[Music] so what are you guys hi guys so before we can get start with anything we need to understand a little bit about the normal distribution so um let me talk briefly about it although you probably already seen this I mean from the last midterm it is good to talk briefly about it just to roll up to speed no.1 distribution looks kind of like this bow not this bad but kind of like this okay and you know the mean the population mean sits right here so the average sits right there but also the median the mode also sit right here so a couple things you have the mean is equal to the median is equal to the mode okay the second thing is he's pretty symmetric so if you look at this guy he's symmetric and you can kind of seen on the pictures say for the fact that I kind of messed it up but number three is of course he's on asymptotic this one's not so big a deal all it means is this guy just keeps going on and on forever and never quite flatlines at zero okay so you might be thinking something like so what good is that gonna do us right cuz how many things are asymptotic like things we're gonna model they're not gonna go infinitely high and implement low right but for all effective purposes once you get outside of 300 deviations above and below the mean you've pretty much gotten this entire curve so in that sense the normal distribution is good okay it's going to model a lot of things really really well and we'll see it is the distribution to go to in a second okay but first how do I use this sucker so these are normal distributions have different means and different standard deviations right but they all have this property that when you look at the mean if you go out one standard deviation to the right or to the left you'll always capture basically 60% of this curve some props have you learned memorize that some don't if you go out to standard deviations so two Sigma right then you're gonna capture about 95% okay and if you go out three like we said get pretty much everything I think you get like ninety nine point seven okay and that's four three standard deviations so three sigma okay so I don't wanna spend too much time in this since you solve this on your last midterm but we should talk to me and of course represents the population mean so this is the average in this standard sense of the word right and Sigma the standard deviation that represents how far from the average is a guy on average okay so how far from the average are you on average so it's a measure of spread okay okay so what's that kinda mean intuitively let's say the meanest a hundred and let's say the standard deviation is like ten that means if you pick somebody random it's not that unusual to find someone between say 100 and 110 and it's not that unusual find someone between say 90 and 100 so you can go up and down by a standard deviation and you're kind of average in a sense okay are pretty normal okay beyond that then you start to get a little freakishly high or a little freakishly low and all that good stuff okay okay it's no big deal so I get it the average is the average the standard deviation is generally how spread out guys are if you're within a standard deviation you're pretty quote normal whatever that means right and if you're more than one standard deviation above or more than one standard deviation below then you're starting to get like a little freakishly high or freakishly low okay so no big deal okay so um good then let's do a problem with this again I'm gonna do it kind of quickly because I've seen from the previous midterm you're comfortable with this but just in case okay so let's say you've got a standard deviation where the average eats 100 pounds of gummy bears per day okay and let's say for example like the standard deviation is 10 okay so that means most people out there will eat between 90 to 110 pounds gummy bears per day okay and I want to know what's the likelihood you pick somebody at random and that person eats more than so let's say more than let's say 140 pounds of gummy bears okay so I don't know what's the likelihood that you pick somebody at random from the population and you find that they eat more than 140 pounds of gummy bears per day okay okay so what's the set up as always we'll draw in that average here for 100 right okay remember this is a frequency distribution so you line up all the different possible scores and technically you're going on the population you're asking people how many pounds gummy bears do you eat right and if a bunch of people get 100 then they get a high mark over here and if not so many people you'd a 150 pounds of gummy bears per day then they're kind of over here you know the mark 4 that's low okay no big deal okay so let's try that so first thing I do is you can talk about numbers right but this would suck because if I had a different normal distribution I'd have a different mean a different standard deviation I have to look each one of these guys up and I'd have to have a separate table for each and that's really a pain in the butt so what I want to do instead is we want to what we want to do is we want to standardize it so it's a standardized it what you do is you take this guy and you convert the actual scores into something called z-scores this is just for archimedes z-scores are a nice nice way of converting these scores we could use just one table to figure things out okay and you know that kind of makes sense because remember the defining property of the normal curve in a way is the fact that once you get one standard deviation above and below you always hit that same percentage like 68% okay and if you go to you always hit 95 cetera cetera okay so let's do that so how do I what's a z-score mean so you guys remember represents the number of standard deviations above or below the mean your score is so the first thing I do is convert so if you look at that 140 right I want to know how many standard deviations above or below the mean is 140 so the first thing is I take 140 and have subtract 100 for a minute everybody agrees so let's just plot it down here so here's 140 and we know it's here and we know this difference is 40 and that's what we got right but let's say you had 24 eggs and I wanted to know how many dozen you have if that's the case you would take your 24 eggs you divide by the number in a dozen to say you have two dozen same thing right so over here you've got 40 points of difference but I want to know how many standard deviations fit in there so you look at your 40 points and you divide by the number of points in the standard deviation or to be ten and that would give you four well that makes sense right is your forty points above the mean right but each standard deviation is worth ten points so if you're forty points above and each standard deviations worth ten you really are four standard deviations above the mean so that's what a z-score represents so if you had a z-score of say plus four that means you are four standard deviations above the mean if you had a z-score say like negative two that would mean you're two standard deviations below the mean okay so this number that we ended up with is ridiculously high but that's fine we can go with that so we now have a z-score is plus four okay we get a couple of tricks depending on the sort of table your book uses you got to remember the fact that the entire curve is 100% and half the curve is of course 50% and you can use tricks for example some books are really nice once we look up the z-score and we convert the z-score to plus four right so that's origines score then you can look this up in a table or just two high number but let's pretend let's pretend for is actually in your book then it would go here maybe I would draw all this right if it gives the area of this shaded region or the percentage for this shaded region then we're good so if this were like for example make up a fake number point zero zero one like that totally fake number but let's say that's what the table gave you then that's the prickly hood that you're going to get a score 140 or more and that's what we want okay however if your table didn't give you this but only gave you this and let's say this was something like point four nine nine like that right then you would know 140 to a hundred well to the mean would be point four nine nine but you want 140 and up and you agree the whole thing over here is 50% so that would be 0.5 0 0 minus 0.49 9 which would be point zero zero one okay just talking about you know you might have to make minor adjustments depending on the sort of table they give you but the procedures still the same take your score convert it to a z-score look up that z-score value in the table and then use whatever they give you the table to figure out your answer okay so let me outline the general procedure again all the way you can run it backwards but since I feel like most people are probably comfortable at this you don't want to take the back will do a poem with this in a second okay but first let's remember the procedure my dad I should have written this out last time least I'm doing it now remember the general procedure z-score you took your score you subtracted the mean right so remember before we had 140 we subtracted the mean is a hundred and then we divided by the number of points in a standard deviation so that's a general procedure for getting the z-score okay