Welcome back. In previous lectures you've heard us use the term bell-shaped for describing distributions. There are a lot of variables that will have a bell-shaped distribution, as well as, a lot of our estimates are going to end up being approximately bell-shaped. So, here we have a bell-shaped distribution, not too many markings except a label on our X axis for the amount of sleep. There have been a lot of studies of college age students and some of the variables that they've measured as how much sleep they get. So, let's talk about the amount of sleep they typically get on a weekday night. Turns out that the average amount of sleep is about seven hours, so we'll mark that mean or average right in the middle of our symmetric distribution. Now, it would be very helpful for us to have a few other values along this axis to get a better feel for how spread out the values are. A measure of spread that's often reported along with the mean is the standard deviation. The standard deviation for the amount of sleep for our college age students, is 1.7 hours. Now, the standard deviation is going to measure how far away our values are from the mean. We like to interpret it as roughly the average distance that our values are from the mean. So on our case here, the amount of sleep that college age students get on a typical weekday night, they do vary from their mean of seven hours by about 1.7 hours on average. So, we want to use this standard deviation, this yardstick, and put a few more values along our distribution. If we were to take our mean and go out one standard deviation each way, so about 8.7 and down to 5.3. We would know that for bell-shaped or normal distributions you would expect to see about 68 percent of our observations in this range. 68 percent of the values are expected to be within one standard deviation away from the mean. Now, if we would go out yet another average distance that would put us out here at about 10.4 and down to 3.6. Well we would expect more observations in that range, and with a bell-shaped distribution we would actually expect about 95 percent. 95 percent of the observations are expected to be within two standard deviations of the mean. Let's go out yet one more standard deviation each way and that's going to put us way out here past 12 hours and down to 1.9, less than two hours of sleep, and there we would expect just about everybody. For a bell-shaped or normal distribution, that would be 99.7 percent, nearly all. So, this really cool fact about bell-shaped or normal distributions is sometimes referred to as the 68-95-99.7 rule, obviously, another phrase that's often used is that it's called the Empirical Rule. So for bell-shaped distributions, we can go out one, two and three standard deviations and expect 68, 95, and 99.7 percent of our observations to fall on those ranges. Very useful for getting a frame of reference for how unusual values are. Hi Reed? Hi. So you're a college student, how many hours of sleep do you typically get on a weekday night? I like to sleep a lot, so I'll get about ten hours of sleep. Ten hours seriously? Yeah. That seems like a lot maybe you can use our distribution here for the amount of sleep and give us an idea of how unusual that is? Sure. So, ten hours of course is quite a bit on our distribution, I would fall right around here just below that 10.4 at 10:00 hours and so how unusual is this? Well, we can see that I'm three hours above the mean, but a good way to get an estimate of how unusual it is, is by calculating what's called a standard score. To do this, we find the distance from our mean, so ten minus seven and then divide by the standard deviation of 1.7. Here we will get a value of 1.76 and that is my standard score or Z-Score when dealing with a normal distribution. So, 1.76 is pretty large, recall that 95 percent of our data is in between the 3.6 and 10.4, so I'm just below that 10.4 mark. So, fairly unusual for me to have that. The equation that I'm using here to calculate this is the standard square equation of observation minus the mean. So, my observation was 10 and I'm subtracting the mean of seven divided by the standard deviation of 1.7. Now, let's take my roommate for example, he's more of a night owl who only get about six hours of sleep a night. So, what would his standard score be? For my roommate who has a usual sleeping of six hours a night, he would be six and using the same equation we're going to subtract our mean of seven and divide by the standard deviation of 1.7. Here we get a value of -0.59. So, this negative that we have here is very important because it tells us that we are below the mean. The 0.59 tells us that we're only slightly below, not too far off from our mean, whereas my positive 1.76 was quite a bit above and the positive tells me it's going to be above the mean. Finally, let's talk about my friend Mark who does not sleep a whole lot. He tells me that he is 2.7 standard score is below the mean. So recall that the 3.6 was our two below, the 1.9 was three below, so he falls somewhere in the middle, we're not quite sure where, so how about we calculate what he gets as an average night of sleep? So, he told me that he was 2.7 standard deviations below, so I'm going to tuck in negative on there, and now I'm trying to find the observation or X for this case, and I'm going to subtract the mean of seven and divide by 1.7 once again. Another way that we can look at this is by doing 2.7 standard deviations, so multiply them together, below our mean of seven, and this would also get us our value or our observation for Mark. Calculating this out we would see an observation of 2.41 hours, so Mark does not get a whole lot of sleep. That explains why Mark is answering some of my emails about this project somewhere between 2:00 and 3:00 AM. So to summarize, if we know that we have a bell-shaped or normal distribution, and we know the mean and the standard deviation. We can use the empirical rule and our standard scores to help us get a nice frame of reference. Awesome Reed, and I think I need a little more sleep tonight.
