Don't trust online reviews!
A fifth are left by people who have NEVER tried the product
Newspaper headlines and the titles of articles in magazines are usually written by editors, rather than the authors of the entitled story. This is why we shouldn't blame writer Victoria Woollaston for the above headline to her story yesterday in England's Daily Mail Online newspaper. As an exercise in critical thinking, first read the article―see the Source, below; read the whole thing, it's short!―and see if you can see what's wrong with the headline above. Only then click on the link below to see if you're right.
Source: Victoria Woollaston, "Don't trust online reviews! A fifth are left by people who have NEVER tried the product", Daily Mail Online, 1/24/2014
In the Mail: Philosophy of Pseudoscience
I mentioned the new anthology Philosophy of Pseudoscience last October―see the Resource, below―and I will post a review of it in the near future.
Update (1/11/2017): Make that the not-so-near future.
Source: Massimo Pigliucci & Maarten Boudry (editors), Philosophy of Pseudoscience: Reconsidering the Demarcation Problem (The University of Chicago Press, 2013)
Resource: New Book: Philosophy of Pseudoscience, 10/14/2013
An Infinitely Paradoxical Proof
Consider the following infinite series.
1 + (-1) + 1 + (-1) + 1 + (-1) + …
What does it add up to? Notice that the series can be written in the following way:
1 - 1 + 1 - 1 + 1 - 1 + …
Furthermore, let's group the members of the series as follows:
(1 - 1) + (1 - 1) + (1 - 1) + …
If we carry out the subtractions in parentheses, we get:
0 + 0 + 0 + …
Clearly, then, the sum of this series is zero. However, going back to the original way of writing the series with negative numbers, there's another way of grouping its members:
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + …
If we carry out the additions in parentheses, we get:
1 + 0 + 0 + 0 + …
Clearly, then, the sum of this series is one. Therefore, 0 = 1.
What's wrong with this "proof"?
How to Misread a Poll
Here's Slate's David Weigel yesterday, in an article titled "Poll: Christie's GOP Approval Numbers Up Since Start of Scandal":
[Christie's]…chief political adviser…[said] that Christie was getting "positive, proactive" feedback from Republicans. You laugh…but then you look at the Monmouth poll, the only one taken in New Jersey since the scandal began."…Republicans are sticking by Christie, giving him an 89% approval rating which is in line with the 85% GOP support he received last month."…The Republican uptick is a better omen for Christie than anything else in the poll.
These claims should set off some warning bells for the following reasons:
- What's the margin of error (MoE) for these results? Weigel doesn't mention it. The alleged "uptick" in Christie's approval rating is a matter of four percentage points, but is that significant?
- This is not a national poll but one limited to the state of New Jersey. State polls often are based on smaller samples than national ones, but this means that the MoE of state polls is usually larger than that for national ones. Remember that the size of the MoE is inversely proportional to the sample size: the smaller the sample, the larger the MoE. The MoE for a typical national poll is plus or minus three percentage points, so it's likely that the MoE for a state poll will be plus or minus four points or more, in which case the result that Weigel trumpets would be within the MoE.
- As if that weren't enough, the result that Weigel points to is limited to Republicans. It's highly likely that this is a subgroup of the sample for the poll, which is a smaller group than the entire sample. Because of their smaller sizes, results for subsamples have higher MoEs than for the entire sample. Thus, it's likely that the unreported MoE for this result is even higher than plus or minus four percentage points.
- That this is probably correct is indicated by the statement from the polling report that Weigel obliviously quoted: "Republicans are sticking by Christie, giving him an 89% approval rating which is in line with the 85% GOP support he received last month." (Emphasis added.) What does it mean to say that the one result is "in line with" the other? It most likely means that the new result is within the MoE of the previous one, so that there is no detectable change in support for Christie among Republicans. In other words, there was no "uptick", but no downtick either.
So, even without checking the polling report itself to see what the actual MoE is, you can tell that Weigel is probably making a big deal about nothing. I'll leave it as an exercise for the reader to check that report―see Source 1, below. It's a worthwhile exercise in not trusting everything you read in the media, not even what you read in The Fallacy Files!
- "New Jersey Riveted to Bridgegate", Monmouth University/Asbury Park Press Poll, 1/13/2014 (PDF). Update (3/16/2016): This source seems to be no longer available, so you'll just have to take my word for it after all.
- David Weigel, "Poll: Christie's GOP Approval Numbers Up Since Start of Scandal", Slate, 1/14/2014
Resource: How to Read a Poll
A Doublespeak Dictionary Puzzle
Can you figure out what an aerodynamic personnel decelerator is?
Source: Hugh Rawson, Dictionary of Euphemisms and Other Doubletalk, Revised Edition (1995)
This looks like a job for Fallacy Man!
Actually, that's not the real Fallacy Man in the Source, below. That's obviously Zorro with an "F" on the front of his shirt for some reason. Also, you can tell he's not the real thing because he calls the appeal to nature "the naturalistic fallacy", among other mistakes. It's the Masked Man Fallacy! However, it's funny and the Fallacy Fallacy at the end is a nice touch. Check it out.
Source: "The Adventures of Fallacy Man", Existential Comics
Via: Corey Mohler, "Have You Met Fallacy Man? Here’s How to Defeat Him.", Slate, 1/6/2014
Why the headline is wrong: The article is a report on a survey done by the YouGov organization―see the Source, below. The survey asked various questions about whether those surveyed pay attention to online reviews of products, and whether they ever reviewed products themselves. Based on this kind of information, how could one determine that what the headline claims is true? Instead, here's the headline of YouGov's own article reporting the survey results:
21% of Americans who have left reviews, reviewed products without buying or trying them
So, the "fifth" in question is not the proportion of online reviews left by people who have not tried the product, but the proportion of people who have reviewed products who have left such a review. These are different things, and there's no good reason to think that these proportions should be the same. For instance, it's possible that a fifth of reviewers at least once review a product that they haven't tried, but that most of the reviews they leave are of products they have used. Thus, the proportion of such fraudulent reviews might be much smaller than a fifth of all reviews. In contrast, it's also possible that a small proportion of dishonest reviewers leave a much larger proportion of all reviews than a fifth, assuming that they work much harder to do so than honest reviewers.
So, the survey result concerns the proportion of reviewers who have left dishonest reviews, which the headline reports as the proportion of reviews that are dishonest. Clearly, the survey result doesn't support the alarming headline warning the reader not to trust online reviews.
Source: Zaraida Diaz, "21% of Americans who have left reviews, reviewed products without buying or trying them", YouGov, 1/22/2014
The series of additions―let's call it "S", for short―in the "proof" is infinite, so adding it up would take an infinite amount of time. More importantly, the usual laws of arithmetic, such as the associative law used in the "proof", are defined only for finite sums. Now, some infinite series converge on a certain value, for instance:
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …
If you start to carry out the additions in this series, you get the following series of sums:
3/4, 7/8, 15/16, 31/32, …
The series approaches closer and closer to the number 1 as you carry out each addition, so it's reasonable to consider this and other such infinite series―they are called "convergent"―to have "sums", namely, the numbers that they converge upon. Moreover, such "sums" have uses in physics, and even in philosophy for explaining Zeno's paradoxes. However, it should be kept distinctly in mind that this is an extension or generalization of the sense of "sum" that applies to finite additions, and it doesn't immediately follow that the laws of arithmetic apply even to convergent series.
However, S is not a convergent series, rather it is "divergent": if you do partial sums you will see that they switch back and forth between 0 and 1, but S never appears to converge on either―such series are said to "oscillate". Now, while the paradoxical "proof" does not prove that 0=1, it does show that the associative law for addition does not apply to series such as S.
Is there any sense in which you can "sum" a divergent series? The answer is "yes", keeping in mind that we're getting farther and farther away from familiar addition. For instance, in the case of series such as S which oscillate between two values, we can take the "sum" of the series to be the average of the two values, so that the sum of S is 1/2 rather than 0 or 1.
However, careful mathematicians always refer to such "sums" as "Cesàro sums" or "Ramanujan sums" or whatever, since there are different methods. There's nothing mathematically wrong with such methods, but it should be kept in mind that "summing" an infinite series in these senses is not the same thing as summing a finite one, and you can't just use the usual laws of arithmetic without further ado.
- Bryan Bunch, Mathematical Fallacies and Paradoxes (1997), pp. 51-69.
- James & James, Mathematics Dictionary (Third, multilingual, edition, 1968), see: "divergent series", "sum of an infinite series" & "summable divergent series".
- Phil Plait, "When Infinity Is Actually a Small, Negative Fraction", 1/17/2014. In this article, Phil "Bad Astronomy" Plait falls for some bad math similar to that discussed above. Understanding the paradox, above, can help in understanding what's wrong with the arguments given in the video Plait discusses. However, see the next Source, which is Plait's correction, for the details.
- Phil Plait, "Follow-up: The Infinite Series and the Mind-Blowing Result", 1/18/2014. Excellent correction of the original, misleading article. Slate, like so many other publications, is hesitant to correct serious errors, though it will correct trivial ones. Plait shows how to set the record straight, with some good advice on how to avoid making similar errors.
Answer to a Doublespeak Dictionary Puzzle: This is military jargon for a parachute.