 Last time, although we used grades, we had percentages and decimals. And the decimals made it so that you might not have any multiple numbers or not many repeated numbers. Whereas in this case, we have it rounded at least it looks like here to just two decimals. I mean, no decimals just rounded two inches. Therefore it's more likely that, that the mode where you have multiple areas of the same measurement are going to happen in that middle range. And therefore the mode is more likely to be a good indication as it being close to the mean that it's a bell curve type distribution. Now, now we might want to graph this. So we're going to say, Hey, look, that looks like it might be a bell curve distribution. Uh, we could also look at the, at the graph of the data. So in this case, I'm looking at, uh, uh, the, the, the data for the bell curve on top of what we'll do as the normal distribution. So you can see that it, it lines up pretty much to like a bell shape type of curve, uh, when we approximate it that way. So now we're getting the idea of thinking this looks like it might conform to a bell shape because the mean is close to the median, the mode. We already had an intuition that it might be due to it dealing with heights. And when we graph it like, like, we can see that it looks somewhat like a distribution that might conform. So now we want to think about how we can actually make a, a bell curve, an actual curve. We could plot our points plotting our X's and the P of X's. Now, when we've plot our X's, we're going to think we're luck. We're looking at heights here. So you might say, well, I don't know how, where should I start? We could start at X of zero, which would be one inch and go up to however many inches high we want to go from there. But it might be better for us to look at and say, well, how high and how low do we need to go to capture the vast majority of the data? So once again, we'll go four standard deviations above and below the mean. So the mean in this case, 73, well, let's do the standard deviation, four standard deviations. So we've got 2.3 times four. That's four standard deviations, 9.2. Let's go below the mean. So minus the 73.7. That means that the low point is going to be a 64.48 about. It's got it's got decimals. It's rounded and the high point is going to be 7.3 times four, four standard deviations plus the 73.7 gets us to the 82.92. So instead of going down to like zero inches, we're going to start at inches at 64 because that's as low as we have to go to capture all the data so we don't have a graph that has a bunch of zeros or nothing really happening up till you get to around that point. So if I look at my exes, we're going to start at 64 and then we're going to go all the way up to the 83. And then we'll do our P of X calculations here. So now we'll do our norm dot dist calculations, which is norm dot dist of the X, which in this case is going to be 64. The mean is what we calculated over here at the 73 standard deviation 2.3. And then does it need to be cumulative? No, not in this case. So we want zero or I believe false. So so now we got the likelihood of us having 70 inches, right? 70 is about 70 70 divided by 12 5.8 feet is about likelihood 4.78 percent of the time. So we're so notice this is to be exactly at the 70 inches, not 70 and above or 70 and below. And then if we go up here, we could say, OK, the likelihood of 75 inches, which is 75 divided by according to the data that we have here, 75 divided by 12 6.25 feet. So there we have it. So we can so we can kind of see the question here. Now, obviously our question might be, what's the likelihood that I have to be like, you know, 510 or above or something like that? So so if I said like 67 and 67 divided by 12 is 5.58 feet. So what's the likelihood that I can be that or below? Maybe what might be a question that I might have if I, you know, if I'm trying to grow up to be a pitcher, those would be kind of questions that might you might be looking at in terms of the data set. So then over here, we might want to compare it to our actual data. So this is the graph of the curve, remembering that it is represented not in whole numbers, but in percentages. But when we get our actual data, it's represented in inches. So what I could do is take my my percentages and multiply it times our test data to get the count. Or we could try to create our actual data converting it into a percent, which is likely more common. So what we're going to do is now is I'm going to take my frequency of my actual data. The frequency means that we're going to say in this bucket of 65, how many of our count in our actual data from these numbers, the counts of pitchers, heights, I believe we're just talking pitchers here, are are above 74 inches up to and including 65 inches. For example, there's zero. Let's go down to where there's some here above 70. 67 inches up to and including 78 inches. There were seven of them counted in our count. And then above 68 inches up to and including 69 inches, 19 of them in the count. Now, if I add all of this count up, it comes out to 1,034, which should be the number of actual data points that we had over here. That'll give us kind of a double check. So if I use my formula in Excel to count all of these items equals count, then it gives me 1,074, which matches what we got over here in our buckets. So it looks like that gives us a good double check that we picked up all of our data. Then I could multiply this number, our actual, our bell curve number, for example, in this case at the 74, I could say, well, I'm going to take that 0.132 times the sample count, which is, which is times 1,034 to approximate. Now I forgot where I was to approximate the actual count on this side. But the way we might want to do it, however, is to take our actual numbers and divide by the total. So for example, this seven is taking seven divided by the 1,034, divided by the 1,034, moving the decimal two places over 6.68%.