Normal distributions and the empirical rule
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Standard normal distribution and the empirical rule (from ck12.org)
We're now on problem number 4 from the Normal Distribution chapter from ck12.org's FlexBook on AP Statistics. You can go to their site to download it. It's all for free. So problem number 4, and it's, at least in my mind, pretty good practice. For a normal, or a standard normal distribution, place the following in order from smallest to largest. So let's see, percentage of data below 1, negative 1. OK. Let's draw our standard normal distribution. So a standard normal distribution is one where the mean is-- sorry, I drew the standard deviation-- is one where the mean, mu for mean, is where the mean is equal to 0, and the standard deviation is equal to 1. So let me draw that standard normal distribution. Let's see, so let me draw the axis right like that. Let me see if I can draw a nice-looking bell curve. So there's the bell curve right there. You get the idea. And this is a standard normal distribution, so the mean, or you can kind view the center point right here. It's not skewed. This says the mean is going to be 0 right there, and the standard deviation is 1. So if we go 1 standard deviation to the right, that is going to be 1. If you go 2 standard deviations, it's going to be 2, 3 standard deviations, 3, just like that. 1 standard deviation to the left is going to be minus 1. 2 standard deviations to the left will be minus 2, and so on, and so forth. Minus 3 will be 3 standard deviations to the left because the standard deviation is 1. So let's see if we can answer this question. So what's the percentage of data below 1? Part a, that's this stuff right here. So everything below 1, so it's all of-- well, not just that little center portion. It keeps going. Everything below 1, percentage of data below 1. So this is another situation where we should use the empirical rule. Never hurts to get more practice. Empirical rule, or maybe the better way to remember the empirical rule is just the 68, 95, 99.7 rule. And I call that a better way because it essentially gives you the rule. These are just the numbers that you have to essentially memorize. And if you have a calculator or a normal distribution table, you don't have to do this. But sometimes in class, or people want you to estimate percentages, and so you can impress people if you can do this in your head. So let's see if we can use the empirical rule to answer this question, the area under the bell curve all the way up to 1, or everything to the left of 1. So the empirical rule tells us that this middle area between 1 standard deviation to the left and 1 standard deviation to the right, that right there is 68%. We saw that in the previous video as well. That's what the empirical rule tells us. Now, if that's 68%, we saw in the last video that everything else combined, it all has to add up to 1 or to 100%, that this left-hand tail-- let me draw it a little bit-- this part right here plus this part right here has to add up, when you add it to 68, has to add up to 1 or to 100%. So those two combined are 32%. 32 plus 68 is 100. Now, this is symmetrical. These two things are the exact same. So if they add up to 32, this right here is 16%, and this right here is 16%. Now, the question, they want us to know the area of everything-- let me do it in a new color-- everything less than 1, the percentage of data below 1, so everything to the left of this point. So it's the 68%. It's right there, so it's 68%, which is this middle area within 1 standard deviation, plus this left branch right there. So 68 plus 16%, which is what? That's equal to 84%. So part a is 84%. They're going to want us to put this in order eventually, but it's good to just solve because that's really the hard part. Once we know the numbers, ordering is pretty easy. Part b, the percentage of data below minus 1. So minus 1 is right there. So they really just want us to figure out this area right here, the percentage of data below minus 1. Well, that's going to be 16%. We just figured that out. And you could have already known just without even knowing the empirical, just looking at a normal distribution, that this entire area, that part b is a subset of part a, so it's going to be a smaller number. So if you just have to order things, you could have made that intuition, but it's good to do practice with the empirical rule. Now part c, they want to know, what's the mean? Well, that's the easiest thing. The mean of a standard normal distribution, by definition, is 0. So number c is 0. d, the standard deviation. Well, by definition, the standard deviation for the standard normal distribution is 1. So this is 1 right here. This is easier than I thought it would be. Part e, the percentage of data above 2. So they want the percentage of data above 2. So we know from the 68, 95, 99.7 rule that if we want to know how much data is within 2 standard deviations-- so let me do it in a new color. Let me do it in a more vibrant color, green. If we're looking from this point to this point-- so it's within 2 standard deviations, right, the standard deviation here is 1-- if we're looking within 2 standard deviations, that whole area right there, by the empirical rule, is 95%, within 2 standard deviations. This is 95%. Which tells us that everything else combined-- so if you take this yellow portion right here and this yellow portion right here, so everything beyond 2 standard deviations in either direction-- that has to be the remainder. So you know everything in the middle was 95 within 2 standard deviations. So that has to be 5%, if you add that tail and that tail together, everything to the left and right of 2 standard deviations. Well, I've made the argument before, everything is symmetrical. This and this are equal. So this right here is 2 and 1/2%, and this right here is also 2 and 1/2%. So they're asking us the percentage of data above 2, that's this tail, just this tail right here, the percentage of data more than 2 standard deviations away from the mean. So that's 2 and 1/2%. Let me do it in a darker color-- 2 and 1/2%. Now, they're asking us, let's see, place the following in order from smallest to largest. So there's a little bit of ambiguity here. Because if they're saying the percentage of data below 1, do they want us to say, well, it's 84%. So should we consider the answer to part a, 84? Or should we consider-- if they said the fraction of data below 1, I would say 0.84. So it depends on how they want to interpret it. Same way here. The percentage of data below minus 1, I could say the answer is 16. 16 is the percentage below minus 1. But the actual number, if I said the fraction of data below minus 1, I would say 0.16. So this actually would be very different in how we order it. Similarly, if someone me asked me the percent, I'd say, oh, that's 2.5. But the actual number is 0.025. That's the actual fraction or the actual decimal. So I mean, this is just ordering numbers, so I shouldn't fixate on this too much. But let's just say that they're going with the decimal. So if we wanted to do it that way, they want to do it from smallest to largest, the smallest number we have here is c, right? That's 0. And then the next smallest is e, which is 0.025. Then the next smallest is b, which is 0.16. And then the next one after that is a, which is 0.84. And then the largest would be the standard deviation, d. So the answer is c, e, b, a , d. And obviously, the order would be different if the answer to this, instead of saying it's 0.84, if you said it was 84 because it's asking for the percentage. So a little bit of ambiguity. If we had a question like this on the exam, I would clarify that with the teacher. But hopefully you found this useful.