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Current time:0:00Total duration:8:16

Standard normal distribution and the empirical rule (from ck12.org)

Video transcript

we're now on problem number four from the from the normal distribution chapter from ck-12 dot orgs flex book on ap statistics you can go to their site to download it it's all for free so problem number four in it it's at least in my mind pretty good practice for normal for a standard normal distribution for a standard normal distribution place the following in order from smallest to largest so see percentage of data below one negative one okay let's let's draw our standard normal distribution so a standard normal distribution is one where the mean is sorry that drew the standard deviation is one where the mean mu for mean is where the mean is equal to zero and the standard deviation is equal to one so let me draw that standard normal distribution C so let me draw the axis right like that let me see if I can draw a nice-looking bell curve the bell curve right there you get the idea and this is a standard normal distribution so the mean or you can kind of view the center point right here it's not skewed this is the mean is going to be zero right there and a standard deviation is one so if we go one standard deviation to the right that is going to be one if you go to standard deviations it's going to be two three standard deviations three just like that one standard deviation to the left is going to be minus one two standard deviations to the left will be minus two and so on and so forth minus three will be three standard deviations left of the standard deviation is one so let's see if we can answer this question so what's the percentage of data below one so the percentage the Part A Part A that's this stuff right here so everything below one so it's all of all of not just that little center portion it goes keeps going everything below one right percentage of delete data below one so this is another situation where we should use the empirical rule empirical rule numbers to get more practice empirical rule or maybe the better way to remember the empiric is just the 6895 99.7 rule and then that 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 normal distribution table you don't have to do this but sometimes in class or people want you to estimate percentages and so it's good to do bit you know 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 one or everything to the left of one so the empirical rule tells us that this middle area between one standard deviation to the left and one 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 one or two 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 percent 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 percent and this right here is 16 percent now the question they want to know the area of everything we do it in a new color everything less than 1 all right 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 one standard deviation plus this left branch right there so 68 plus 16% which is what that's equal to 84% so this part a is 84% they're going to want us to put it in order eventually but it's good to just solve it 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 delete 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 were just have to order things you could have made that intuition but you know 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 zero so number C is zero D the standard deviation well by definition the standard deviation for the standard normal distribution is one so this is one right here this is easier than I thought it would be part-ii the percentage of data above two above two so they want the percentage of data above two so we know from the 68 95 99 point seven rule that if we want to know how much data is within two standard deviations so let me do it in the new color so if we're looking for from this form we do in a more vibrant color green if we're looking from this point to this point so it's within two standard deviations right the standard deviation here's one if we're looking within two standard deviations that whole area right there by the empirical rule is 95% within two 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 two standard deviations in either direction that has to be the remainder so you know everything in the middle was 95 within two standard deviations so that has to be 5% if you add that tail and that tail together everything to the left and right of two standard deviations well I've made the argument before everything is symmetrical this and this are equal so this right here is two-and-a-half percent and this right here is also two-and-a-half percent so they're asking us the percentage of data above two that's this tail just this tail right here the percentage of data more than two standard deviations away from the mean so that's two and a half percent doing a darker color two and a half percent two and a half percent now they're asking us is he 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 one or do they want us to say what's 84 percent so should we consider the answer to Part A 84 or should we consider that they said the fraction of data below one I would say 0.8 4 so it depends on the how they want to interpret it same where 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 you someone asked me the percentage 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 should fixate on this too much but let's just say that they're going with the the decimals so if we wanted it to do it that way the they want to do it from smallest to largest the smallest number we have here is C right that's a 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 0.8 4 and then the largest would be the standard deviation D so the answer is C bad and obviously the order would be different if the answer to this instead of saying is point 8 4 if you said it was 84 because they're asking for the percentage so a little bit of ambiguity if you had a question like this on the exam I would clarify that with the teacher but hopefully you found this useful