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Author: Jonathan Osters

Identify results of a given situation as Simpson's paradox.

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Source: Tables and Graphs created by Author, Benford's Law, Public Domain: @2:26

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This tutorial is going to speak to you about statistical paradoxes. Now, paradoxes are apparent contradictions in what you see versus what you expected to see. Specifically, the one that we're going to talk about in this tutorial is called Benford's Law.

So statistics allows us to draw conclusions about things that we see. But sometimes, the phenomena that we see are counter to what we thought would happen. So these seeming contradictions are called paradoxes. And we can understand them better. And when we understand them better, we can improve as statistical thinkers.

So suppose that you were going to create a phony checking account. And you wanted to set it up so that you could steal some money from people. I don't know. Would it matter what number you started with? Think about that for a second.

Probably what you thought was no. It really wouldn't matter. All the numbers one through nine are equally likely to be selected for the first number. So if the account number is going to be randomly selected, it doesn't really matter. That's your intuition.

What's really the case, though, is that our intuition that all the numbers one through nine are equally likely to be selected for the first number is actually wrong. What's really the case is they're most likely to start with one. Checking account numbers, they're most likely to start-- pretty much any number you see-- is most likely to start with a one in published data.

And this is the idea of Benford's Law. Benford's Law says that the first number of-- what he said was financial accounts-- but most any real life data follow a pattern with the number one being the most likely to being the next most likely, specifically in this order. So only about 10% of account numbers will start with a four, whereas about 30% will start with a one.

And apparently, what people do who try and steal identities is they'll actually load it up on fours, fives, and sixes, because they think those are the middle. But really, it's the number one that's the most likely.

And Benford took a look at lots of these. The area of rivers, populations of different countries. Just regular old constants that are used in physics and things like that. Numbers that happen to appear in the newspaper. The specific heat values of different things.

And just took a look at all sorts of these different values and saw that almost across the board, one is the most likely, two is the next most likely lead number. And nine is the least likely lead number. It wasn't here, but in the vast majority of cases it is.

So once you wrap your mind around that, you might think to yourself, oh, all right, well if the lead number follows that law, then the second digit must also follow that law. But again, your intuition leads you astray. The first digit follows Benford's Law.

The second digit is, if you take a look, approximately equally likely to be any of the numbers zero through nine. Apparently, a little bit favoring the lower numbers, but all of these are about 10%. So the second digit, it's about equal frequency. It's approximately a uniform distribution here, whereas this Benford's Law distribution was very, very heavily skewed if you remember that histogram.

Now, the reason that you see something like this has to do with exponential growth, where if you take a look, these are the number 100, 200 to 300, 300 to 400. These are, you can see, equally spaced.

But if you take a look at the numbers that create them on the curve, there's a lot more numbers here on the x-axis that create a value between 100 and 200 versus create a number between 200 and 300. And you can see that that amount diminishes as you move along to the right.

So these are the numbers that create a one. That's about 30% of these numbers that are highlighted. About less than that create a two. Less even still create a three. And less create a four, all the way down to nine. These are the ones that create numbers between 900 and 1,000.

And so to recap, a paradox is a seeming contradiction between what you think should happen verses what's actually happening. The First Digit Law, which is Benford's Law, is one of these paradoxes. We thought that we would find a uniform distribution among first digits of certain numbers that we see. But apparently, one is a lot more common to lead than anything else.

Not all numbers occur with equal frequency as the lead digit. And once we understand paradoxes more, as we'll see in other tutorials that will go into more paradoxes in greater detail, we'll hone our statistical thinking and get more and more precise. So the terms we used were paradox and Benford's Law in specific. Good luck, and we'll see you next time.

Simpson's Paradox


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This tutorial's going to teach you about a specific statistical paradox called Simpson's paradox. And we'll start with an example, very famous example. In 1973, UC Berkeley had a sex discrimination lawsuit filed against them, and this is why. They said that they were favoring men over women substantially in the admissions process for their grad schools.

It looks like 977 men applied to two of the departments, and 492 were accepted. That's a little over half. Versus of the 400 women who applied, well under half, or 148, were accepted. In fact, the proportions are 50.3% versus 37%. That is a huge difference, and that's why the lawsuit was filed.

So in an effort to see exactly where the women were being discriminated against, the lawyers looked into the admissions by department. And you would expect there would be some large discrepancy in one or both departments. So they looked at-- I don't know that these are, in fact, the engineering and English departments. The case is real, but I just made these names up.

So look at the engineering department, and you can see that for the men, about 63% of men were accepted to the engineering department, versus 17 of the 25, which is 68% for women. Women were accepted at higher rates to the engineering department. All right, in that case, you would assume that the discrepancy, then, occurs in the English department.

Well, when you look at the English department, women were accepted at higher rates to the English department as well-- 34.9% versus 33.3%. So women were accepted at higher rates to the engineering department and the English department, but way lower overall. And this is what Simpson's paradox is. It's a relationship that's present in groups but reversed when the groups are combined.

The reasoning behind it is that if you take a look at how the men's application rates were distributed, their 63% was weighted for a lot more in into the weighted average of admissions rates, versus the 68% for the women. Look, only 25% of women applied-- or not 25%. 25 women applied to the engineering department of the 400. That's not very many.

And so that 68%, even though it's a high percentage, doesn't count nearly as much in the weighted average as the 34.9% does. So this 63% is weighted heavily for the men, versus the 68% is weighted hardly at all for the women. And that's why you see that reversal of association.

And so to recap, Simpson's paradox is an association that the data show when you group them in specific ways, and the association gets reversed when the groups are combined. And you'll see several paradoxes like this. Simpson's paradox is one of them. As we learn more about paradoxes, we can hone our statistical thinking and become better statistical thinkers. Good luck, and we'll see you next time.

Terms to Know
Benford's Law

A law that shows that most of the numbers that are published, regardless of topic, begin with smaller numbers, and very few of them lead with larger numbers. The most common first digit is 1, the least common is 9.


An apparent contradiction between what our intuition tells us, and what is true in reality.

Simpson's Paradox

When two sets of data are subdivided, the means for the first data set can be consistently higher than the second, but when looked at as a whole, the mean of the second set is higher than the first.