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The Financial Times did some analysis of Donald Trump's controversial immigration restrictions (which were claimed to be about protecting Americans from terrorism).

They focus on analysing risk from refugees since (my emphasis):

President Donald Trump will indefinitely block Syrian refugees from resettling in the US and will temporarily suspend the entire US refugee programme

They put the refugee issue in context:

Those resettling in the US are interviewed and screened for terrorism and crime links by several US agencies, including the National Counterterrorism Center, the FBI, the Department of Homeland Security and the State Department.

Mr Trump cited terrorism risks as his reason for limiting the number of refugees the US takes. However, since the US refugee programme began in 1975, more than 3.2m refugees have entered the US and only three have carried out a deadly terrorist attack.

The FT analysis includes the chart below:

FT analysis

They claim that Americans are more at risk from death caused by vending machines than death caused by terrorist attacks committed by refugees.

Are their estimates of the risks credible?

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    Please do not attempt answers here. Please do not post blatantly racist comments. Please do not try to explain statistics in 500 chars. Yes the claim should say "incidence" and not "probability". Let's move on. – Sklivvz Feb 3 '17 at 11:12
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    The premise of the question is somewhat arbitrary and ridiculous, why not compare the number of people killed by refugees to the number of people killed by spatulas, or chocolate chip muffins? – Robert S. Barnes Feb 4 '17 at 17:08
  • @RobertS.Barnes a fair point. Why not make the comparison? It would be an even better way to emphasis just how small the historic risk from refugee terrorists is. – matt_black Feb 5 '17 at 10:36
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    ´Are Americans more likely to be killed by vending machines than terrorist refugees?´ I'll go out on a limb and say yes. Refugees tend to interact much less with vending machines [citation needed], hence they have a lower risk of getting harmed by one. – xDaizu Mar 13 '17 at 15:13
  • @xDaizu very funny. – matt_black Sep 26 at 11:24
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Yes those numbers appear to be correct. The Cato instituted published "Terrorism and Immigration A Risk Analysis" in September 2016. From that report:

Chance of Dying in an Attack by a Foreign-Born Terrorist

NPR article on the chance of winning the lottery from 2012 notes that it's more likely to be killed by a vending machine. The odds given : 1 in 112 million.

And that means it's more likely, at least judging from one sort-of-old but widely cited statistic, that you'll be crushed to death by a vending machine as you try to shake loose a stubborn candy bar.

The supposed odds of such a death? About 1 in 112 million.

The link to the source of their data is dead, but it seems likely that 1 in 3.6 billion odds are lower than a whole lot of very unlikely causes of death (struck by lightning while drowning: 1 in 183 million).

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    Why don't they count the 1993 World Trade Center bombing under "asylum". Why don't they include the 11 September 2011 killing of 3 people by the Boston Marathon bomber? They also leave out one of the police officers killed by the Marathon bombers after the initial bombing. – DavePhD Jan 31 '17 at 15:58
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    Likely they used the visa status used to enter the country: the Tsarnaevs entered on a tourist visa, then applied for asylum after the fact (and one of them later became a citizen and so might not be included in the above table anyway, depending on how they considered it). Either way I think it's reasonable to remove those two from the "refugee/asylum" category - given they were here for almost fifteen years, and with no evidence they were bad actors until very near to the event. – Joe Jan 31 '17 at 16:10
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    I can't find any example of someone dying from a vending machine since 1998. darwinawards.com/darwin/darwin2001-25.html And the 1998 death was in Canada. Do people really still die from vending machines, or has the design been changed. Are the statistics limited to the United States? – DavePhD Jan 31 '17 at 16:16
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    Some context is always nice. You're more likely to die from lightning than a Terrorist Attack at all (let alone form a refugee), or a public shooting. However, also be mindful that this data is being wielded for political story control, and is designed to make one side seem preposterous. – SnakeDoc Jan 31 '17 at 16:18
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    Please remember the Be Nice policy, and make sure your comments aren't going to be misconstrued. – Oddthinking Feb 1 '17 at 2:00
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According to US Citizenship and Immigration Service webpage Refugees & Asylum, last updated in 2015:

Asylum status is a form of protection available to people who:
•Meet the definition of refugee

In other words, not all refugees are asylees, but all asylees are refugees.

By not including asylees in the definition of "refugees" the Financial Times disregarded attacks such as the Boston Marathon Bombings.

Tamerlan and Dzhokhar Tsarnaev (the Boston Marathon bombers) were refugees, but the FT did not included them.

Ibragim Todashev, who allegedly helped Tamerlan Tsarnaev kill 3 Jewish men on 11 September 2011, was a refugee, but the FT did not include him.

1993 World Trade Center bomber Ramzi Yousef was a refugee, but the FT did not include him.

The FT left out 15 terrorism deaths: 7 from the 1993 World Trade Center bombing, 3 from the 11 September 2011 killings, and 5 from the Boston Bombing an aftermath.

So instead of the 3 deaths that FT counted based on the source Terrorism and Immigration, there were really 18.

The vending machine statistic is based upon data before a 1995 campaign to improve the safety of vending machines. It does not indicate the current odds of being killed by a vending machine. 37 people are known to have died due to vending machine accidents from 1978-1995. Also, there is no indication of how many of the 37 were in the United States. In fact, according to the Guardian article The (mainly) men who have fallen under sway of drinks vending machines about Dr. Michael Q. Cosio who was the main researcher of this issue, in his initial study of vending machine injuries and deaths:

The alarming incidents all happened at American military bases in Germany and Korea.

Also, though the OP labels the graph "per billion people", the data, at least for terrorism, is per billion people per year. On the same basis the odds of winning the mega millions jackpot is much higher than what the OP graph says. 8 people per year with a population of 320,000,000 is 25 per billion people per year, for example.

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    Also the Tsarnaev brothers were in the country for over 10 years before they got radicalized, their asylee status has nothing to do with what they've done. – ventsyv Feb 3 '17 at 15:40
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    @DavePhD Which is basically what ventsyv said. They were homegrown terrorists, not terrorist refugees. They became radicalized after arriving in the U.S., not before. – aroth Feb 4 '17 at 1:56
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    @bain The law is very clear: "the alien may be granted asylum in the discretion of the Attorney General if the Attorney General determines that such alien is a refugee within the meaning of section 101(a)(42)(A)" gpo.gov/fdsys/pkg/STATUTE-94/pdf/STATUTE-94-Pg102.pdf To be granted asylum one must meet the definition of "refugee" and be physically present inside the United States or at a land border. – DavePhD Nov 2 '17 at 13:02
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    @bain I agree there are two different processes, one ("asylum procedure", section 208) for refugees in or at the land border of the United States, and another (section 207) for refugees outside the United States. Pew is directly contradicting the law, which says "The Attorney General shall establish a procedure for an alien physically present in the United States ... if the Attorney General determines that such alien is a refugee" – DavePhD Nov 3 '17 at 10:58
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    @bain and contradicting themselves, as they say "Asylum status is a form of protection available to people who: •Meet the definition of refugee •Are already in the United States ..." uscis.gov/humanitarian/refugees-asylum – DavePhD Nov 7 '17 at 19:22
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According to this analysis on CrossValidated.SE, we don't really know.

The first problem with this data is non-stationariness. These numbers hold only if the number of refugees and vending machines stay constant. These 3 deaths are not "deaths per 1M of refugees", it's a total. Let 10 times more refugees in, and you'll have to multiply that number by 10.

Taking that chart data to the extreme, I could say that illegal immigrants are 3 times less dangerous than refugees, so why have border control at all?

The second (and IMO more important problem) is statistical significance. 3 deaths over 35 years is simply too little to predict similar deaths in the future.

However, the author acknowledges that the original article's point may still be valid, despite flawed statistical proof.

There are legitimate issues with the chart, but the FT's broader point is correct that terrorism in the U.S. is quite rare. Your chance of being killed by a foreign born terrorist in the United States is close to zero.

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    While correct, this is off-topic for Skeptics.SE. On-topic is citations for the numbers. Off-topic is criticizing (or agreeing with) the argument being made. – Brythan Feb 2 '17 at 18:03
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    @Sklivvz then the question itself must similarly be ruled off topic, as the question is about a theoretical model. Without a model, no conclusion whatsoever can be drawn from the numbers. – Chris Stratton Feb 4 '17 at 17:59
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    @ChrisStratton the question is about some incidence numbers, which are measurable facts, and certainly not what we think the implication of showing those numbers is. – Sklivvz Feb 5 '17 at 0:00
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    No. "Likely" from the very title, and "analysis" and "estimates" from the body are not facts, but rather the conclusions of a model, and cannot be determined without considering the validity and applicability of that model. Your mistake may be common, but it is still a mistake, and severe one. The question is not just about historical facts, and it is disingenuous to try to claim that it is when the very wording makes it obviously otherwise. – Chris Stratton Feb 5 '17 at 1:22
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    @ChrisStratton You make a more important error in delving into the detail. The point of the chart is not to posit a complicated risk model. It is to challenge the widespread public perception that refugees are a major terror risk. The simple historic numbers are a perfectly good way of doing that. – matt_black Feb 6 '17 at 10:24

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