which is what we'll do here. So now I'm gonna say, all right, let's take like this, if I take this 14 and divide it by 1,000, 14 divided by 1,000, then I get, if I move the decimal two places over 1.4. Now that 1.4 is in a percentage format. So I can now compare it to this percentage format. So I'm just doing that all the way down. So this 19 is 19 divided by the count of 1,000. So move the decimal two places over 1.9%, which is pretty close to the 1.97 that we got with the norm.dist function. So then I can compare these. I can look at the differences. This is what I got with the function. This is my actual percent data of the total and you can see we're pretty close on the data. So that would give us, it's somewhat close. That would give us another indication that are how far away from a normal distribution we are. And then here we got the Z score. Now the Z score is another way to represent our data. And it's trying to say, I'm gonna represent the data in terms of how close it is to the middle point, which is the mean of our normal distribution. So if I look at my information over here, the middle point on our graph is gonna be 74.92, standard deviation 10.09. So what I could do is start to say, well, if I look at my graph and I say, well, this is the middle point, I could start to measure how far away above and below that middle point, the normal is. How far away are you from the normal? Would be the Z score, right? The lower the Z score, the closer to zero the Z score, then the closer you are to normal, the mean, the further away you are in a positive sense, the higher you are from the middle and on a negative, the lower you are. So over here, how do we calculate that? We're gonna say that we're gonna take each point, in this case, 34, 34 minus the mean, which was 74 minus this number, 74.92, and then I divide that by the standard deviation, the standard deviation. So divided by the 10.09, and that gives us our, in this case, negative 4.06. So that's gonna be our Z score, and I can look at all these, and each of these data points then, remember oftentimes when we're looking at comparing something like job performance or performance in schools, sports performances and stuff, then we have to represent things in percentages, a percentage, we're not saying how many we got correct or incorrect, we're taking a percent of the total, which is even more useful if you're talking about things that are uneven, like batting averages or job performance, because you had a different number of tries at something. That's why the percent is a useful tool in and of itself to make more comparability. And then we can also compare kind of with the Z score, which is gonna give us our, how close we are to that middle point or middle when we're talking about a normal or bell curve