Normal distributions and the empirical rule
Qualitative sense of normal distributions
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 is I've taken some sample problems. This is from ck12.org 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 up these same problems. So this first problem, which of the following data sets is most likely to be normally distributed? For the other choices, explain why you believe they would not follow a normal distribution. So it's a choice. So this is 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 that's the index finger. And then you got the middle finger. And then you got your ring finger. And 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 how much you hung from your pinky from a bar while you were growing up. So I would think that it is a sum of a huge number of random processes. So I would guess that it is roughly, normally distributed. You know, 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 roughly nine inches or so. I play guitar. Maybe that helped me stretch my hand. But 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 maybe it has a mean of eight inches or nine inches and is distributed something like this. It's distributed something like that. So maybe it probably does look like a normal distribution. But probably won't be a perfect, in fact, I can guarantee you it won't be a perfect normal distribution. Because one, no one can have negative length of that span. This distance could never be negative. So they're going to have, I guess you could have no hand so that would maybe be counted as 0. But the distribution wouldn't go into the negative domain. So it wouldn't be a perfect normal distribution on the left hand side. It would really just end here at 0. And even on the right hand side, there is some physically impossible hand lengths. No one can have a hand that's larger than the height of the atmosphere or an astronomical unit. You would you start touching the sun. There's some point, which is physically impossible to get to. And 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 a row. It's almost 0. But there's some probability. But in the case of hand span, there's no way out here, you know, the probability of human being who happens to be a high school senior, having a one-mile length hand span, that's 0. So it's not going to be a perfect normal distribution of the outliers or as we get further and further away from the mean. But I think it'll be a pretty good, in our everyday world as good as we're going to get, approximation. The normal distribution is going to be a pretty good approximation for the distribution that we see. And I guess one thing that you know-- It's funny. This is high school seniors, when I did this, it was kind of from my point of view as a guy. And I would argue that high school seniors, guys, probably have larger hands than women. So 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 peak for women at seven inches. And then the distribution falls off like that. So it's also possible it could be bimodal. But in general, a normal distribution is 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, we have minimum wage laws whatnot. so I would guess that 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 down there at the low end of the scale. And then you have your different middle level managers and whatnot. And then you probably have this big gap. And then you probably have your true executives, maybe your CEO or whatnot. If this mean, right here, is maybe $40,000 a year and this is probably $80,000 where some of the mid-level managers lie. But this out here, this will probably be-- Actually if you were to draw it real, the way I've scaled it right now, this would be about $200,00, which is actually 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 chart. Let's say the CEO made $5 million in a year because he cashed in a bunch of options or something. So it would 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 it definitely wouldn't be a normal distribution. And it would be bimodal. You would have another peak over here for senior management. Well, if we were maybe in Europe, this would 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, a right skewed distribution. Since it has two humps right here, one there, and one there. We can also says it bimodal. I mean, it depends on what kind of company this is. But that would be my guess of a lot of large shipping company's salaries. Let's look at choice C or problem part C. 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, the gender gap has not been closed fully, and there is some discrepancy. So if I was just purely 50 CEOs of major companies, I would say it's probably close to a normal distribution. It's probably something like, once again, there's going to be some level below which no CEO is willing to work for, although you have 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 value, 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 humps here, which would be a bimodal distribution. So if you assume there's some gender gap, this is part C right here, then maybe there's one hump for women. And if you assume that women earn less than men, then another hump for men. And there are 25 of each so there wouldn't necessarily more men than women. And then it would skew all the way off to the right. And in fact, I think there would probably be chance that you have this other notion here where you have these super CEOs, or mega CEOs who make millions, while most CEOs probably just make, I'll put it 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 it could even be a trimodal distribution here. So that's choice C. And then so far, choice A looks like the best candidate for a pure, or the closest to being a normal distribution. Let's see what D is. The date of 100 pennies taken from a cash drawer in a convenience store. 100 pennies. So that's actually an interesting experiment. But I would guess, and once again, this is really a question where I get to express my feelings about these things. 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 they get pressed at the little tourist place into those little souvenir things. I'm not even sure if that's legal, if you could do that the money legally. So my guess is that if you were to plot it, you would have a ton of pennies that are, within the last few years, So the date of 100 pennies, not their age, so the dates, so if 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 a ton of 2009 pennies. And then it probably just goes down from there. And of course, you're not going to find pennies that are older than the United States or before they even started printing pennies. So it's obvious this tail isn't going to go to the left forever. But my guess is you're going to have a left skewed distribution. Where you have the bulk of the distribution on the right, but the tail goes off to the left. That's why it's called a left skewed distribution. Sometimes this is called a negatively skewed distribution. And similarly, this right skewed distribution, or this right you distribution, sometimes is called positively skewed. And if you have only one hump. You don't have a multimodal distribution like this, in 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 some place 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-- the other choices explain why you believe they would not follow-- or 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 a matter of opinion, at least in this question.