Twice, or a hundred times, whatever …
« previous post | next post »
"Figures reveal ethnic variations in prevalence of alcohol-related diseases", The Scotsman 5/9/2016 [emphasis added]:
Irish people living in Scotland are more than twice as likely to end up in hospital or die from alcohol-related diseases as white Scottish people, research has found.
The risk for women from a mixed ethnic background is almost 100 times that of white Scots, scientists concluded. […]
Compared to the white Scottish population, women of mixed ethnicity are 99% more likely to to [sic] require hospital stays or die from alcohol-related disease.
This is enough to warn us that we're dealing with a hard-to-believe level of numerical illiteracy. Apparently the writer interpreted "99% more" as meaning "almost 100 times" more — and no editor caught it, either at The Scotsman or at any of the other papers that reprinted the story.
The optimistic interpretation here is that the writer knows what "100 times" means, and also knows what "100 percent" means, and that the substitution of "times" for "percent" in the second sentence of the article was just a Fay-Cutler malapropism. But the fact that the sentence still reads that way, at least 24 hours after the story was posted, suggests that optimism might be unwarranted.
And as you should expect, looking at the original scientific paper reveals that the reporting was problematic in deeper ways.
The source is Neeraj Bhala, Genevieve Cézard, Hester J.T. Ward, Narinder Bansal, and Raj Bhopal, "Ethnic Variations in Liver- and Alcohol-Related Disease Hospitalisations and Mortality: The Scottish Health and Ethnicity Linkage Study", Alcohol and Alcoholism May 2016.
Here are a few key sample rows and columns from their Table 3, "Age-adjusted rates per 100,000 PY [Population Years] and relative risks (RR) for first ARD [Alcohol-Related Disease] hospitalisation or death for the population ≥ 20 years by sex and ethnic group":
| Sex and Ethnicity | First ARD Events |
PY at risk | Poisson Rates for 100,000 PY |
Age- SIMD- and COB-adjusted Poisson Relative Risk |
|
|---|---|---|---|---|---|
| White Scottish Men |
47,085 | 11,764,486 | 391.7 | 100.0 | |
| White British Men |
2036 | 1,064,515 | 346.9 | 94.2 | |
| White Irish Men |
694 | 142,867 | 881.4 | 184.2 | |
| Men of any Mixed Background |
51 | 17,397 | 620.6 | 141.5 | |
| Indian Men | 76 | 35,152 | 433.1 | 136.7 | |
| Men of African Origin |
30 | 17,602 | 360.1 | 83.2 | |
| White Scottish Women |
21,640 | 13,613,525 | 159.0 | 100.0 | |
| White British Women |
1212 | 1,169,381 | 199.7 | 128.2 | |
| White Irish Women |
244 | 165,477 | 289.2 | 154.5 | |
| → | Women of any Mixed Background |
38 | 21,793 | 347.6 | 199.4 |
| Indian Women | 21 | 31,590 | 128.3 | 93.3 | |
| Black Women | 21 | 15,446 | 264.2 | 153.8 |
(The arrow pointing to "Women of Any Mixed Background" is my addition. Note that the original Table 3 gives data on 22 ethic/sex combinations, and the article contains two other tables of similar size — keep this in mind when evaluating the decision to feature "women of a mixed ethnic background" in the second sentence of the article.)
The first point to note is that rates for men are overall more than twice the rates for women — somehow the reporter managed to miss that. So the "Poisson Rate per 100,000 PY" for the featured "Women of any Mixed Background" was 347.6 — compared to 391.7 for "White Scottish Men" and 159.0 for "White Scottish Women".
This indeed yielded an "Age-adjusted Poisson Relative Risk" of 199.4 for the mixed-background women — relative to the "White Scottish Women" as the nominal standard of 100. And a relative risk of 199.4 compared to 100 is about twice as high, NOT 100 times as high. But compared to "White Scottish Men" at 391.7 per 100,000 population-years, those mixed-background women at 347.6 were actually LESS likely to experience Alcohol-Related Disease.
So when the article says that "The risk for women from a mixed ethnic background is almost 100 times that of white Scots", it's promoting two falsehoods. One piece of idiotic nonsense is directly asserted, namely that a ratio of 1.99 to 1 is "100 times". But another falsehood is implied, by omitting the biggest effect in the study's data, namely the sex difference.
And there's a third issue. The second column in the table tells us that during the period of time surveyed (which was nine years from 2001 to 2010), the total number of cases of ARD among "Women of any Mixed Background was 38, or an average of about 4 cases per year. This compares with 47,085 cases among White Scottish Men during the same time period, and 21,640 cases among White Scottish Women. Yet the second sentence of the article is
The risk for women from a mixed ethnic background is almost 100 times that of white Scots, scientists concluded.
Amazing.
I've noted in the past that the lack of numerical concepts among the Pirahã is paralleled by the lack of statistical concepts among members of our advanced industrial societies (see e.g. "The Pirahã and us", 10/6/2007). But I was wrong to hang this on the failure to understand an important but relatively subtle aspect of statistical analysis, namely how to compare group distributions of features.
Let's start by insisting that science writers for major publications should understand simple ratios — a ratio of 199 to 100 is about twice as great, not 100 times as great. And they should be able to evaluate large differences in counts — something that happened 38 times in 9 years (ALD among "women from a mixed ethnic background") shouldn't be featured in a context where it's being compared to something that happened almost 70,000 times during the same period of time (ALD among "white Scots"). Or, at least, you should have some reason for featuring that vanishingly rare class of events, other than sensationalism and an implicit appeal to ethnic and gender prejudice.
[h/t Eric P Smith]
Screenshot:
George said,
May 10, 2016 @ 7:05 am
But, quite apart from the ludicrous "100 times more likely" (suggesting a deficiency not only in numeracy but in basic common sense), surely there's already a problem with "99% more likely"? "1.99 times more likely" – OK. "Almost twice as likely" – OK. But "99% more likely"? Let's round it up and say "1 more likely". Would that make any sense to anybody?
Ralph J Hickok said,
May 10, 2016 @ 7:17 am
A usage that always throws me of is "x times less likely" and similar statements. A quick Google search brings up this headline (and many similar statements):
Women in communication three times less likely to hold top management positions
A quick look at the article seems to reveal that men are three times more likely to hold top management positions. But does that really mean that women are three times less likely? I can't seem to wrap my head around that, at all.
At best, it seems ungrammatical. At worst, it seems unmathematical.
Ralph J Hickok said,
May 10, 2016 @ 7:17 am
I'm sure "ungrammatical" is incorrect, but I hope you know what I mean :)
RP said,
May 10, 2016 @ 7:46 am
I agree that it's curious how people say "three times less likely" or "three times smaller" when they mean "a third as likely", etc. It's hard to make sense of it.
I also find it odd how "three times more likely" means the same thing as "three times as likely", whereas it sounds like it might mean the same as "four times as likely".
Still, if that's how it's used by most people and they know what they mean, it obviously can't be called incorrect. (Then again, do they know what they mean?)
Jerry Friedman said,
May 10, 2016 @ 8:21 am
As the number of first ARD events for British men is much less than that for white Scottish men, and likewise for women, does "British" not include "Scottish" in that table? Would it mean "English and Welsh" (and possibly "coming from those islands with charming legal statuses that foreigners aren't meant to understand")?
Jerry Friedman said,
May 10, 2016 @ 8:25 am
Ralph J. Hickok and RP: I speculate that most of the people who use "three times less likely" know what it means and believe that it's more understandable than "one-third as likely" because many other people don't understand fractions.
D.O. said,
May 10, 2016 @ 8:32 am
I guess, interpreting 1.99 ratio as 100 times is an example of what is known by technical neuro-psychological term "brain fart". As for sensationalism and overinterpretation of small sample differences, unfortunately, journalists are not the only guilty profession…
Mark P said,
May 10, 2016 @ 9:58 am
Jerry Friedman, I think you're right about the assumption that "three times as small" is more understandable than "one-third as large," at least in the minds of some people. I worked in the physical sciences, and that expression is jarring to me. I recognize that it has become common, sometimes even among physical scientists who aim their comments at a lay audience, but if I were reviewing a technical paper (as I sometimes did when I worked), I would change that expression to one that makes mathematical sense.
As to percentages, the example MYL's cites rises (or sinks) to an entirely different level from ordinary misunderstanding. I facepalm so hard it gives me a concussion as well as whiplash.
unekdoud said,
May 10, 2016 @ 10:30 am
In my opinion, the whole "X times less likely" phrasing is just a can of worms. From the usage that I've seen, "4x less likely" is "3/4 less likely", but "4x more likely" is "4x as likely", and you can substitute 400% for 4x in the last two cases but not the first. Which is crazy and standard.
(And if you want to think of these probabilities or frequencies as percentages, you'd have to wrestle with the "0.16% plus 99%" construction, which can be challenging even for the numerically literate.)
Certainly a "99% less likely" is exactly a "100 times less likely" (according to most common usage of these terms), but I don't believe the sense of the statement got reversed twice before making it to its current form.
Instead, it seems plausible to me that somewhere in the writer's notes a "99%" got mistaken for a "99x". In either case, there's no reason to keep the 99 number, since rounding it to 100 isn't wrong, doesn't lose much (if any) precision, and is much easier to understand.
J. W. Brewer said,
May 10, 2016 @ 10:33 am
https://en.wikipedia.org/wiki/Demography_of_Scotland#National_identity gives some additional color on what "British" may mean in this context. Separately, the percentage of current residents of Scotland who are identified as "Irish" for census purposes is barely 1% , which is roughly consistent with the numbers given above, but the percentage who are largely/entirely descended from 19th or 20th-century Irish immigrants as signaled by ethnic markers like surname, religious affiliation, soccer-team loyalties and the like may well be at least an order of magnitude higher than that. (>900% higher, one might say …) I expect that if you were interested in ethnic factors relevant to epidemiology, it's the larger population rather than the much smaller census-stats population you ought to want to focus on. Or at least you ought to want to look at both measures before figuring out which is most appropriate for the context.
[(myl) The original article considers other demographic variables, include COB (country of birth).]
Theophylact said,
May 10, 2016 @ 11:06 am
And what about the distinction between "Irish people" and "white Scottish people"? Of course, there are non-white people living in Scotland, but surely there must be some in Ireland as well. There's an implication that Irish equals white and Scottish doesn't. I'm sure that the study was merely controlling for race, but still… .
RP said,
May 10, 2016 @ 11:07 am
@unekdoud,
I don't think 4x more likely necessarily means the same as 400% more likely.
Presumably, we're agreed that 100% more likely means twice as likely. In which case, is it also the case that 200% more likely must mean three times as likely, and 400% more likely must mean five times as likely?
When I hear people say "four times more likely", usually they seem to mean "four times as likely" rather than "five times as likely".
J. W. Brewer said,
May 10, 2016 @ 11:37 am
Re Theophylact's question, the "What is your ethnic group" question on the questionnaire used for the most recent (2011) census of Scotland gives both "Scottish" and "Irish" as subcategories of "White." It is probably less confusing to report the resultant category as e.g. "White: Scottish" than "Scottish White." This is another example of the classic difficulties that arise when a language lacks good lexical distinctions between "native/citizen/resident of country X" and "member of the dominant/namesake ethnicity of country X."
The census does provide a chance for non-whites to self-identify as Scottish, but in kind of a half-assed way. E.g., there's a single box you can check if you identify as "Bangladeshi, Bangladeshi Scottish or Bangladeshi British" (and the same for various other racial/ethnic options) but no way to further specify if you want to stress that you identify as one of those three things to the exclusion of the other two. "Bangladeshi Irish" is not an explicit option (nor is, e.g., "Bengali-American" – which in my limited experience seems more common in U.S. than a specification that distinguishes as between Bangladesh and West Bengal), although there's always "Other, please write in."
RP said,
May 10, 2016 @ 11:49 am
@Jerry Friedman,
Usually ethnicity is self-identified. In this case "British" might not mean simply English or Welsh. It might include those of mixed Scots/English heritage (though some might identify as Scottish instead), possibly a small minority of those of purely Scottish heritage, and some Northern Irish.
"Scottish" and "English" aren't legal statuses. Everyone's passport says British. Insofar as there are different rights, responsibilities, applicable laws, they are based on location. For example, in the Scottish independence referendum, all British citizens resident in Scotland could vote, and none who weren't (even if they were born in Scotland to Scottish parents) because the distinct legal status doesn't exist and would be controversial to create.
DWalker said,
May 10, 2016 @ 12:09 pm
I, also, don't understand what "three times less likely" means.
I can't resist repeating a memory from the olden days, when the company behind the mainframe product SyncSort ran a full-page ad in a trade magazine. The ad quoted a programmer saying "Our production jobs run in 140% less time now that we use SyncSort!".
Sheesh.
Gregory Kusnick said,
May 10, 2016 @ 12:49 pm
My guess is that the sentences containing "99% more likely" and "almost 100 times" were written by two different people.
Writer A saw the 1.99 ratio in the table and wrote it down as "99% more likely". Editor B saw "99%" in A's copy and interpreted it as "99 out of 100 cases", leaving just 1% for everybody else; hence "almost 100 times".
Not that this excuses anything.
Guy said,
May 10, 2016 @ 1:10 pm
It seems pretty apparent to me that the mistake was interpreting the 99% as 99% of the larger number, rather than 99% of the smaller number. This shows a failure to interpret English statements about math in the conventional way, but not necessarily a mathematical error in and of itself.
I can't agree that "1.99 times more likely" is a good way to express "99% more likely". I would say "1.99 times as likely". To me "more likely" usually communicates that we are expressing the amount in excess of equality, so I would consider the proposed statement at best ambiguous.
unekdoud said,
May 10, 2016 @ 1:28 pm
@RP You're right, 400% isn't quite substitutable in that case… so it's even worse: 4x more, 4x as, 400% as, 300% more.
The "increased by 400%" version of this example is also discussed in this xkcd thread ( http://forums.xkcd.com/viewtopic.php?t=80166 ) where "increased to 400%" is considered as an unambiguous wording.
So in both cases, we have an unambiguous and direct way of specifying multiplication by a factor of 4, but not for adding 4 times of a base value.
Eric P Smith said,
May 10, 2016 @ 2:31 pm
Thanks for airing this, Mark. I'm impressed by your diligence in drilling down to the original paper (to which I didn't have access). I think you're right about the "implicit appeal to ethnic and gender prejudice".
Oh, and I loved your cheeky colour coding (blue for a girl and pink for a boy).
Chandra said,
May 10, 2016 @ 2:57 pm
@Theophylact – That statement surprised me as well, but for a different reason, as I took it on first read to mean that the authors were making a distinction between "Irish people" and "white people".
Stephen Hart said,
May 10, 2016 @ 3:23 pm
I had the same reaction as Chandra, at first, that the comparison was between people of Irish heritage (considered nonwhite) vs. white Scots.
From the article.
"Irish men living in Scotland showed an 82% higher risk compared to white Scots, while for Irishwomen in the country it was 55% higher."
And, call me a picky retired science writer, but why suggest that a "science writer" wrote this? The only attribution is to Press Association Ltd. If that's anything like AP, it's likely some poor, young writer pounding out five stories a day, quite possibly on many topics.
[(myl) All I can say is that the story is about science, and so was written by a "science writer" in the compositional sense of someone who writes about science. You might well be right that the author was an underpaid young drudge harassed about their clickbait quota. But I wish it were true that designated "science writers" could be counted on to do better.]
hector said,
May 10, 2016 @ 3:30 pm
@ Chandra – Back in the 60s, I read a book about racism that I found in the university library. It had a very illuminating chapter on English colonists' descriptions of the Irish. They were described as having a natural sense of rhythm, having a happy-go-lucky and childish nature, being lazy, unreliable workers, and so on. Sound familiar?
It made a deep impression on me. It taught me that racism isn't about race, it's about power relationships.
Andrew (not the same one) said,
May 10, 2016 @ 3:48 pm
One can see from the table that the full description is 'White Irish'; 'Irish' is here being contrasted with 'Scots' rather than with 'White'. One thing that puzzles me, however, is that men are 'of African origin' while women are 'black'. It's true, of course, that black people are of African origin (so is everyone else if you go back far enough, but you know what I mean). But 'African' in the UK normally means someone who comes, or whose family comes, directly from Africa rather than from the Caribbean, so I would not have naturally read these expressions as picking out the same group of people.
Mark said,
May 10, 2016 @ 7:12 pm
"Science writers" are often not very well informed about science, especially at smaller newspapers. Sometimes even at larger papers they are journalism graduates, some perhaps with a lot of general experience, who are assigned science stories. I remember the reporter at the major Atlanta newspaper who was assigned to cover the Y2K story explaining how she knew nothing about computers or coding or much of anything else. She said all she was trained to do was read (public) records and ask questions in general. In other words, she was completely unqualified to write a story about supposed Y2K problems in any way other than quoting anyone who claimed to be an expert. That newspaper at one point mentioned predictions of airliners falling from the sky. That's responsible journalism.
Jerry Friedman said,
May 10, 2016 @ 11:14 pm
J. W. Brewer and RP: Thanks for shedding light on my question. I gather that "British" as distinct from "Scottish" could mean either "other British" or "possibly Scottish, but preferring to identify oneself as British". It still seems very strange to me, like a sample that contains more Californians that Americans. But I've never been to Scotland.
R. Fenwick said,
May 10, 2016 @ 11:16 pm
@hector:
It taught me that racism isn't about race, it's about power relationships.
Or maybe it's just that the English have no sense of rhythm, have a boring and stoic nature, and are tedious and monotonous workers, so everyone else seems lazy and happy-go-lucky and eurhythmic by comparison? ;)
In seriousness though, do you happen to remember the book, or anything about it? It'd be interesting to find that so that I can cite it as an example of racism's roots in establishing power hierarchies. Not that citing sources often helps when arguing against prejudice, but every once in a while one gets lucky.
tangent said,
May 10, 2016 @ 11:16 pm
I have only seen the term "relative risk" used as a ratio, never indexed to 100 = nominal. Is 100-based relative risk normal in some fields?
Too bad the authors of the paper didn't include confidence intervals. That probably would have prompted them to remove those low-count rows of the table as being embarrassingly contentless.
[(myl) They did include confidence intervals — I left them out in order to make the table fit and be readable. Their full table 3 is here:
The number of cases is large enough that the (95% CI) confidence interval for the "Women of Any Mixed Background" category is just 152.1 to 261.4 — or "a risk between 52 and 161 times that of white Scots", in the news story's absurd representation. The absurdity of that misunderstanding was my main point, of course. But my second point was not about statistical significance but rather about social significance — in a story that covers more than 70,000 cases in total, why feature a subcategory that involves just 38 cases over 9 years? The confidence interval on the relative risk estimates is relevant but much less important to the interpretation of the story.]
mollymooly said,
May 11, 2016 @ 2:23 am
I don't see a problem with "a is three times less than b"; the only interpretation I can see for it is "b is three times more than a". Is there some other interpretation that might be intended?
Jon said,
May 11, 2016 @ 2:59 am
Hector – when I worked on a London building site in the 1970s, one of the English workers used to refer to blacks as 'sunburnt Irishmen'. There was plenty of casual racism, of the ethnic joke rather than fear/hatred variety.
Gregory Kusnick said,
May 11, 2016 @ 3:26 am
mollymooly: That particular case may be unproblematic (apart from the dubious use of "times" to signify division). But elaborating the example quickly gets you into trouble. For instance, what would it mean to say that A is more than three times less than B? A > B/3, or A < B/3?
He said, she said,
May 11, 2016 @ 7:04 am
Although I am not a linguist, I hate zombie rules just as much as the typical reader of Language Log. I admire and endorse descriptivism. Nevertheless, I feel a burst of finger-wagging comin' on.
My casual understanding of the descriptivism/prescriptivism debate suggests that zombie rules solve problems that don't exist. But here, I see a genuine problem.
(Aside: My amateur's understanding of descriptivism/prescriptivism is not so casual that I fall into the trap of believing that there is no middle ground: that prescriptivists are snoots and descriptivists are hippies. I'm happily occupying the middle ground here, endorsing descriptivism and simultaneously pointing out a problem with the way people speak and write. I think we need some non-zombie rules for talking and writing about percentages and ratios.)
Anytime I read a construction like "X is three times smaller than Y," I lose confidence in the writer. This construction constitutes a speed bump in the prose for me; it interferes with the smooth flow of ideas in my head while I contemplate set of things the turn of phrase might mean, the set of things the writer might intend to mean, and whether those sets overlap. The problem is not merely a loss of confidence — the way a snoot might lose confidence upon encountering a harmless split infinitive. The problem is that I sincerely do not know what the writer means.
In conversation, EVERY TIME I hear someone say "X is three times smaller than Y," I ALWAYS ask for clarification. Sometimes the speaker means that X is 33 percent of Y. (That is, X is 2/3 smaller than Y.) Sometimes the speaker means that X is 25 percent of Y. (That is Y = X + 3X.) Sometimes the speaker says something akin to "whatever."
When we allow utterances like "X is three times smaller than Y" to pass without demanding clarification, preposterous consequences ensue. Gregory Kusnick describes some of them. Another, which I find especially intolerable, is that two different speakers could describe a given situation with these two sentences:
X is one-half smaller than Y.
X is two times smaller than Y.
but neither of those speakers would consider these sentences equivalent:
X is one-half bigger than Y.
X is two times bigger than Y.
I find this intolerable — especially in a culture that grows more saturated in data every day.
He said, she said,
May 11, 2016 @ 7:22 am
The example I gave at the end of the previous post contains a mistake. It should read:
Two different speakers could describe a given situation with these two sentences:
X is fifty percent smaller than Y.
X is two times smaller than Y.
but neither of those speakers would consider these sentences equivalent:
X is fifty percent bigger than Y.
X is two times bigger than Y.
ohwilleke said,
May 11, 2016 @ 7:24 am
"The confidence interval on the relative risk estimates is relevant but much less important to the interpretation of the story."
The lack of confidence intervals is the second most important rather than the biggest error in the story, but it is still a huge one because the touted headline finding that mixed women have the highest rate is not statistically significant and is simply a product of a tiny sample size.
[(myl) In fairness, the article didn't say that "women from a mixed ethnic background" had the highest relative risk — it just featured (and misrepresented, in several ways) their relative risk.
And many of the differences are "statistically significant", to use that pernicious phrase. The problem is that the relatively small group size (21,793 population-years, compared to about 70 million in the overall study) means that the social significance is negligible. ]
Jerry Friedman said,
May 11, 2016 @ 9:15 am
Thanks for posting the table, Prof. Liberman. My question about "Scottish" versus "British" is now answered.
J. W. Brewer said,
May 11, 2016 @ 9:36 am
FWIW, although this doesn't directly explain the mismatch between the male and female categories in the table myl cut-and-pasted above, the Scottish census questionaire treats "African" as distinct from "Caribbean," which makes sense to me in a UK context as those are two rather culturally-distinct groups of immigrants (or children/grandchildren of immigrants) despite similarities of skin color. But then bizarrely it seems to treat "black" as a synonym for "Caribbean" rather than a macro-category encompassing both "African" and "Caribbean." http://www.scotlandscensus.gov.uk/documents/Householdpre-addressed27_05_10specimen.pdf (scroll down to question 15). Note also that there's a separate question about "national identity," so you can be "Scottish" (or "British" or various other things) for national-identity purposes but something else for ethnic-group purposes.
Linda said,
May 11, 2016 @ 10:01 am
In recent history the people who moved to Britain from the Caribbean have had a 50 year head start over those coming directly from Africa and so gain proprietary rights to the term "black". Which has left us with the situation of most of those claiming a black identity are paler than those who came from Africa.
And going back to Irish identity, I have a vague recollection of joining a library in the 70s and part of the form was a tick box list of ethnicities for council statistics. The options were grouped, European was grouped with English, Scottish and Welsh, but Irish was with Afro-Caribbean, Indian and Pakistani. I think the argument was that Ireland was a foreign country, like India etc.
Rodger C said,
May 11, 2016 @ 11:41 am
@R. Fenwick: I don't know if it's the book hector read, but William Ryan's Blaming the Victim comes to mind.
Daniel Barkalow said,
May 11, 2016 @ 4:55 pm
Looking at epidemiological data in a paper involving fatalities primed "Cause of Death" for me. Then I was disconcerted to see they'd controlled for "COB". I suppose that would be one way to gauge genetic (or epigenetic) predisposition, but not something that I expect most people's medical records include.
Chris C. said,
May 11, 2016 @ 6:04 pm
This discussion has confirmed for me that the difference between people who prefer expressions such as "three times less often" and those who prefer "one third as often" is that the latter understand basic math, while the former do not.
Rodger C said,
May 12, 2016 @ 11:32 am
http://www.amazon.com/Twice-Less-Performance-Students-Mathematics/dp/0393317412
tangent said,
May 12, 2016 @ 1:14 pm
I assure you, Chris C., plenty of people of all phrasing preferences and phrasing tolerances have a limited ability to reason with frequencies or probabilities.
I doubt you can show a statistically significant difference between the math abilities of the two phrasing populations. But if it exists, the effect size is not large enough to be dispositive about anything.
Eneri Rose said,
May 13, 2016 @ 3:27 pm
As did Chandra and Stephen Hart, I read the first sentence to imply that the Scottish are white and the Irish are non-white.
I too dislike the construction x is some% less than y. It is confusing and shows a lack of numeracy on the part of the writer. I think a reason for the construction could be that the writer/editor insists on x being the subject of the sentence to grab attention. If my subject is men, that might be boring. If my subject is poor, disadvantaged, women of color, now that will garner attention!