11 September 2010

The Normal Distribution and the 68-95-99.7 Rule

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all right in this video I'm going to

talk about the normal distribution and what I'm going to talk about is what's known as the 6895 99.7 rule for for normal distributions or bell curves and what it says if you have a normal distribution with a mean mu and a standard deviation of Sigma it says the bulk of your observations which in this case 68% of the observations are going to fall within one standard deviation of the mean mu it says ninety-five percent of your observations are going to fall within two standard deviations of the mean mu and it says ninety-nine point seven percent of the observations will fall within three standard deviations of the mean mu so let's uh let's try to make some sense out of that so there's a popular IQ test and it turns out that the scores for these IQ tests are approximately normally distributed with a mean of 100 and a standard deviation of 15 we're going to use the 68 95 99 point 7 rule to figure out the following so we're going to do three things we're going to figure out about what percent of people have IQ scores above 100 above 145 and below 85 so I went ahead and made a little a little normal

distribution here pretty rough again the peak is where the mean is so we know the means at 100 that's given to us so the first question is you know about what percent of people have a score above 100 well again the idea for a normal distribution 50% of the observations roughly are going to be you know at or below the mean and again roughly about a hundred fifty percent of the observations are going to be at or above the mean so in this case since our mean is at 100 we know that 50 percent of the observations are going to be to the left 50 percent of the observations are going to be to the right so to answer Part A we don't have to do much at all we don't have to really use this rule so above 100 again half the observations will be you know roughly again half the observations will be below 100 half of them will be above 100 so I would say we'll say about 50 percent so about 50% of the observations would be above 100 the second question was what about above 145 you know what percent of the observations and this is where we're going to start using our rule and let's go ahead and talk about it on this example anyway so our

standard deviation was 15 so I'm going to add 15 to 100 that will give me 115 and I'm also going to subtract 15 from 100 that would give me 85 and again what this rule says is we're now one standard deviation away from the mean this is supposed to be symmetric but it's you know obviously I'm doing this by hand so not quite perfect what we could conclude here is we could actually say and we didn't ask this question at all but we could say roughly or approximately 68% of the IQ scores are between in this case between 85 and 115 okay so we've gone one standard deviation away if we go two standard deviations away okay so if I go two standard deviations I'm going to have to add another 15 that's going to put me at 130 and I'm going to subtract 15 that'll put me at 70 so if I look at the region between the 70 and the 130 again now I'm two standard deviations away so I could say that 95% of my observations 95% of the IQ scores are going to be at or above 70 and at or below 130 likewise if we do three standard deviations okay so if we subtract 15 from 70 that will give me 55 and if I add 15 to 130 I'm going to get 145 and we'll use this to answer

Part B but again now it says ninety-nine point seven percent are of the IQ scores are going to be above 55 at or above 55 but at or below 145 so again we're going you know sort of one standard deviation to go to 85 to 115 two standard deviations to go from 70 to 130 and then we're going three standard deviations to go from 55 to 145 okay so to address this question you know again what percent of the scores are above 145 well again we're now three standard deviations away so if I kind of look at the area inside of there again at this point I have got ninety-nine point seven percent of the observed values well what that means is if there's ninety-nine point seven percent of the observed values that means there remaining so if we take ninety-nine point seven and subtract that from 100 that means there's point three percent left over and that's going to be in the tails of our normal distribution so really it says if we take point three percent and divide it by two it says there's 0.15 percent of the population that has an IQ score below fifty five and likewise there's going to be a point one five percent of

the population that has scores above one forty five so I would say to answer this question about one five excuse me about point one five percent of the population is going to have an IQ score above one forty five so you know very very rare here our last question here was you know what percentage of the people had an IQ score below eighty five well okay so I think if we looked at between 85 and 115 that's going to be one standard deviation away so again we said there's 68% of the observations fall within one standard deviation okay well how much is left over well since there's 68% within one standard deviation that means the remaining 32% must be outside and kind of the same thing if we take our 32% and divide it by two that's going to give us 16% so it says if we if we look outside of this region the 68% region it says there's going to be 16 percent of the observations are going to be smaller than that and likewise the other 16 percent are going to be above that so for Part C here when we talk about below 85 I would say in this case about 16 percent of the test scores are going to be below 85

likewise if they said you know if the question was how many of the you know what percentage of the test scores are above 115 again it would be the same answer roughly 16 percent of the test scores would be above 15 percent so just kind of little useful information to know you know all you need to know is the mean the standard deviation and then just remember this little rule and it gives you kind of a lot of information about you know the scores that fall within that range