All right. So in this question, we want to compare the empirical formula to actual data and then we're going to look at the natural history am and decide whether the empirical rule is actually useful in this case. But for part a right you the empirical rule to estimate the percentage is Well, the empirical rule tells us that within 12 and three standard deviations of the main, respectively, You are going to have a party, you're going to have 68%,, 95 in 99.7 of the observations. And this is respective 2, 1, 2, three standard deviations from the mean. All right. So that is our empirical formula estimation. And now what we're going to do, we're going to take this data and we're going to see what percentage of the actual data fall within 12 and three standard deviations of the mean. So what I've got here, I've got a number line which is centered at 45.3, so it's centered at our mean, and we're just going to determine essentially these buckets. Okay, we're going to determine these boundaries for 12 and three standard deviations. So in order to do that, all we have to do is add Well. So, for example, for one standard deviation above the mean, all we're going to do is add 45.3, we're going to add one standard deviation above the mean. So that's 4.16. Now, when we do that, we're going to get 49 0.46 So this is 49.46 and we can keep right, we can keep doing this. This one is going to be 49.46 plus 4.16 49.46 plus 4.16 Right, And that's going to give us 53.62, 53.62. And if we do that once more, that's going to give us a third boundary. Just 57.78 57.78. And likewise, on the left hand side, we could do this by subtracting the standard deviations and right. Why am I using, Why am I using the standard deviations as these buckets? Well, we want to compare it to the empirical rule which uses the standard deviations as buckets as bench works essentially. All right. So 45.3 45 .3 -4.16 gives us 41.14. And then if you subtract 4.16 again, um That gives a 36.98 and once more To get the three standard deviations below the mean gives us 32.2. All right. Now, our data here are organized from flow, like the biggest risk is quite nice, um which gives us the ability to uh drop them in various buckets. So 36.3 is going to be in this bucket, 37.7 in this bucket, 38 in this bucket. Uh 38.8 38.9, 39 39.3. Right, everything up to um 41.1 39.3. So we have 40.9, we have 41.1 Then 41.3 is in this bucket. 41.5 41.8, 40 to 40 to 42.1 42.5. Uh 42.5 again, 42.8. Just to be clear, you could also um simply, you know, mark them off as you go. So we've done all of this first row, we've done all of the second row now. 42.9 43.3 43.4 43,45, 44 44.4. There's a lot of here and there at 44.7 44.8, 45.2 45.2 45.2. And then, right, We have done all of those, okay? And then the rest of them. So 45.4, what I'm actually gonna do now, I'm just going to count. I'm literally going to count in our table. Um up to 49.6 so 49.6 happens right over here. So all of these guys here, 1 2 3 4567, uh plus nine, seven plus nine. So at 16 plus another nine, That's 25. Fall within here. Um and then between 49.46 and 53.62, 53.62, that's going to be here. So then there's four and then to Sorry about that, then there's four and then to and I suppose we could have also done this for these guys. So this guy would have been one um 12345678 This guy would have been eight and 123456789 10 11 12 13 14 15 16 17 18 1920. So then you have 20 here. All right, Yeah. Um so let's compare. All right, so how many are within one standard deviation? While 20 plus 25 That's that's equal to 45. And how is that as a percentage? Well, how many data do we have in total? 123 So this is nine rows 1234569 times six is 54 plus 2, +3456 So they're 60 in total. Okay, so 45 out of 60 so 45 out of 60 Gives us our uh percentage or proportion rather. So 45 out of 60 is 75%. The 75 are within one standard deviation. Alright, Now, let's look at within two standard deviations. So within two standard deviations, you're gonna have eight plus 20 plus 25 plus four, so eight plus um 20 plus 25 plus four. So we're just adding essentially 12 to the previous ones. That's gonna be 57. So then 57 out of the 60 total observations are going to be within Um two standard deviations, which is 95 Actually matches quite nicely, so this is 95%. Within two standard deviations, and finally, how many are within three standard deviations? Well, in this case, actually, all of the data notice all of the data fall within three standard deviations of The mean, so 100 of data of observations, all within three standard deviations. Alright. Um so actually when we when we put these together, so 75 95 100 that's not too far off from the empirical world. Alright, And the empirical rule, of course, is using a perfectly smooth mathematical model, and if we were to look at the actual data in a history RAM and you can do that by flipping to page 76 looking at figure 2.100 um it's not perfectly smooth, right? It's it's a bit jagged, but it is pretty symmetrical, it's pretty bell shaped and for this reason um in conjunction with the data that we've discovered, I would argue that the empirical rule is quite useful here because the data is pretty bell shape.