module 13 introduction to normal normal probability distributions 12 of 19. the empirical rule gives us a method for identifying likely and unlikely values but it is limited to probability questions involving x values that fall exactly one two and three standard deviations away from the mean we will now consider probability questions that cannot be answered using the empirical rule let's take a look at this example so let's go back to our example of male foot length and ask a question that cannot be answered using the empirical so the question is what is the probability that a man's foot length is more than 13 inches so we know that the distribution of the foot length follows the normal distribution as we learned from the previous example so because in this case 13 inches is not exactly one or two or three standard deviations away from the mean we can give only a very rough estimate of the probability at this point so we are now looking at this point right here so this one right here and recall we still have a mean of eleven and so we want to be able to give we want to be able to find the probability that a man's foot length is more than 13 inches so in other words we are looking for the probability that x is greater than 13. so to find the probability we will determine the number of standard deviations 13 is above the mean and then use technology to determine the probability so we will start by practicing how to find the number of standard deviations an x value is from the mean so currently the x value that we have that we are interested in is x equaling 13. so how many standard deviations above the mean is a mouse foot length of 13 inches and again notice how 13 does not fall exactly within one two or three standard deviations from the mean right so here's the mean and it doesn't fall exactly within those empirical rule values that we know so we have to use a different method so we know that 13 inches is 2 inches above the mean how many standard deviations is this so right we know it's two inches above because we have um 11 is our mean so we have to go one two so 12 and 13 so 2 inches above the mean so now the question is how many standard deviations is this now since the standard deviation is 1.5 inches 2 inches is 2 divided by 1.5 with it which is about 1.33 standard deviations above the mean so what did we just do right what did we just do so we found the distance between 13 and the mean we found what the distance was that distance is just 1 2. so here we would have 12 so 1 two so we found that distance and we got that distance was two and then we divided by the standard deviation and so once we did that computation now we know that 13 inches is 1.33 standard deviations above the mean the other way we can take a look at this is by taking 13 subtracting 11 right 11 is the mean dividing by the standard deviation so then that gives us 13 minus 11 which is equal to 2 and we will have 2 divided by 1.5 which again gives us 1.33 standard deviations in the language of statistics we found the z-score for a male foot length of 13 inches to be z equaling to 1.33 so that is um this is called the z-score or to put it another way we found we we have found um we have actually standardized the value of 13 into standard deviation units a foot length of 13 inches is 1.33 standard deviations above the mean so here is the formula in general so in general the standard the standardized value z tells us how many standard deviations below or above the mean uh the mean original value is the formula for the z-score is this right here so it's the value that we are interested in this case we were interested in 13 we subtract the mean and then we divide by the standard deviation so now using familiar notation we have this formula here which is z equaling x minus mu over sigma for values of x above the mean the z-score will be positive for values of x below the mean the z-score will be negative so when we um when the z-score is positive right when the z-score is positive then that means that x is above the mean when the z-score is negative then that means that x is below the mean okay let's take a look at a different example so let's practice by finding another standardized value so we have um what is the standardized value for a male foot length of 8.5 inches what does this number tell us so now the current x value that we are interested in right so the x value that we are interested now is x equaling 8.5 so using my formula for the z-score i have 8.5 minus mu mu is the mean divided by the standard deviation so putting that into our calculator we get that it's negative 1.67 and now notice that it is a negative value right it is a negative value which means that this x value is below the mean below the mean a z score allows us to compare x values from different normal probability distributions here are several examples male foot length revisited women's foot lengths have um also have a normal probability distribution but the mean is 9.11 inches and the standard deviation is 1.2 inches sam's foot length is 13.25 inches and candice foot length is 11.6 inches which of the two has a longer foot relative to his or her gender group we cannot just compare the foot wings and this is important so the question is why why can't we just compare the footlines and that is because these footlinks come from different distributions canda's foot length is barely above the mean for men but her foot is much larger than the mean for women to make the comparison we standardize the values using z-scores so we will standardize the foot length for sam and also for candidates so we're going to be using our z-score formula so we have the x-value of interest in this case uh sam's foot length which is 13.25 minus the standard deviation oh sorry minus the mean of the male distribution for footlink divided by the male standard deviation of for foot length so once we put it into our calculator this is 1.5 so we say that sam's foot length is 1.5 standard deviations above the mean foot length for men candice z score when we plug it in we have 11.6 and because that is her foot length minus the mean for the distribution of women's foot lengths divided by the standard deviation for women's foot lengths so once we calculate that we have that that is equal to 1.75 so candace foot length is 1.75 standard deviations above the mean foot length for women okay right and remember how do we know that it's above right because it's positive the z-score is positive so we know it's above the mean so now how how can we compare these two values right so the question remember was which of the two which of the two has a longer foot relative to his or her gender group so even though sam's foot is longer than candace candace's foot is longer relative to other women than sam's foot is relative to other men so because her z-score is larger then we know that she has a larger foot based on her own gender so she has a larger for relative to other female let's take a look at another example test scores the test scores in maria's class have a normal probability distribution with a mean of 70 and a standard deviation of 5. the test scores in lamar's class have a normal probability distribution with a mean of 85 and a standard deviation of 10. maria scored a 76 and lamar scored an 80. why did why did better who did better on the test relative to their classmates okay so when we are comparing two people and we know that the information that they're giving us came from different distributions that we know we have to standardize at that specific value so again notice that maria's class the scores for her class had a distribution with a mean of 70 and a standard deviation of 5 but lamar's class had a mean of 85 and a standard deviation of 10. so we need to standardize their their individual scores in order to be able to compare them and actually say who did better okay so we find the z-score for maria computing that we get 1.20 so we say that maria scored 1.2 standard deviations above the mean right it's positive so we know we um are above the mean and lamar has a z-score of negative 0.5 so negative right so then lamar scored 0.5 standard deviations below the mean for his class so now we compare who did better on the test so that was the question who did better on the test so relative to the respected classes maria performed better on the test even though her score was lower than lamar's so the way to think about it is that once we standardize it right and we're talking about test scores once we standardize this then the z-score has the z-score has a normal distribution with a mean of zero and standard deviation of one so here we have a mean of zero i know that standard deviation is one so maria is um 1.2 standard deviations above the mean so 1.2 sds above the mean uh the lamar is actually below the mean right below the mean so lamar is actually um 0.5 below the mean and when we are talking about test scores right um just think about it as your own experience right when you take a test and they tell you what the um what the mean is what the average is you want to score above average right you want to score above the mean so if someone scores below the mean that means that they didn't do that well so the way that we are comparing this is is saying okay well who actually scored above the mean and that's gonna be maria