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CCSS.Math: you can never have too much practice dealing with the normal distribution because it's really one of those super important building blocks for the rest of statistics and really a lot of your life so what I've done here is I've taken some sample problems this is from ck-12 dot orgs open source flex book their AP statistics flex book and I've taken the problems from their normal distribution chapter so you could go to their site and actually look at look up these same problems so this first problem which of the following datasets is most likely to be normally distributed for the other choices explain why you believe they would not follow a normal distribution so let's see choice a so this is really this is you know my beliefs come into play so this is unusual in the math context it's more of a what do I think it's kind of an essay question so let's see what they have here a the hand span measured from the tip of the thumb to the tip of the extended fifth finger so I think they're talking about let me see if I can draw a hand so if that's it's the index finger and then you got the middle finger and then you got your ring finger and then you got your pinky and the hand will look something like that I think they're talking about this distance from the tip of the thumb to the tip of the extended fifth finger which is a fancy way of saying the pinky I think they're talking about that distance right there and they're saying if I were to measure it of a random sample of high school seniors what would it look like well you know how far this is this is a combination of genetics and environmental factors maybe how much milk you drank or you know how much you hung from your pinky from a bar when while you were growing up so I would think that it is you know a sum of a huge number of random processes so I would guess that it is roughly normally distributed and if I look at my own hand and my hand I don't think has grown much since I was a high school senior it looks like I don't know it looks like roughly nine inches or so I play guitar maybe that helped me stretch my hand but if that's you know it's really an essay question so I just have to say what I feel so I would guess that the distribution would look something like this I don't know I've never done this but you know maybe it has a mean of eight inches or nine inches and it's distributed something like this it's distributed something like so maybe it probably does look like a normal just abuse but it probably won't be a perfect in fact I can I can guarantee you it won't be a perfect normal distribution because one no one can have it can have negative length of that span this distance could never be negative so they're going to have a I guess in you know you can have no hand so that would maybe be counted as zero but the distribution wouldn't go into the negative domain so it wouldn't be a perfect it wouldn't be a perfect normal distribution on the left hand side it would really just end here at zero and even on the right hand side there is some physically impossible hand lengths no one can have a hand that's that's larger than the than the height of the Earth's atmosphere or you know an astronomical unit you would start touching the Sun there's some point which is physically impossible to get to and you know in a true normal distribution if I were to flip a bunch of coins there's some very very small probability that I could get a million heads in the row in a row it's almost zero but there's some probability but in the case of hand spans there's no way out here you know the probability of a human being who happens to be a high school senior having a I don't know one mile length one mile length hand span that's zero so it's not going to be a perfect normal distribution at the outliers or as we get further and further away from the mean but I think it'll be a pretty good you know in our everyday world as good as we're going to get approximation the normal distributions going to be a pretty good approximation for the distribution that we see I guess one thing that you know it's hot you know this is high school seniors what I just when I did this I was kind of from my point of view as a as a guy and I would argue that you know high school seniors guys probably have larger hands and women so they're it's possible you actually have a bimodal distribution so instead of having it like this it's possible that the distribution looks like this that you have one peak for guys maybe at eight inches and then maybe a slight another peak for women at you know I don't know at seven inches and then it just the distribution falls off like that so it's it's also possible it could be bimodal but in general this is going to distribution it's going to be a pretty good approximation for Part A of this problem let's see what Part B what they're asking us to describe the annual salaries of all employees of a large shipping company so if we're talking about annual salaries you know we have minimum wage laws whatnot so I would guess at any corporation if we're talking about full-time workers at least there's going to be some minimum salary that people have so I would say and probably a lot of people will have that minimum salary because it'll be probably the most labor intensive jobs you probably have most people are down there at the low end of the pay scale and then you have your different the middle level managers and whatnot and then you probably have you know this big gap and then you probably have your your your true executives maybe your CEO or what's not with a with you know if this if this mean right here is maybe 40 $40,000 a year and you know this is probably 80,000 where you know some of the mid-level managers lie but this out here this will probably be you know actually if you were to draw a real you know the way I've scaled it right now this would be eighty to me this would be about two hundred thousand which is actually you know a reasonable salary for a CEO but the reality is is that this actually might get pushed way out from there it might look something like that it might be way off the charts you know let's say the CEO made five million dollars in a year because he cashed in a bunch of options or something so it'd be way over here and maybe it's a CEO and a couple of other people the CFO or the founders so my guess is would it would it definitely wouldn't be a normal distribution it and it definitely would have a second it would be bimodal you would have another peak over here for senior management up at the unless where well they're not saying you know if we're in maybe in Europe this would probably be closer to the left but it won't be a perfect normal distribution and you're not going to have any values below a certain threshold below that kind of minimum wage level so I would call this when you have a tail that goes more to the right than to the left call this a right skewed distribution right skewed right skewed distribution since it has two humps right here one there and one there we call sits bimodal it depends on what what kind of company this is but that would be my guess of a lot of large shipping companies salaries let's look at choice see or problem part see the annual salaries of a random sample of 50 CEOs of major companies 25 women and 25 men the fact that they wrote this year I think they maybe are implying that maybe men and women you know the the the gender gap has not been closed fully in there there is some discrepancy so if it was just purely 50 CEOs of major companies I would say it's probably close to a normal distribution it's probably something like well you know once again there's going to be some level below which no CEO is willing to work for although I've heard of some cases where they work for free but they're really getting paid in other ways if you include all of those things there's probably some base salary that all CEOs make at least that much and then it goes up to some it goes up to some value you know the highest probability value and then it probably has a long tail to the right and this is if there were no gender gap so this would just be a purely right skewed distribution where you have a long tail to the right now if you assume that there's some gender gap then you might have two two humps here which would be a bimodal distribution so if you assume there's some gender gap boom this is part C right here then maybe there's one hump for women and if you assume that women are less than men then another hump for men and they're 25 of each so there wouldn't be more necessarily more men than women and then it would skew all the way off to the right and in fact I think there'd probably be a chance that you have this other notion here where you have these you know super CEOs or mega CEOs who make millions while most CEOs probably just make you know just put in quotation marks a few hundred thousand dollars while there's a small subset that are way off many standard deviations to the right so you could even be a trimodal distribution here so that's choice C and then and so far choice a looks like the best candidate for a pure or the closest to being a normal distribution let's see what Troy what D is the dates of a hundred pennies taken from a cash drawer in a convenience store a hundred pennies so that's actually an interesting experiment but I would guess once again this is really a question where I get to express my feelings about these things you know there's as long as your answer is reasonable I would say that it is right most pennies are newer pennies because they go out of commission they get traded out they get worn out as they age they get lost or you know or they get put you know pressed it at the little tourist place into those little souvenir things I'm not even sure if that's legal if you can do that to money legally but so my guess is that if you were to plot it you would have a ton of pennies that are you know within the last few years so if we were so the dates of 100 pennies not their age so the date so if we're sitting here in 2000 so this is 2010 I would guess that right now you're not going to find any 2010 pennies but you're probably going to find you're probably going to find a ton of 2009 pennies then it probably just goes down from there and of course you're not going to find a pennies that are older than say the United States or before they even started printing pennies so it's obviously this tale isn't going to go to the left forever but my guess is this is going you're going to have a left skewed distribution left skewed where you have the bulk of the distribution on the right but the tail goes off to the left that's what's called a left skewed distribution sometimes this is called a negatively negatively skewed skewed distribution and similarly this right skewed distribution could all or this right skewed distribution sometimes it's called positively skewed and if you have only one hump you don't have a multi-modal distribution like this and a left skewed distribution your mean is going to be to the left of your median so in this case maybe your median might be someplace over here but since you have this long tail to the left your mean might be someplace over here and likewise in this distribution your median your middle value might be someplace like this but because it's right skewed and for the most part only has one big hump this hump won't change things too much because it's small your mean is going to be to the right of it so that's another reason why it's called a right skewed or positively skewed distribution so to answer the question you know these are my feelings about all of them but I would say you know for the other choices explain why I believe they would not followed or although they said which of the following data sets is most likely to be normally distributed well I would say choice a but it's really you know a matter of opinion at least in this question

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