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Is this the Worst Pie Chart Ever?
December 15th, 2017 (Permalink)

Charts & Graphs:
A Half-Baked Pie Chart

Before you read any further, take a look at the pie chart to the right. What do you make of it?

I came across this chart in a column by mathematician Keith Devlin1, who focuses on the fact that the chart seems to show a pie cut into two pieces, one representing the richest 1% and the other the remaining 99%. However, the "1%" piece looks to be about 23% or 24% of the pie, leaving the "99%" piece representing the remaining 76% or 77%.

First, here's a brief explanation of what the chart is actually trying to convey―if you need more, see Devlin's discussion. A careful reading of the title of the chart reveals that the pieces of the pie actually represent the share of taxes paid by each income group, not the income groups themselves. In other words, the richest 1% pay about 24% of taxes, whereas the remaining 99% of us pay only the remaining 76%.

This combination of two sets of percentages in one pie chart―percent of the population in terms of wealth, and percent of federal tax paid―is confusing, but I doubt that anyone would be seriously fooled by it. Since the percentages and the sizes of the slices are so at odds, one is led to read the title of the chart to resolve the confusion. I want to focus, instead, on a couple of other problems that Devlin does not comment on, one of which I think is more seriously deceptive.

One problem with this chart that Devlin does not discuss is that a graph of any kind is unnecessary to convey this information. With only two income groups being compared, it is easier and clearer to do so in words. As it is, the chart just confuses matters. One of the earliest questions you should ask yourself before making a graph is: Can this data be more easily understood in a graph than in words? If not, don't do it. Charts shouldn't just be pretty pictures to adorn your words, and they especially shouldn't be misleading pictures unless your goal is to mislead people.

The pie chart was put out by a group called "Tax March"2, which organized a number of marches earlier this year to protest President Trump's refusal to make public all of his tax returns. In other words, it's an anti-Trump political pressure group. Now, this doesn't necessarily mean that the pie chart was intended to mislead people, even if it had that effect, but it does mean that it deserves no benefit of the doubt. This is a group with an agenda, and anything they put out should be treated with skepticism. As Ronald Reagan should have said: "Don't trust; verify!" Is this the Worst Pie Chart Ever?

The original graph was an animated GIF that if clicked on would produce the following additional chart3. Here, you can see the point of the chart: the piece of the pie representing the tax share of the wealthiest 1% shrinks from about 24% to approximately 20%, with the remaining piece increasing a proportional amount. If you don't get the point, the second chart provides the helpful pointers: "The rich pay less" and "We pay more".

But do we non-rich folks really pay more? Now, this is not a fact check, it's a logic check, so I'm just going to assume that the percentages shown by both charts are correct.

What worries me is the words: "We pay more": "more" what? This a dangling comparative, that is, a comparison lacking one of the items compared. "We pay more" could mean that we, the 99%, pay more taxes than we used to. In fact, that is what I expect most people who see the chart will believe. However, what the chart shows is that we, the 99%, will pay a larger share of the total tax burden, which is not the same thing as paying more taxes.

If this point isn't sufficiently clear, think about it this way: Suppose that the entire amount of federal tax paid is a billion dollars, and that the top 1% in income pay 240 million of that, which is 24% of the total. That means that the remaining 99% of taxpayers would pay 760 million, or 76%.

Suppose, further, that an across-the-board tax cut reduced the total amount of tax paid to 900 million, of which 180 million―20% of the total―is paid by the richest 1%, and 720 million comes from the rest of us―80%. In this scenario, everyone is paying less tax, but the share paid by 99% has increased. For all that we can tell from the chart, this is what will happen under the proposed tax plan.

How many people who look at this chart will come away thinking that the rich are getting a tax cut, but the rest of us will actually end up with a tax raise? I expect that it's probably less than 99%, but much greater than 1%.


  1. Keith Devlin, "Clash of representations", Devlin's Angle, 12/14/2017.
  2. See: Matt Stevens, "The Tax March Explained: Protesters Hope to Pressure Trump Into Releasing Returns", The New York Times, 4/15/2017. I'm pleased that they didn't decide to call themselves "the Million Person Plus One Giant Inflatable Chicken March", or the like; see: The Million "Million X March" March, 6/18/2014.
  3. I was unable to find the original of this chart, so I'm copying only the unclickable version supplied in Devlin's column.

December 2nd, 2017 (Permalink)

New Book: Dollars and Sense

Thankfully, I have no complaints about the title of Dan Ariely's new book with Jeff Kreisler. But, what's up with this subtitle: "How We Misthink Money and How to Spend Smarter"? How can we "misthink" money if we can't think it? I think I understand what the subtitle is getting at: this is a book about the mistakes we tend to make in thinking about money, and how to avoid them.

Ariely is a psychologist and author of the previous book Predictably Irrational, which I've read and referred to in passing here. It's been almost ten years since I read that book, so unfortunately I don't remember much about it. In the intervening years, Ariely has written some other books that I haven't read, including The Upside of Irrationality―whose title I was put off by, as I don't think irrationality has an "upside". I am interested in reading the new book because I could benefit from learning how to avoid financial mistakes. Couldn't we all?

December 1st, 2017 (Permalink)

What's New?

This weblog has been rather quiet for the last month or two because I've been engaged in the long-delayed project of revising all of the fallacy entries. As part of that project, I've added a new fallacy: The Fallacy of the Heap! More accurately, it's sort-of new, because this is the same fallacy of vagueness that used to be listed under the name "Slippery Slope", along with the causal fallacy of the same name. In other words, it's a new entry for an old fallacy under a new name.

I'm in the process of splitting the old Slippery Slope entry into two distinct entries, one for the fallacy of vagueness, and the other for the causal fallacy. The first part of that task is now done. As mentioned in the new entry, the Fallacy of the Heap is sometimes referred to as "the slippery slope fallacy". However, "slippery slope" is most commonly used to refer to the causal mistake. Though, as explained in the new entry, the vagueness and causal fallacies are sometimes related, they are distinct fallacies and for that reason should have distinct entries and distinct names. There's reason to believe that some readers were confused by two different fallacies being discussed in the same entry under the same name, so this change should clear that up. For the time being, the entry for Slippery Slope is still the same old entry, but will soon be revised to cover only the causal fallacy.

New Fallacy: The Fallacy of the Heap

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November 24th, 2017 (Permalink)

New Book: How to Think

Not to toot my own horn, but I think I already know how to think, though I'm sure there's always room for improvement. That's what I'd like to get out of Alan Jacobs' new book, so I think a better title for it would be: "How to Think Better".

In these "New Book" posts, I'm always complaining about the titles and subtitles of the books, despite the fact that I know that they may have been forced on the author by the publisher. However, I haven't read the books yet, so there's not much else I can say about them.

So, to continue my carping, the subtitle of the U.S. edition is: "A Survival Guide for a World at Odds". I'm not sure what that means. The U.K. edition has the same title but a different subtitle: "A Guide for the Perplexed"1. That's better, but perplexed about what? I'm perplexed by quantum mechanics; is Jacobs going to explain that to me? I'm often perplexed by the titles and subtitles given to books. Perhaps Jacobs can begin by explaining his, and I'm completely willing to hear that they weren't his idea.

Judging from what I've been able to read about the book so far, it appears that Jacobs is primarily concerned about political, ethical, and perhaps religious thinking, as opposed to thinking about subatomic physics. More specifically, he seems especially worried about controversial issues about which people disagree, often contentiously. For instance, Jacobs writes: "I have admitted that the chief impetus of this book was the ever-increasing hostility and (often malicious) misunderstanding of one another that became one of the chief themes of the 2016 Presidential election here in the U.S. and of the debate over the Brexit referendum in the U.K."2 This is a limited type of thinking, though no doubt an important one.

Another thing that perplexes me is that Jacobs refers to "the error of believing that we can think for ourselves"1. I was not aware that that is an error. This suggests to me that the book's title ought to be: "How to Not Think"; subtitle: "Let Someone Else Do it!" However, even if you don't think for yourself, someone else is going to have to do your thinking for you3, and that person will have to think for him or herself4.

I hope this isn't a case of making an outrageously false claim just to get people's attention, then rapidly retreating when challenged5. I'll be very disappointed if Jacobs isn't at least serious, but I suspect that his notion of what "thinking for yourself" means must be different from my own. However, I'll probably have to read the actual book to find out what he does mean.


  1. Alan Jacobs: "Welcome!", How to Think, Accessed: 11/24/2017
  2. Alan Jacobs: "Can politicians think?", How to Think, 10/19/2017
  3. Perhaps in a few decades we'll be able to let computers think for us, but they're not there yet.
  4. Unless someone else does that person's thinking for him or her, but this regress can't go on forever: at some point, some individual person is going to have to think.
  5. À la Stanley Fish.

November 23rd, 2017 (Permalink)

A Thanksgiving Recipe Puzzle

This Thanksgiving I'm planning to make my famous turkey gravy. Thankfully, I have a full 12-ounce container of turkey stock, which is hard to come by. For the recipe, I need exactly 6 ounces of stock. Unfortunately, I have only two measuring cups, one of which holds exactly 5 ounces and the other 7 ounces. What's more, the cups are so old that all of their markings have worn off. So, it appears that I can only measure out exactly 5 ounces or 7 ounces of stock.

Now, of course, I could just pour the stock into the 7 ounce cup and estimate when it is about an ounce less than full. Or, I could double all of the ingredients and make twice as much gravy. Or, I could run out to a store and buy a new and improved measuring cup. But is there any way, using only the two measuring cups I have, to measure out exactly 6 ounces of stock?

If you think you know the answer, click on the solution, below.


November 23rd, 2017 (Permalink)

A Traditional Thanksgiving Thank You!

On this day of thanksgiving, thanks to all of those who have supported The Fallacy Files since last year, whether by clicking on ads, donating directly via the PayPal button on your right, or just reading it! Also, The Fallacy Files is an Amazon Associate and, with the holidays approaching, please consider doing any shopping at Amazon by way of one of the links from this site. It won't cost a penny extra and will help keep The Fallacy Files busting fallacies for another year. Thanks!

Weird Science-Fantasy
November 6th, 2017 (Permalink)

Java Jive: A Refill

A recent headline reads:

Study: Coffee may help kidney disease patients avoid early death1

In a previous entry2, I discussed how three classes of words and phrases may help you understand science and health reporting. The above headline contains two of them: "may" and "help". "May" is one of a class of related words that indicates that the results of the study are less than definitive. Coffee may help; but, then again, it may not. No one now would say that cigarette smoking may cause lung cancer because this would suggest a reasonable doubt about it. "Help" is a word from a different class that indicates that the supposed healthful effect of coffee is weak.

The article beneath the headline begins:

Can caffeine help people with chronic kidney disease [CKD] live longer? That's the suggestion of a new study that found that among more than 2,300 Americans with chronic kidney disease, those who drank the most caffeinated drinks reduced their risk of premature death by 24 percent.1

"Suggestion" is another word, like "may", that indicates that the results of the study are less than conclusive. Also, we now learn that it is "caffeinated drinks", rather than coffee, that supposedly "reduced" the risk of death. So, presumably, you could get the same effect by drinking tea, or one of those highly-caffeinated "energy" drinks, instead of coffee.

When reading news reports of scientific studies, I strongly suggest reading the whole thing or at least skipping down to the end and reading the last few paragraphs: it's often there that we get the bad news. In contrast, in this article we get the bad news as early as the fourth paragraph:

"…[O]ur observational study cannot prove that caffeine reduces the risk of death, but only suggests the possibility of such a protective effect," [lead researcher Dr. Miguel Bigotte Vieira] said.1

This, despite the fact that a couple of paragraphs previously we were told that "those who drank the most caffeinated drinks reduced their risk " of death. Now, the lead researcher tells us that the study only "suggests" such a "possibility". It's a good thing that he didn't say that it may suggest such a possibility!

However, the study found a "dose-dependent inverse association"3 between caffeine consumption and death, which does indeed suggest a causal relationship between the two.4 However, since it was only an observational study, it cannot establish such a relationship, which is why Vieira goes on to note:

The findings also need to be replicated in a trial that compares caffeine consumption with no caffeine consumption before patients are counseled to drink more coffee or other caffeinated drinks….1

I think this is exactly right, but it comes close to contradicting what Vieira said in a press release:

These results suggest that advising patients with CKD to drink more caffeine may reduce their mortality. This would represent a simple, clinically beneficial, and inexpensive option, though this benefit should ideally be confirmed in a randomized clinical trial.3

This suggests that, while it would be ideal to have a randomized clinical trial, it's not too soon for CKD patients to increase their caffeine intake. In contrast, Dr. Leslie Spry, a spokesman for the National Kidney Foundation, is quoted at the end of the original article:

This is yet another observational study, Spry noted, where only an association was found, not cause and effect. Given the relatively small size of the study, and the small reduction in death risk, Spry said he's not willing to tell kidney patients that the more caffeine they drink, the longer they'll live. "I would rather say that compared to little or no caffeine intake, those people with the highest intake of caffeine as estimated by dietary recall, may have a lower mortality, but the reason for this lower mortality is not proven by this association research," he said.1

This is actually a better-than-average news report on a new study, for it includes something that most such articles leave out: a second opinion from an independent expert. Moreover, Spry's cautions are not included in the original press release, so this is not just a rewritten press handout, which is all too common nowadays.


  1. Steven Reinberg, "Study: Coffee may help kidney disease patients avoid early death", UPI, 11/4/2017
  2. See: Cuckoo for Cocoa Puffery, 6/11/2017
  3. American Society of Nephrology, "Caffeine consumption may help kidney disease patients live longer", EurekAlert!, 11/3/2017. This is the press release that probably started it all. A "dose-dependent inverse association" between caffeine intake and death means that the more caffeine the study's subjects consumed, the longer they lived.
  4. This is an application of John Stuart Mill's "method of concomitant variation", see: Philosophy of Scientific Method (Hafner, 1963), Book 3, Chapter 8, Section 6.

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