 Hello, and welcome back to Statistics. It's Monica Wahee, your Library College lecturer, and you've made it to Chapter 7. I broke up Chapter 7 into bite-sized pieces, and we're going to start with Chapter 7.1, talking about the normal distribution and the empirical rule. So here are your learning objectives for this lecture. At the end of this lecture, you should be able to state two properties of the normal curve, state two differences between Chebyshev intervals and the empirical rule, and explain how to apply the empirical rule to a normal distribution. So remember distributions, we learned about them a while back, but I'll remind you a little bit about them. And then we're going to talk about properties of the normal distribution or specifically the normal curve, that shape that comes out of making a histogram of normally distributed data. Then we're going to remember Chebyshev intervals. We're going to talk about what Chebyshev did for us, and what Chebyshev really did do for us. And then we're going to move on to the empirical rule, which works very well, better than Chebyshev intervals, when you have normally distributed data. And then I'm going to show you an example of how to apply the empirical rule to that normally distributed data. So remember the normal distribution, in fact, remember distributions at all, right? So to get a distribution, and a lot of people sort of forget this by the time we get to chapter seven, but I just wanted to remind you, this was from an earlier lecture. We had a quantitative variable, which was how far patients had been transported. And we determined classes, and we made a frequency table. So remember that. And then after that, we made a frequency histogram, and it made a shape. And as you could see that shape, which is the distribution, that shape in this one was skewed right, see that light on the right. Okay, but that's an example of something we cannot apply the empirical rule to, because the empirical rule only applies to normally distributed data. So I had to give you an example of that. And here's my example. So when I was in my undergraduate in costume design at the University of Minnesota, they made us take a chemistry class in one of those big lecture halls. So I was in a very large class that probably had about 100 people. And we were given this really difficult test. It was 100 point test, and I was used to getting like A's. And so when they were done with the test, the TAs were handing the test back to everybody so they could see they're great. While the professor was writing on the board and was writing the frequency of all the different scores. And I remember the TA handed me my test, and it said 73 on it. And I'm used to getting like 90s up to 100. And I remember stating out loud saying 73. That is an awful score. I can't believe I did so badly. I was talking like that. But at the same time, the professor was writing the frequencies on the board. And what I realized is the top score was in the 80s. And I had the third top score was 73. That's how hard the test was. And that's when I shut up. Because I noticed everybody giving me dirty looks because they had scored actually below me. So I wanted you to imagine that class. And I imagined what the normal distribution would look like for that class with the distribution of the scores. And the reason why I thought it would be normal is because we all did badly, right? And so nobody got 100. So we were all below the 100. So I imagined this curve here for you. And I imagined my class had 100 people just to make it easy. Of course, the test was difficult, and nobody got 100 points. And the mode, the median and the mean were all near a C grade because you remember how when you have a normal distribution, the mode, median and mean are all on top of each other. So we all did pretty badly. So I'm going to use this example of the fake chemistry test scores to exemplify these properties of the normal curve. So there's five I'm going to talk about. The first is that the curve is spell shaped with the highest point over the mean. And so you can see I drew a scribbly little curve, put a little arrow there to show you that that's where the mean of the scores were. And then I also wanted you to notice that the curve is symmetrical with a vertical line through the mean. So there's like a mirror image of the curve on the other side. Now it's not perfect, obviously, but it should be roughly like that. And you know, this is not true of skewed or bimodal or these other things we've been talking about. Okay, and the third property is that the curve approaches the horizontal axis but never touches it. You don't have to memorize this, but remember asymptote or asymptotically close. That's when a line gets really close to another line, but they never touch. It's so romantic. But anyway, that's a very Bollywood thing to say by the way. But so the curve approaches the horizontal axis and never touches or crosses it. Then also there's this inflection or these transition points between cupping upward and downward. And these transition points occur at about the mean plus one standard deviation and about the mean minus one standard deviation. And this is a little hard to explain. But imagine you're on a roller coaster, and you're going up this normal curve. There's this part where you're just mainly going up. Well, the part where it seems to kind of level out and you're at the top of the curve, you start to relaxing. That's that inflection point. And so as you're going over in your roller coaster, and you're in that flat part, and then you start kind of going down, that's the second inflection. So that's where what it's saying about is the property of this curve is that you have these inflection points like that. And they roughly occur at plus or minus one standard deviation above and below the mean. Then finally, and I colored this in just so you could see it, the area under the entire curve is one. So think 100%. So it would be nice if that were a square or rectangle or even a triangle, something that we're used to in geometry, but it's not. It's this goofy shape, right? But still, you need to get it in your head that that shape is worth 1.0 in proportion land, or 100% in percent land. And what I mean by that is, let's say we cut that shape in half, the each side would have 50% or 0.5 on it. Then let's cut it a different way. So the part of the curve on the right side of that line is a fourth of the curve or 25% of the curve even though it's goofy shaped. And the part on the left side is 75%. So that's what we're trying to get you to think like is that, yeah, you can just declare that all the area under the curve equals one or 100%. But the reason why we're declaring that is because we're going to cut it up and say different amounts of percents of the curve. Now we get to the empirical rule since we reviewed this whole curve thing. And I'm going to make you remember Chevy chef, I'm sorry. But you know, let's talk about Chevy chef. Chevy chef helped us get some intervals, right? And intervals have boundaries or limits. They have a lower limit and an upper limit. That's how you know what bounds the interval. So when we were doing Chevy chef intervals, what we would do is we'd figure out a lower limit and upper limit. And we'd say at least so much percent of the data falls in the interval, right? So when we would choose a lower limit of mu minus two times the standard deviation, and the upper limit was mu plus two times standard deviation, we would say at least 75% of the data were in the interval. So I wanted to just show you a demonstration using my fake class. So remember, there were 100 students in the class, I actually came up with a mu for them. And their mu on the test was 65.5. So my 73 was better than the mean, but not much better, right? So the mu for that class was 65.5. And the standard deviation was 14.5. So I calculated these Chevy chef, this Chevy chef interval for 75% of the data. So I took 65.5 minus two times 14.5. And I got 36.5, which is a pretty bad grade. And then the upper limit was pretty good, right? 65.5 plus two times 14.5 equals 94.5 on 100 point test, that's a pretty good grade, right? So if you had 100 data points or 100 students, at least 75 would have scored between 36.5 and 94.5. So you're probably already realizing, okay, that doesn't really help Monica, who scored 73. And this is a really wide range, we say at least 75% of people scored there, you could have probably guessed that without even knowing about Chevy chef intervals, right? So it didn't really help me narrow down like how well is this class doing? If I had had the mu in the standard deviation, I could have calculated this and said, Okay, I'm no better off. So Chevy chef's theorem on the left side, it applies to any distribution, you don't need a normal distribution, you can use that skewed distribution. Also, you'll notice it says at least. So like this was at least 75% of the data fell in there, maybe even 100% fell in there. So it doesn't really help us. And as you go, let you start with two standard deviations. If you go out three, it's 88.9%. And four, it's 93.8%. You know, you might as well start at the beginning and say almost 100% of the data falls in this interval. And if you're saying that it's not very useful, right? But it kind of gets stuck doing that because Chevy chef's theorem applies to any distribution. The empirical rule is much more elite. It only applies to the normal distribution. And you'll see why if you are lucky enough to get a normal distribution that you want to use the empirical rule instead of Chevy chef. Okay, because secondly, the empirical rule says approximately, it doesn't say at least. So it's saying basically, not at least, it's saying about exactly this. So you can trust it. Okay, you don't have like this unknown, like maybe 100% is in there. So it says, this is what it says, and I'll show you a diagram of it. But it says that 68% of the data are in the interview interval, mu plus or minus one standard deviation. So mu minus one standard deviation all the way up to mu plus one standard deviation, 68% of the data are in there. And you'll notice that Chevy chef didn't even say anything about one standard deviation. And so already we've got something way more useful if we apply the empirical rule, right? So next we go to 95% of the data are in the interval mu plus or minus two standard deviations 95%, approximately 95% are in there. Now, if we had bought Chevy chef, we'd be saying about this to we'd be saying 75%. Okay, we'd be saying at least 75%, which could be 95%. But here, if we're using the empirical rule, we're relatively sure that it's 95% between mu plus or minus two standard deviations, you can like bet on it, right? Finally, if you get out to three standard deviations, you're kind of running out of data, because 99.7% almost all of them fall in that interval. So as you can see, the empirical rule is going to give you a more specific answer. But again, you can only use it if you have a normal distribution, but which we do. So let's go look at that. Okay, this is a diagram that I'm going to help I made it myself actually, because I thought it was the other diagrams I saw were not pretty. This one is very pretty in my mind. But let me unpack this diagram for you because there's a lot going on in it. First of all, I want you to notice the shape of it. It's a normal distribution. Okay. And then I want you to notice that I put this black line down the middle. And I put a little arrow that says mu. So this is where we want to imagine mu is no matter what your what your actual numbers are for me, like in our case, this is 65.5 for our points. Just imagine whatever your mu is and whatever your standard deviation is, this is where you would put the meal, right? Then you'll notice that each of these sections that's colored has a little standard deviation symbol in it, because that's representing that that the width of that is one standard deviation. So if your standard deviation was like five, then mu would be plus plus or minus five, like the green one would be mu plus one standard deviation. So it'd be mu plus five. And then you draw that parallel line there and see that arrow that says mu plus one standard deviation that would be there. And of course, I can't I just had to use these symbols because I don't know how big the standard deviation really would be or what the me really would be. But whatever it was, mu plus one standard deviation, if you go up there, you would see that that green area represents 34% of the data. And if you're lucky enough to have exactly 100 people like I did in my demonstration, that would mean that between mu and mu plus one standard deviation of these test scores would be 34 people scores, right? So you can really figure that out. Same with the yellow section, only that's mu minus one standard deviation. And 34% of the scores would be between those two numbers. Now you'll see as you get up into the blue, that's between one and two standard deviations above the mu. You'll see that because the roller coaster is a lot lower to the ground there. That section is really small. It's only 13.5% of the data. And the same with the orange one, that's on the other side of the mu. So that's below the mu. And that's only 13.5. And then you'll notice that at three standard deviations between two and three, there's a little tiny piece, right? The purple piece and the red piece. Those are only worth 2.35% of this shape. And then I wanted to point out there is some stuff at the end, in the little black part beyond three standard deviations on either side. There's 0.15%. And a lot of times people forget that. But one way you can make sure that you got to remember that it's there is that if you add up all these percent on the slide, you'll get 100%. Because remember, I promised you that the whole the whole curve is worth 100%. And this is how we split it up. I also want you to notice that there's kind of a cheat, right? If you just add up the green, blue, purple, and then the little black part at the end, if you just add up those percent, you'll get 50%. Right? Because it's half the curve. And the same, you'll get the same thing if you do the yellow, orange, red, and the little part in the black at the bottom. If you add those up, you'll get 50% too. So that's how you want to just conceptualize this whole empirical rule diagram. But now we'll apply. So I put the empirical rule diagram on the left. And then I put our class frequency histogram on the right. And look, I put the meal, and I put the standard deviation so we could have it there. Now the first part of this section, I'm just going to show you how to fill in the numbers under the diagram. Okay, and then after we fill in the numbers, I'm going to talk to you about how to interpret those numbers. So let's start with easy, let's write the meal underneath the symbol for me, which is 65.5. So we just wrote that was simple. Okay, now let's do the plus or minus one standard deviation. So you'll see 65.5, which is our meal minus and I put one times 14.5. I know I just did that for demonstration purpose. So you see we're doing one times the standard deviation. So if you subtract that from the meal, you get 51. And so I wrote that 51 underneath the meal minus one standard deviation. And if you go the opposite way, and you add on 14.5, you get 80. So I put that up there. So that's I just labeled those two, you can kind of guess what we're going to do on the next slide. Surprise, we're going to do almost the same thing. Oh, we're doing the mu minus two times the standard deviation to get the 36.5. And the meal plus two times the standard deviation to get that 94.5. And you probably already were ahead of me with this one. This is where we do 65.5 the meal minus three standard deviations, and we get 22. And then we add three standard deviations, and we get 109. And now we're all labeled. So what does this all mean? Well, remember our n equal to 100 just out of convenience. So what does this mean? It means that 34% of the scores are between 51 and 65.5. So that's a yellow bar, right? So 34 scores were that because I had 100 people in the class. So I'm standing there in that class. And I've got a 73. But I don't 34 those people I'm looking at have a score between 51 and 65.5. I also know that another 34% or another 34 in this class because there's 100 have a score between 65.5 and 80 and my 73 is somewhere in there, right? So already I'm getting an idea that 68 people or 68% of the scores are going to be between 51 and 80, right? And so I'm right there with 68% of the class. So I'm going to go through some fake test questions for you to just show you how to come up with the answer. So let's say the question was, what percent of the data students scores are between 36.5 and 80? So think about how you would answer that question. So see where 36.5 is it's on the lower limit of the orange part and see where the 80 is it's on the upper limit of the green part. So what you would do is you would add up the percents in between right 13.5 plus 34 plus 34. And the answer to what percent of the data are between 36.5 and 80. The answer would be 81.5%. Here's another question. What cut point marks the top 16% of the scores? So already, you know, you're up in that area probably where the purple or the blue are, right? And so what would make the top 16%. Well, if you actually add together that 0.15% from the little black part, the 2.35% from the purple and the blue 13.5% you'll get 16%. So the cut point then for that all the scores above 80 that would constitute the top 16% of the scores. Here's another quiz question. What percent of the scores are below 94.5? So we see 94.5 is at the upper limit of the blue section. So you could kind of say, Well, let's just add up everything below it, right? We'll add up everything below it and that percent of scores will be below 94.5. And so we do that we add up everything below it. But remember how I said that they're that the yellow, orange, red, and the little black part there that that equals 50%. If you just wanted to say, Okay, that's 50% plus the green part plus the blue part, you could do that and you get the same answer. So what are the cut points for the middle 68% of the data? I just wanted to show you an example. What if they say middle, right? Well, you're going to have to be centered around me, right? So the middle 68% means 34% above the mean and 34% below the mean. So the cut points would be 51 to 80. Okay, now I'm going to ask a similar question, but I'm going to use different words. Okay, what is the probability that if I select one student from this class, that student will have a score less than 80. Okay, so notice I'm using totally different terminology. I'm saying what is the probability yet? The only the actual answer is what you would probably guess, which is where you add up all the percents below 80. So the point of me giving you this quiz question is to point out that percent and probability mean the same thing when you talk. So either I'm going to say what percent of the data are below 80, the score of 80 or what is the probability that if I select one student, that student will have a score less than 80, that is actually the same question. So the answer is going to be I use that 50% trick here. That answers to be 50%, which is the whole bottom half that curve plus 34% gets up to 84%. Right. So so the probability that if I select one student that student will have a score less than 80 is 84%. And that's the same as what percent of the data is below 80 is 84%. Okay, here's another probability question. What is the probability I will select a student with a score between 36.5 and 51? Well, that's as if I was asking, it's the same question as what percent of the data are between 36.5 and 51, which you would know the answer that that would be 13.5. That's the orange part, right. But even if I say what is the probability I will select a student with a score between 36.5 and 51, it's 13.5%. So let's say that we were at a casino and we were betting, right. And I'm like saying, Okay, there's 100 students, I'm going to just grab a score out. And I'm betting a lot of money that I'm going to grab somebody between 36.5 and 51. And you'd probably be like, you don't want to bet on that, because you only have 13.5% probability of selecting one, you probably want to bet if you're going to bet on something in in the yellow section or something in the green section, because they have higher probability. So that's how you would think about probability and percent, even though they're kind of the same thing. I just wanted to show you how they were the questions differently. But it means the same thing. So now I want you to just sit back and think for a second. So think about what would happen in a different class taking the same hard test, meaning nobody's getting 100%. What if the mu was the same, meaning everybody's doing badly? But the standard deviation was larger than 14.5. What would that do to the intervals? So let's just stare at this for a second. Let's say the mu was still 65.5. But the standard deviation was like 30. Okay, that there was a lot of variation in the class. That would already mean that where the 80 is right now, that that would actually be 95.5. Right. And where that 51 is there. Now, if we had a standard deviation of 30, that would actually be 35.5. I mean, that'd be a way bigger interval, right. And so the class I was in in chemistry was an undergraduate class, I was in costume design. This was a whole bunch of different kinds of people in chemistry. And that's probably why we even had kind of a big standard deviation of 14.5, even though I made that up. I mean, in reality, we probably did have a big standard deviation. I knew in the chemical engineering department, they had chemistry classes for chemical engineering majors. I'll tell you their standard deviation was probably a lot smaller, because they were probably more alike and got more similar grades as each other. But with this diverse class, we probably had a pretty big standard deviation. So that gets to my last question. What if the standard deviation was actually smaller than 14.5? So if we were like in the chemical engineering class, and they were taking chemistry and they had a smaller standard deviation, maybe they might have had the same mean 65.5. Well, let's say their standard deviation was like five, then where the 80 is now would be a 70.5. And where the 51 is would be a 60.5. And we'd have way more confidence of where we knew the scores fell. Like as I was standing there with my 73, I would be saying like, Oh, you know, my 73 is pretty high, if everybody has a small standard deviation, right? Whereas it's not that high here, because we had kind of a big standard deviation, it's in the first, the green part. So the reason why I wanted you to think about that is, that's why this shape goes by mu in standard deviation, because it really matters how big the standard deviation is, how big each of those areas are with the different colors. So I just wanted to remind you that percent area and probability are all related. The percents literally refer to the percent of the area of the shape. Okay, and imagine the whole thing is 100%. So just to remind you the orange part is 13.5% of the area of the whole shape. But it also is the probability that an x like a student and acts falls between mu minus one standard deviations and that if I select one x from a group, this group, that 13.5% is the probability that I will get an x in that range. And so it means both things. So in conclusion, the empirical rule helps establish intervals that apply to normally distributed data. And it's more useful than Chebyshev because it's more specific. These intervals have a certain percentage of the data points in them. And they also refer to the probability of selecting an x in that interval. And these intervals depend on the mean and the standard deviation of the data distribution. So if those change, then exactly where the numbers are on those intervals change. Well, I hope you enjoyed my explanation of the empirical rule. And now you can practice doing it yourself at home.