To be fair, it appears they fixed the math later, or at least roughly. (2 out of every 1M) x (the US population) [they cite ~311M]. That gives 622. Not far off the 620 they report in your quote.

This is like having a weighted coin that comes up heads with probability 2⋅10⁻⁶, flipping it 311 million times, and seeing 0 heads. That's astronomically unlikely.

To see this, observe that the number of heads follows a binomial distribution with n = 311 million and p = 2⋅10⁻⁶. This can be well approximated¹ by a normal distribution with mean μ = np = 622 and standard deviation σ = Sqrt[np(1 - p)] = 25.

99.7% of the time², when you sample from this distribution, the sampled value will be within 3 standard deviations of the mean, i.e., between μ - 3σ = 547 and μ + 3σ = 697. Results further from the mean are more unlikely. For example, seeing a value more than 7 standard deviations from the mean (i.e., less than 447 or more than 797) is about a 1 in 2 trillion event³. Since 0 is about 25 standard deviations from the mean, the probability of seeing 0 heads is on the order of 10⁻¹³⁸.

[1] https://math.stackexchange.com/questions/2021801/conditions-...

[2] https://en.wikipedia.org/wiki/68–95–99.7_rule

[3] https://www.johndcook.com/blog/table-of-normal-tail-probabil...

> This is like having a weighted coin

Tangent: there is no such thing. You can weight a die, you cannot weight a coin.

Intuitively this should make sense because even if you made one side of the coin from lead and the other from balsa wood, all you are doing is changing the center of gravity of the coin. The coin spins about its center of gravity, not the geometric center of the coin, so this makes no difference.

https://www.stat.berkeley.edu/~nolan/Papers/dice.pdf

Possibly, though if we want to be pedantic "weighting" has shifted from merely meaning "increased in weight" to meaning "biased through manipulation" - colloquially you can describe a situation as "weighted against someone" if you think that person may be unfairly disadvantaged - like in a multi-talent contest a continued draw of activities that favor one party using a hidden method that is suspected to be directed to the end of favoring that party.

You absolutely can manufacture unfair coins either using a two surfaced approach (like a weak magnetic field acting on a magnetized coin) but even a one surfaced approach is possible if you make use of carvings on the surface of the coin to get a favorable result from air resistence - lastly you can even achieve it through density, if one side of a coin is significantly more dense then the other then it will tend to land face down - you can play with this a bit by trying to flip a weighted cylinder and observing the landing pattern - I might suggest taking a coin roll (like you get in a bank) and gluing some coins into one end of it - then try and flip it in the air so it lands coin-side up.

It is, however, very hard to bias a coin significantly without skewing the dimensions or having clear alterations visible on the coin.

To add to your point: I ran some experiments a while back with curved coins. The coins have to be absurdly shaped before the shape affects the outcomes:

https://izbicki.me/blog/how-to-create-an-unfair-coin-and-pro...

Falling below the noise floor is different from proving no correlation. Negative results for a given experiment simply mean it’s below an the experiments ability to detect.

Aka, you need to pick a threshold for bias that’s interesting before designing the experiment. A casino might care a great deal about say 0.5% bias which would take a lot of trials to detect.

I respect the level of free time taken to run and publish this experiment!

Interesting results, I thought they'd be more skewed.

The parent comment that a coin of lead and balsa wouldn't be biased, but to clarify, the linked research paper states it can't be biased unless allowed to bounce/spin. Maybe your weighted cylinder flipping is not biased if you throw+grab instead of letting it land?

> Grandparent understood it as well.

Considering you were the grandparent, I guess that should be somewhat reassuring.

FWIW, I interpreted your use as rigid in the same manner as the GP (of this post).

As far as I’m concerned a electronic bistable generator that is biased through manipulation is effectively a “weighted coin”.. those exist.. and are counter to your claim unless you restrict “weighted coin” to the completely literal.

> Considering you were the grandparent

God no, GP here means the contextual grandparent of the topic, as it always does. GP means the person I responded to. Why in the world would I respond to myself just because "GP"?

If you haven't read the rules, you need to be giving the most gracious reading of the comment chain here. Pretty obvious what the implication was. Try reading again.

> As far as I’m concerned a electronic bistable generator that is biased through manipulation is effectively a “weighted coin”

Oh cool you're trying to impose a change of topic from the ability to literally bias a coin when a normal person flips it randomly, into telling me to defeat... all attempts of people manipulating all RNGs ever?

Super weird angle for a discussion about flipping coins. Not really interesting.

That paper is not good support for your pedantic argument. (More pedantically, any coin without uniform density is "weighted" by definition, regardless of toss bias)

In fact, most methods of coin toss will be influenced by an unbalanced coin in some way. The paper only demonstrates that if you flip a coin with a certain precisely specified method (and catch it midair) - can you be reasonably assured a weighted coin will be unbiased.

See their own referenced book Jaynes, 1996 pp 1003-1007, which I think gives a much clearer explanation of the possibilities.

Note that the NFL for instance does not catch the coin, so there's at least a real world where a coin could be biased.

The important part of that paper is this:

"Examples of how others have flipped and tossed coins show the students how essential it is to carefully describe the experimental process." not just the one detail about angular momentum and CoG.

That paper redefined what it meant to toss a coin making their concussion meaningless in practice. For a more in dept real world analysis.

http://statweb.stanford.edu/~susan/papers/headswithJ.pdf

PS: Of note they where detecting bias in the range of 1% that’s difficult to detect by hand.

You're presenting it like your article substantially rebuts the point. When the absolute best bias you can do is 1% above random that's pretty much "fair" as far as practical usage.

It's not even clear that's a real bias or just random noise, or an anomaly of their testing setup.

>Tangent: there is no such thing. You can weight a die, you cannot weight a coin.

Yes but you don't need such a coin. You can use a perfect coin and consider flipping it many times.

Example : https://www.youtube.com/watch?v=Vo9Esp1yaC8

There are examples in that paper of scenarios of effectively influenced coin tosses...

I feel like it's not super difficult if you're catching the coin, if you're able to position the coin in the same place and apply the same force consistently you'll largely get the same height and speed of rotation. When I was a bored kid I was able to get a pretty consistent coin toss where it would land on the opposite of the side at the beginning. [0]

[0] Of course I didn't do any statistical analysis or a huge number of trials for this to really tell if I was able to do it but it felt pretty consistent.

It could be made to make a difference in air. Imagine a styrofoam coin with one side padded with an extra layer of lead.

(For best results, make the styrofoam thick relatively to the lead, and/or attach extra aerodynamic surfaces :).)

A magnetic coin, might be the "trick coin" that was being posited.

I wonder if you could build a (admittedly thick) coin with some sensing/actuation going on in order to make it land on an arbitrary side.

The same is true for a die :-) It could make a difference for the coin if you “rolled” it instead of catching it in the air.

That's only true if you test all 311 mio for both. While I think that blindness is self-analyzing, that's not true for schizophrenia.

So how many blind people were evaluated for schizophrenia?

This might actually be a lower percentage than for normal-seeing persons. People are less experienced with the behavior of blind people, so it's harder for surrounding people (and probable even for the blind themselves) to recognize it and push people to go get diagnosed.

> Since 0 is about 25 standard deviations from the mean, the probability of seeing 0 heads is on the order of 10⁻¹³⁸.

Note: the above figure comes from the normal approximation to the binomial, which loses accuracy towards the tails. The exact probability of seeing 0 heads is (1 - p)^n = (1 - 2⋅10⁻⁶)^(311⋅10⁶) ≈ 10⁻²⁷⁰ [1].

[1] https://www.wolframalpha.com/input/?i=%281+-+2+*+10%5E-6%29%...

Another way to get there is that, given N shots at a 1/N event, you expect 0 hits 36% of the time (1/e).

You can divide the population up into into 600 groups with 500000 people in them, and each of those has a 36% chance of not ever hitting.

So seeing no cases is like flipping a 36% coin 600 times and hitting 600 times.

Your 'approximation' is doing a lot of the work here.. Since you have a binomial distribution why not just use it directly?

(1 - 2e-6)^(3e6) ≈ 0.002

So about 0.2%. Still highly unlikely but orders of magnitude more likely than what your normal distribution-detour gave.

> (1 - 2e-6)^(3e6) ≈ 0.002

Using the binomial directly is a good way to get the probability of 0 heads. Note, though, that the U.S. population is in the neighborhood of 300 million, not 3 million (as you seem to have used).

(1 - 2⋅10⁻⁶)^(3⋅10⁸) ≈ 10⁻²⁶¹

https://www.wolframalpha.com/input/?i=%281+-+2+*+10%5E-6%29%...

Except we don’t have perfect information. Schizophrenia is misdiagnosed fairly frequently. I couldn’t find stats on undiagnosed schizophrenia, or schizophrenia diagnosed without visual hallucinations (which is probably the more relevant metric)

Wowsers! That's quite the elementary math mistake in a journal article. I'm tempted to track down the authors and point this out to them.

Or the journal editor, you definitely should. This is the kind of thing that can easily get cited in future work on the matter.

I sent an email to editorial.office@frontiersin.org. If they reply, I will post their response here.

Subject: elementary math error in journal article

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3615184/ was linked to and discussed today from Hacker News (a popular discussion board for software engineers) here https://news.ycombinator.com/item?id=22307445.

In the ensuing discussion a poster noticed that the journal article contains an elementary math error:

"if schizophrenia occurs at a rate of 0.72% in the population (McGrath et al., 2008) and congenital blindness occurs at an estimated rate of 0.03% in people born in the 1970s and 1980s (based on Robinson et al., 1987), then the joint probability of a person having both conditions, if the two are independent, would be 0.02% or 2 out of every 10,000."

0.72% * 0.03% is actually .0002% or 2 out of every 1,000,000.

Despite the article being several years old, I thought you might want to know.

-harryh

Great job; we need more good citizens like you!

I was unaware that they do at least have the implied 620 figure for the entire US correct, so there is hope that this is just a localised typo.

I can't get over how stunningly-similar this is to the infamous Verizonmath!

Well it's been a day and, sadly, I've received no response. The error also still appears on the publication's own website:

https://www.frontiersin.org/articles/10.3389/fpsyg.2013.0015...

Thank you for your effort. For someone who is in academia and a firm believer in the advancement of knowledge via peer-review, these things matter. Cheers!

it is probably not a math mistake since, when they applied the probability the 620 people is correct. Might be a typesetting (formatting) mistake. The 0.0002 is already a percent, but someone overlooked that and turned it again into percent.

Still it should definitely be corrected as the 2 out of 10,000 is a value that sticks in the mind.

I once helped setting an article in tex (originally written in word) and made the mistake of not noticing that rich text 10^20 got converted to 1020 in plain text. We only noticed post print but nobody else seem to have stumbled on it even though it was at the core of the articles thesis.

They also used the 2 in 10,000 figure when picking out the comparison syndromes. I believe Retts is ~1 in 10,000. It's definitely not 1 in a million.

My guess is the editor somehow messed up the per thousand sign ‱

https://en.wikipedia.org/wiki/Basis_point

edit: My bad. Correct character, incorrect description. It is indeed "ten thousand".

Somewhat ironically, I think you're off by one character here.

Unicode U+2031 ‱ is actually the per ten thousand sign (check its character name).

Unicode U+2030 ‰ is the per thousand (or "per mille") sign.

The one thing worse than off-by-one errors are off-by-zero errors.

Weirdly, as ChrisRR points out, they get the total number of predicted blind + scizophrenic people in the US correct. So it almost looks like more of a typo than a math error.

But still.

It makes sense, because as a scientist you often only write it as e.g. 0.2% for the article, but 0.002 in your actual calculation.

Percentage calculations being off by 2 zeros I would say is one of the most common mistakes I saw in, e.g. statistics exams.

So even then, 600 people in the US should be both born blind and scizophrenic

Obviously there's no guarantee, and there's also the fact that we may not have every case documented, but if it is true that we have 0 case documented, that's pretty far from 600. Even if it's not "impossible", then at the very least it's less likely.

It's a simple binomial probability calculation. The probability of one or more blind schizophrenic people born in the US, assuming they're independent variables, is 1-0,999998^330000000. I don't have a calculator on hand that can calculate that, but it's more than 1 - 10^-87. So the odds that there is no link between the two is close to the odds of guessing a specific bitcoin wallet's key in one try.

You can't just ignore the percentage signs.

0.72% expressed as a decimal is 0.0072. 0.03% expressed as a decimal is 0.0003. 0.0072 x 0.0003 = 0.000002. Expressed back as a percentage, that's 0.0002%.

> it's 0.0072 x 0.0003, which is 0.0002, but that's 0.02%

No. 0.0072 x 0.0003 = 0.00000216‬ ~= 0.000002 = 0.0002%

You're right. Not sure how I fell into that trap so easily.

I wonder if it's better to treat percentage as a unit or variable. So a-percent x b-percent = c(percent-squared), or c divided by 100 twice.

You could also replace % with 10⁻² and use scientific notation:

      0.72% ⋅ 0.03%
    = 0.72 ⋅ 10⁻² ⋅ 0.03 ⋅ 10⁻²
    = 7.2 ⋅ 10⁻³ ⋅ 3 ⋅ 10⁻⁴
    = 7.2 ⋅ 3 ⋅ 10⁻³ ⋅ 10⁻⁴
    = 21.6 ⋅ 10⁻⁷
    = 2.16 ⋅ 10⁻⁶
    ≈ 2 in 1 million

It's because we usually dont multiply percentages, but instead work with normal decimal figures.