 data because of that randomness involved in it. And then we have the median, the one in the middle, and that's the equals the median of this data. Now note that the mean and the median are pretty close together. That is one indication that this might conform to a normal distribution or bell curve. You also might calculate the mode, which is the number that has multiple times the same number comes up. And if that is also similar to the mean or median, it would be another indication that you're close. You might be, I have a bell curve distribution. Now in this case, the mode might not work exactly because notice we're not representing the grades as just like a 72 or a 90. We've got decimals. So the fact that we have decimals means it's a lot less likely that we're going to have multiple numbers that are exactly the same. However, if we did not have decimals, if we rounded this to the whole number, then it would be quite likely that the mode would be a useful tool and it would probably be similar to the mean something like a 74. All right, and then we've got the now, if I want to plot this, the next thing we're going to do is say, it kind of looks like this data conforms to a bell curve due to the mean and the median being the same, we could make a curve from it to further test that. So we could. So if I make a histogram of this data, it looks like this. Now the histogram is where we have the buckets down below. So this is going from 40 to 43 63. And the middle point is in here. Remember, the mean is like a 74. So this is going from 71 to 75. And so and so you can see it kind of looks somewhat like a bell curve. Notice that it's not going to be exactly like a bell curve because we don't have a whole lot of data. I can't remember exactly how much, how many data points we put here. But the more data points, the closer you would think that it would conform to a bell curve, if you're using something that doesn't have as many data points, you would expect it to be a little bit more jagged. Also, it will be impacted by the bucket sizes that we're going to use down below, which we talked about in prior presentations. But we're saying, hey, look, the median, the mode look like they're pretty close. And it looks like it's kind of conforming to a bell curve. And we have an intuition that this data might conform to a bell curve. So we might then want to graph this thing as a bell curve. So we're going to choose the X's and we're going to choose the P of X's. So we're going to then plot this thing out, plot our data points so that we can then create a chart from this information. So then the question are the X's. When I start my X's, I can think about this and say, well, look, this is grades. So I would think it would be going from zero up to 100 100 representing 100%, zero representing zero, one representing 1%. So I could do it that way, I could just go from zero to 100. But I might want to, in other examples, I might say, I might not have that convenience. I might say, well, where, where should I start my beginning and ending X's when I want to plot my graph, and then it's going to show up on the X axis. Well, we know that in a bell curve, that the vast majority of the data will be within three or four, the vast, vast majority of the data will be in four standard deviations. So we could say, let's take four standard deviation both above and below, and that should be encompassing all of the data. So for example, the standard deviation is 10.09. If I take that times four, there's four standard deviations. If I'm starting at 74.92, the middle point minus that 74.92, then I can say that the lower X should be should be 34. So I can really just go down to 34 and still be picking up all the data. I don't have to go down to zero, in other words, because it's unlikely that you're going to have test scores all the way down to zero. That would be quite badly performed test people could guess and you probably do better than that. If I did this again, 10.09 times four, and then go above the mean for standard deviations above the mean plus 74.92, then I get to 115. Now, you might be saying 115 doesn't make any sense, because it's over 100%. But, you know, in some cases, you might have such like extra credit or something. But in the bell curve scenario, note that you can go on forever when you look at the theoretical concept of a bell curve. So it might be useful even though in practice, it's not going to go over 100 to plot it out to 115 so I can see the entire bell curve tapering off as bell curves do. And then I can also see like the total adding up to 100%. Let me show you what I mean. We've got then the Xs. So here's our Xs going from 34 on down. And then we're going to do our P of X calculation to do this. We're going to use the norm dot dist function within Excel. So this is going to give this is going to be kind of similar to some of the Poisson distribution, for example, or binomial distribution we talked about in the past, but now we got the norm dot dist. The X is going to be 34 in this case representing that grade. And then so what's the likelihood of getting a 34 for talking about the bell curve with the mean and standard deviation defined over here. So the mean is going to be the 74 and then comma the standard deviation is going to be the 10.09. And then is it cumulative or not? This cumulative is similar to what we saw with the Poisson distribution, for example, although here we're representing the area under the curve. So it's so it's a little bit different in terms of kind of like calculus involved the area under the curve. But the concept is basically the same. Do we want the cumulative up to that point? Or do we want just that point? In this case, we want just that point. So it's false or zero. And then we get to around 0%. So if we take this down, we're going to say, okay, what's the likelihood of getting a 46? We're going to say 0.07%. Right? So that's the questions that we might ask with this. Now, most of the time, if you're a student, for example, you're probably going to be asking what's the likelihood that I get like a 70 or above that kind of question, which you might say, well, I can go down to 70 down here and then add up everything from 70 on up. But you can't usually do that exactly because again, although you might have been able to do that with a Poisson distribution here, we're talking about the area under the curve that might give you an approximation, but you'd want to use a formula with the cumulative formula, which we'll talk about a little bit in a little bit here. But there's our P of x. So that'll give us our approximated curve. So if I was to look at it, then it would look something like this. Here's our approximated