Video Transcript: Distribution of Continuous Data - Part 2

Welcome. Let's continue our conversations around distributions of continuous  data by focusing on what is probably the most popular distribution in all of  statistics. We call that the normal distribution. The normal distribution is one of  the most common and important distributions for describing a continuous  random variable. In all honesty, the normal distribution is the foundation of  statistical inference, hypothesis testing, confidence intervals, regression  analysis. All these things hinge on the normal distribution. We'll be talking about  all three of these things later on in the course, which is why we have to  understand the normal distribution now, because again it will be the  underpinning of all these other concepts. The most interesting part about the  normal distribution as well is that it appears all over the place in nature, in real  world data. Again, it's amazing to see the patterns that God has put out there in  the world. This is one of the most popular ones, the bell-shaped curve, the  normal distribution. You have probably seen it before. This is the shape of the  normal distribution. Now, this shape, this bell-shaped curve, where it has a big  hump in the middle, and it sort of gets smaller and smaller and smaller, the  further out in the tails you go. Does have an actual equation to it. Now, you do  not need to memorize or understand this equation in all its details, but some  people like seeing these things, and so I wanted to make sure I did bring it  forward. This is the equation to calculate that nice, pretty bell-shaped curve, you  see there. The big thing I want to focus about this equation is two different things that are involved in the equation. The first is the mean. Wait a minute, that's mu,  that's the mean. We've seen that before. So the middle of your data, the  average of your data, the mean of your data plays a big role in the normal  distribution. We also have the standard deviation, sigma. Again, the standard  deviation plays a big role in the normal distribution. In fact, these are really the  only two things you can define in this equation, the pi that you see here is the  good old 3.1415 number that you've probably learned about before. The e that  you see there is what we refer to as the exponential function. So the only thing  that you have that's there that is unknown would be mu and sigma, the x's that  you see there, all the x's are just your data itself. So, really, how your data looks  in terms of its average and how spread out your data is is all you need to know  to completely define this normal distribution. In fact, that's one of the many  important characteristics that the normal distribution has. So, starting at the  bottom again, the normal distribution is completely defined by the mean and the  standard deviation. We'll get to more of that here in a moment, but let's talk  about some of these other points as well, the normal distribution is perfectly  symmetric, or more formally, we would say it has a skewness of zero. It's not  skewed one way or the other, and it looks like that here in the picture as well,  right? You have a symmetric distribution. If we were to draw a vertical line down  the middle, you would see the same thing on both sides. The normal distribution  is also what we refer to as unimodal. It basically means you have one big 

collection of data in the middle, so if you had bimodal data, you would see two  humps in your data curve. If it was trimodal data, you would see three humps in  your data curve, but because you have one big hump of data here, with the tails  just getting smaller and smaller and smaller away from that collection of data in  the middle, we call this a unimodal distribution. Another fun characteristic about  the normal distribution, the mean, the median, the mode, they're all equal to  each other, and they're all equal at the exact middle. So, the median is the peak  of the curve, the mean is the peak of the curve, the mode is the peak of the  curve. So, when you look at the average of your data, it would be right in the  middle of everything that we see. Let's talk about another one of these  characteristics: asymptotic to the x axis. Whoa, okay, hold on a second. There's  some big words there that are a little bit more mathematical. Let's try and  understand what they mean. Asymptotic to the x axis means that. Literally, a  normal distribution can take any value, any value from negative infinity all the  way up to positive infinity. I know what you're thinking. Well, wait a minute, it  looks like the curve kind of ends on the left hand side and ends on the right  hand side at some point. Well, actually, that's not the case. The curve goes off  into infinity in both directions. It's just there's so small a chance of actually  seeing things far away from the middle of the distribution that it's basically very,  very unlikely. And in fact, that's really what the normal distribution is. Kind of  think about it like a bar graph, but a lot of bar graphs stack side by side, so the  height of your normal distribution is where things are more likely to take place,  so the more probable values of your data are going to take place closer to the  middle of your distribution, the less probable values are going to take place far  away from the middle of your distribution. Kind of think about it like heights of  people. Let's say the average height of people is six feet tall. That means that  most people would be around six feet tall. That doesn't mean that there aren't  people that are much taller or much shorter than six feet tall, but that they occur  with a less probability of happening. So, again, there are also people who are  seven feet tall. There's just not as many of them, and so they are less probable  because they're further away from the middle the mean of your data. Let's talk  about that last point that we led with as well, completely defined by the mean  and standard deviation. What do you mean? Well, let's talk about the mean. The mean, the average of your data basically defines where this normal distribution  is located, like I said, the mean is the middle of the distribution, so you can see  here three different normal distributions, all of them have the same spread, they  have the same shape, however, they're centered at three different points, one of them is centered at 0.One of them is centered at five, and one of them is  centered at negative 10. So you can see that shifting the same distribution can  be done with the mean. So if we had something with a mean of zero and I  wanted to shift the whole distribution to the right, I could add five, for example, to every value, and it would shift the whole distribution to the right, and we can do 

the same thing going the other direction by subtracting things, but really, again,  the average of your data defines the center of this bell-shaped curve. The  spread of your data defines how spread out your data is in terms of the bell shaped curve. So, again, here are two different normal distributions. Now, both  of these normal distributions have the same center; they're both centered at the  same mean. However, they have different spreads. The one that is more spread  out does not have as high of a peak in the middle, and more of its data is spread out into the tails in terms of the idea of just width, whereas the one in the smaller standard deviation is a little bit more narrow in terms of the hump itself, so  again, don't want to get this too confused with the last concept of asymptotic to  the x axis. Both of the normal distributions you see here on this screen can take  any value from negative infinity to positive infinity. However, it's a matter of  where most of the data is located, so if your data is more spread out, it's not as  located tightly around the mean as another data set might be, that's the idea.  So, let's summarize real quick. Well, the normal probability distribution, again, is  one of the most common in nature, as well as one of the most important  distributions in mathematics, the normal distribution is the foundation of all  statistical inference. Later chapters in this course, like hypothesis testing,  confidence intervals, even introductions to regression analysis, are all going to  hinge on the normal distribution, and this normal distribution has some  wonderful mathematical characteristics. Again, it's symmetric, it's asymptotic to  the x axis, it's unimodal, it's completely defined by the mean and the standard  deviation. The mean and the median and the mode are all equal to each other.  All of these are wonderful characteristics that we can take advantage of. In fact.  We're going to take advantage of some of them now as we talk about what we  call the empirical rule. So, what is the empirical rule? Well, the empirical rule is  basically the idea that the normal distribution has a very predictable shape, and  because it's predictable, we can use that shape to our advantage, and this all  goes back to the world of probabilities, so the probabilities for a normal random  variable are determined by the area underneath that bell-shaped curve, so again the total area under the curve is one. Now, since the total area underneath that  bell-shaped curve that I showed you previously is one, and we know that the  normal distribution is perfectly symmetric around the mean, which is also the  median. Then the area of the curve below the mean and the area of the curve  above the mean are both point five half of your data is below the average, half of your data is above the average. That's one of the beautiful aspects of the normal distribution. So, if I were to sit there and ask you a question about, well, what's  the probability you get someone that is greater than six feet tall and you know  that the average of your people are six feet, then well, half of them are going to  be greater than six feet tall, because they follow a normal distribution. Now,  again, this only works if it follows a normal distribution. Without that normal  distribution, we wouldn't be able to say what we're going to be talking about 

here, but again, that split of 50/50 isn't the only thing we can do when it comes  to the normal distribution, like I mentioned, we have something that we refer to  as the empirical rule. Well, what does the empirical rule state? Well, the  empirical rule has three pieces to it. The first piece is that 68% roughly more  exact, it's 68.26% but we say roughly 68% or if you want, that's roughly two  thirds of your data, but with the normal distribution, roughly 68% of your data is  contained within one standard deviation of the mean. Well, what do I mean by  that? I mean that if we were to look one standard deviation below the mean, so  the mean minus one standard deviation, and then we were to look one standard  deviation above the mean, the mean plus one standard deviation, then  everything in the middle of that, so everything in between one standard  deviation below and one standard deviation above. If we were to look at all of  that data and say how much of that data, or how much of our data exists in that  range. Well, if your data follows a normal distribution, then no matter what the  mean is, no matter what the standard deviation is, 68% of your data is within  one standard deviation of that mean, isn't that neat? So, again, no matter what  the mean or standard deviation is, if you have a normal distribution, we know it  has this characteristic, but, like I said, there are three parts to the empirical rule.  This is just the first. The second part of the empirical rule is that roughly 95% of  your data, again to be more exact, 95.44 but roughly 95% of your data is within  two standard deviations of the mean, or in other words, if we were to look two  standard deviations below the mean and two standard deviations above the  mean, and look at all the possible values in between those two, then that would  encompass 95% of our data, and again, it doesn't matter what that mean and  standard deviation are, if your data follows a normal distribution, this holds true.  The last component of the empirical rule is that 99.7% of your data, almost all of  your data, not quite all of it, but almost all of your data is within three standard  deviations of the mean, so again, if we were to look three standard deviations  below the mean, the mean minus three standard deviations, and three standard  deviations above the mean, mean plus three standard deviations, if we were to  look at all the possible values of our data in between those two ranges, those  two boundaries, then that would basically have 99.7% of our data, almost all of  our data is within three standard deviations of the mean, so all three of these  pieces together form the empirical rule, or as some people like to call it, the 68  95 99.7 rule again has a completely understandable name, it just helps them  remember what the values are. So, again, we more formally call it the empirical  rule, but like I said, it could be called the 68 95 99.7 rule, and if you think about  it, because our distribution is symmetric, because of the fact that we know how  many or what percentage of the data falls within certain ranges in our normal  distribution, then we can know a lot of things, right? So, if you were to look  between the average, the mean, and one standard deviation above the mean,  that little slice is 34% Well, why? Because if you go one standard deviation on 

either side of the mean, that's 68% So, if we were to split that in two, that would  be 34% and so on and so forth. We can actually fill in all these little blocks. So,  again, if you were to know that 95% of your data is within two standard  deviations of the mean, and 68% is within one standard deviation of the mean.  That means there's about 27% in between those two ranges. Again, divide that  by two, and you've got 13 and a half percent in each of those pieces. So, like I  said, we can figure out what all of these little pieces of the normal distribution  are now I know you might be thinking this just sounds like math for the sake of  math, but this can actually help us analyze data. Let me show you. Let's imagine that new employees at a company have previous years of professional  experience that follow a normal distribution where the average is seven and a  half years, and the standard deviation is two and a half years. Okay. Well,  because it follows a normal distribution, because I know the mean, because I  know the standard deviation. Then I can answer a question like this: What's the  probability that any random new employee has between five and 10 years of  experience. Well, let's take a look at our normal distribution. So our normal  distribution would have this shape again, same shape, but the center is 7.5 the  mean is 7.5 and the standard deviation is 2.5 Well, that means that if I were to  go 7.5 down to five and 7.5 up to 10, one standard deviation below, one  standard deviation above, that between five and 10 years of experience  contains 68% of my data, or in other words, there's a 68% chance that a random new employee has between five and 10 years of experience, isn't that neat? and we can answer a variety of other questions, because we can fill in this normal  distribution again, we start with a mean of 7.5 in the middle, then we go down by standard deviations, so 7.5 then we go down to five, then 2.5 then zero,  because of the fact that again our standard deviation is 2.5 so we can do the  same thing above the mean, 7.5 plus one standard deviation is 10 plus another  standard deviation is 12.5 plus another standard deviation is 15, so again we  can answer a lot of different questions. Let's ask another same distribution.  What is the probability that any random new employee has between two and a  half and 10 years of experience. Well, again, if we were to fill out this chart, we  could basically isolate where the two and a half is, where the 10 is, and we can  just add those pieces together. We would have about 13 and a half percent of  our people between two and a half and five years, about 34% of our people  between five and seven and a half years, another 34% of our people between  seven and a half and 10 years. Well, that's around 81 and a half percent of our  people have between two and a half and 10 years of experience, or in other  words, the probability of any one random new employee having between two  and a half and 10 years of experience is 81 and a half percent. This is the power of the normal distribution. If we can show our data follows a normal distribution,  we can answer so many questions about it. It's kind of like those distributions we were playing with, with the discrete distributions. Remember, how we were 

dealing with binomial situations. We were just sort of filling in the equation to get  an idea of what was going on. Same idea here, except the normal distribution  has so well-defined areas that we could look at a variety of possibilities. All right, let's summarize. The empirical rule, also known as the 68 95 99.7 rule, is good  for quick, fast, and rough analysis of data that follows a normal distribution. Now it's not exactly the best for exact analysis, unless our interests are only in the  integer of standard deviations, so for example, if I wanted to go back and you  said, well, what about someone with three years of experience all the way up to  14 years of experience, I can't do that as easily here. So that's a downside of  this empirical rule approach. So we'll need to find another way to quickly  calculate area under the curve, when we have fractions of standard deviations  that we go away from the mean, but that's the next lecture. For now, that is the  end of this lecture, and I look forward to seeing you in the next one.