In the free speech / dogfighting video case, United States v. Stevens, the majority opinion said:
The demand for hunting depictions exceeds the estimated demand for crush videos or animal fighting depictions by several orders of magnitude. Compare ibid. and Brief for National Rifle Association of America, Inc., as Amicus Curiae 12 (hereinafter NRA Brief) (estimating that hunting magazines alone account for $ 135 million in annual retail sales) with Brief for United States 43-44, 46 (suggesting $ 1 million in crush video sales per year, and noting that Stevens earned $ 57,000 from his videos).
It’s pretty clear that Chief Justice Roberts was using the normal definition of “by X orders of magnitude”, which is “by a factor of roughly 10X, give or take” — to quote the example from the American Heritage Dictionary, “The masses of Earth and the sun differ by five orders of magnitude” (the Sun being 3 x 105 times bigger than the Earth). But this led me to wonder: Is the term really understood this way by nearly all lawyers?
So I ran a quick survey, asking only our law student and lawyer readers the question,
You read that
The demand for product A exceeds the demand for product B by three orders of magnitude.What do you think this means about the ratio between the demand for A and the demand for B?
Obviously, the respondents are a highly unrepresentative sample of all lawyers. But I thought that if there was a significant minority of our readers who didn’t understand the term in its dictionary sense, there would also likely be a significant minority of all lawyers who would likewise misunderstand. If anything, I’d expect the misunderstanding fraction among lawyers at large to be greater, especially since I expect that people are more likely to respond to the survey if they are confident in their views, and people who are confident in their views on this are more likely to be so because they actually learned the official definition.
So here’s what the 1325 responses over the past 17 hours yielded:
- 68% responded 1000 (or 1001 or 999, which are close enough, since the usage contemplates an approximation).
- 18% responded 3 or 4.
- The rest gave other responses, such as 27 (there were nine such responses), 30 (twenty-three responses), and 100 (twenty-seven responses).
So all in all nearly a third of respondents didn’t understand “orders of magnitude” in the way the dictionary suggests, in the way I’ve seen it used by mathematicians and engineers, and in the way Chief Justice Roberts was using it.
Of course, that might not matter to the Chief Justice, who isn’t much worried if a few lawyers misunderstand at first, and are thus unpersuaded. But it might matter to us lowly folk, who might care a lot about immediately conveying our message to our readers (judges, partners, clients, and so on). If I think “three orders of magnitude greater” sends the message “1000 times greater,” but the judge who reads my motion happens to be someone who reads this as “3 times greater,” that means my phrasing was ineffective. And even if I follow this up (as did the Chief Justice) with the specific data, the judge will be reading that data with the wrong assumption, and might either misinterpret the data or at least be confused and distracted from my argument.
So my advice: If you want to stress that something is (say) more than a thousand greater than something else, say “more than a thousand times greater,” rather than “greater by three orders of magnitude.” At least do this unless you’re sure that your audience is going to understand you — though even if you are sure of that, I think the first phrase is at least as effective as the second.
UPDATE: Some commenters are upset that lawyers are so “innumerate” or “ignorant of fundamental mathematical and scientific concepts.” That may well be true as to other things. But I think that all that this example shows is that they are not clear on a particular mathematical term. They may perfectly well understand the underlying mathematical concepts (multiplication and powers of ten) but just not know which concepts “orders of magnitude” refers to.
Not knowing that the Earth orbits the Sun would be ignorance of a fundamental scientific concept. Not knowing that the term “heliocentric theory” refers to this understanding of the Solar System is just ignorance of a particular phrase. Likewise, not knowing what ten to the third power is would be ignorance of a fundamental mathematical concept (unless you were somehow taught this concept without hearing the phrase “to the Xth power,” which is unlikely). Not knowing that “three orders of magnitude” means “greater by a factor of ten-to-the-third” is ignorance of a particular phrase.
FURTHER UPDATE: Commenter Butters writes: “I could’ve told you this! During a moot court final in law school, before judges including a former state chief justice and a current federal appellate judge, I used the phrase ‘orders of magnitude.’ One of them stopped me and asked what it meant (yes, I was able to give a satisfactory answer), which of course wasted time, broke my concentration, and broke the flow of my argument. Lesson learned.”
Praetorius says:
Terrible that law students and attorneys are so innumerate.
July 2, 2010, 12:31 pmZach says:
You’re saying that a third of lawyers are ignorant of fundamental mathematical and scientific concepts. I’m shocked!
July 2, 2010, 12:37 pmClayton E. Cramer says:
Yes, terrible that law students and attorneys are so innumerate–but still better, I’m guessing, than the majority of Americans.
I’m reminded of the story of when Gen. Leslie Groves (the engineer in charge of the Pentagon building project) first got together to talk to the scientists when Groves was put in charge of the Manhattan Project. When the subject of the size of critical mass for the bomb came up, the scientists were talking about being able to know the size within a couple of orders of magnitude. Being a civil engineer, Groves didn’t know what they were talking about: he was used to tolerances measured in percentages–and usually one (or at worst) two digits of percentage.
July 2, 2010, 12:40 pmCareless says:
As a non-lawyer, I find this significantly less surprising than the number of people who were unfamiliar with the word “aegis”
July 2, 2010, 12:44 pmTamerlane says:
The good news is that over 2/3rds of respondents got it right. But, as someone in the previous post pointed out, there’s probably a really large response bias: many of the persons who would have gotten it wrong had they responded probably chose discretion over valor.
July 2, 2010, 12:46 pmMark Horning says:
As a physicist, I would have answered 316 to 3162 times, i.e. 1000 scaled by error bars of a factor of 10^.5 to the original umber.
But I tend to overalalyse things.
July 2, 2010, 12:50 pmToday's Tom Sawyer says:
Isn’t it generally bad practice to use technical language in legal documents due to their technical and easily miscontrued nature (especially if there is a common usage outside of the technical meaning)?
July 2, 2010, 12:53 pmerp says:
I think the lawyers did a good jobs answering the question posed. Afterall law schools don’t usually draw from math or engineering programs.
I’d like to see the replies from random people on the street under 40 years of age to questions like what’s three times three or asking them to tell the time on an analog clock.
I’d be willing to bet a lot less than 60+% of them would answer correctly.
July 2, 2010, 12:55 pmDavid Welker says:
I would also say that, to me (1) “more than a thousand times greater” sounds more dramatic than (2) “greater by three orders of magnitude” even though I know the definition of the phrase “order of magnitude.”
Sure, when you think about it, there is no difference. But look at the two sentences. With the first, you get to the big number 1000 in the beginning of the sentence. In the second, you get to the little number 3 in the beginning of the sentence. There is a translation process that one’s mind must go through to transform the small number 3 to the big number 1000 even if you know the definition. So, the fuzzy subconscious “feel” one might have even after going through that translation might be to think of it is as a smaller than it really is. (And yes, I agree that it this is speculative. I am no psychologist. Also, I don’t think this is going to be very true for people who use scientific notation all the time.)
So, I would argue that there is reason to prefer “more than a thousand times greater” over “greater by three orders of magnitude” even when you know your audience understands the latter phrase. On the other hand, if we are talking about really large orders of magnitude (above one quadrillion where the word trillion is not conveniently used) it would tend to be inconvenient to not use the “order of magnitude” phrase as most people will not know what you are talking about if you use the terms quadrillion, quintillion, sextillion, or septillion. And even if the reader knows what these terms mean, order of magnitude are easier to comprehend at that point.
July 2, 2010, 12:57 pmAJK says:
What alternate usage would you suggest? “A lot”?
July 2, 2010, 12:58 pmButters says:
I could’ve told you this! During a moot court final in law school, before judges including a former state chief justice and a current federal appellate judge, I used the phrase “orders of magnitude.” One of them stopped me and asked what it meant (yes, I was able to give a satifactory answer), which of course wasted time, broke my concentration, and broke the flow of my argument. Lesson learned.
July 2, 2010, 1:00 pmHouston Lawyer says:
Back when I took the LSAT, there was a math section. The LSAT people told me I got 24 out of 25 questions right. The next year, they changed to test to eliminate the math section. The math portion of the test was geared more towards geometry and logic.
I explain to clients all the time that lawyers don’t do math. While I often check the math in prospectuses, I make sure that they know that numbers are not my responsibility.
July 2, 2010, 1:01 pmDJR says:
How prescriptive of you, Eugene!
What if the Chief Justice just meant “a lot more,” which I suspect is how “orders of magnitude” is most often used and understood?
July 2, 2010, 1:04 pmDavid Welker says:
Compare “three orders of magnitude larger” and “orders of magnitude larger.” The first has a precise meaning, the latter does not.
July 2, 2010, 1:07 pmMike P Wagner says:
Since I am not a lawyer, I avoided responding earlier. But the nitpickers were pretty much wrong here. When used as it was used, arguing that 1000 was incorrect and 1001 (or 999) was correct showed a clear misunderstandingof the term. “Order of magnitude” is always used an an approximation – as in the remark by Clayton E. Cramer makes clear:
The very point of the term “order of magnitude” means that difference of less than 10x are unimportant, unknown or inconsequential. It doesn’t make any sense to use that term if the difference between 999x and 1000x and 1001x is known and of consequence.
Mike
July 2, 2010, 1:07 pmToday's Tom Sawyer says:
I would suggest “1000 times greater,” but, from an economic theory standpoint, “a lot” is more accurate to the truth of the relationship model than some numerical derivate created by mathematical fetishism. I would also like to point severe economics fail on the part of Stevens, as he read annual retail sales to be “demand,” which annual retail sales are a point measurement (and isn’t even a measurement of quantity supplied or quantity demanded) instead of a function, and as such, are derived from the interaction of the supply and demand curves.
July 2, 2010, 1:08 pmAsterix says:
Probably the larger concern is that approximately 1/3 of the surveyed population either falsely believed they understood the usage and failed to independently verify, or they recognized they didn’t understand, and still didn’t look it up. Admittedly this was a very informal poll with no real incentive to getting the ‘right’ answer, but perhaps the bigger question is whether this ratio would carry over to subjects more immediately relevant to the practice of law.
July 2, 2010, 1:10 pmEugene Volokh says:
DJR: I’m certainly happy to “prescribe” in the sense of advising people about what usage is likely to be more effective. I oppose prescriptivism only to the extent that it purports to say that some usage is “wrong” even if it’s widely used. Here the usage is clearly not wrong — it’s just one of the many standard usages that are nonetheless ineffective in certain circumstances.
If you mean “a lot more,” say “a lot more.” That way you’ll see that the phrase is vague and unpersuasive, and this will lead you to say something more specific and therefore more effective, such as “several hundred times more.” (Or if the context is such in which the vagueness of “a lot more” is just fine, or is inevitable, then “a lot more” strikes me as simpler, clearer, and more effective than “orders of magnitude more.”)
July 2, 2010, 1:12 pmBob K says:
Interesting, but I’d pay good money to see the response from Congress and the Senate. Better yet, see how they respond to basic Econ 101 questions.
July 2, 2010, 1:14 pmAnthony says:
If you say ‘X is orders of magnitude higher than Y’ without specifying how many orders of magnitude, that could be anywhere from around fifty times more on up; it’s pretty much just an emphatic to indicate that X is dramatically more than Y (verbiage such as ‘a lot more’ might be seen as only something like 2-3 times; I’d probably favor something like ‘vastly more’ as an alternative).
July 2, 2010, 1:18 pmPatent Lawyer says:
In law school, I took a class on Supreme Court litigation where I had to argue for the parents in the Parents Involved in Community Schools v. Seattle School Dist. No. 1 case (which concerned remedial racial preferences in school districting.) One of the anticipated questions was, “If the school district simply used a random lottery, couldn’t that lead to segregated schools by random chance?” Unfortunately, the simplest response, “That would be a 12 sigma event”, was unavailable to me for the reasons described in this post. :)
July 2, 2010, 1:19 pmDavid Welker says:
To expand on this, I would say “of consequence” at that point in the discussion. Maybe the difference is known and does matter, but maybe you just don’t want to talk about it right now.
Also, I would object to the idea that one could say “greater than three orders of magnitude” when something is only 999 time larger. So, in this sense, I would say that in this context, greater than three orders of magnitude does have a precise meaning of greater than 1000.
July 2, 2010, 1:21 pmDan Weber says:
Don’t worry, the other half understand completely.
July 2, 2010, 1:40 pmUrso says:
How many mathemeticians can explain what consideration is — a “basic and fundamental” legal concept if ever there was one. But, rightly, no one cares, because that’s not the kind of thing mathemeticians are supposed to know.
July 2, 2010, 1:46 pmRoger says:
Your experiment proves nothing. Since both the brief and the court opinion included the numbers, it is possible that all the readers understood the phrase “orders of magnitude” accurately in the context where it was actually used.
July 2, 2010, 1:46 pmFub says:
I could believe the story if it actually came from a percipient witness. But I don’t find it particularly plausible on its face alone as an apocryphal story.
I could believe Groves (or any civil engineer at the time) would have been surprised to learn that they were initially working with such broad estimates. But I find it difficult to believe that he (or any other civil engineer) “didn’t know what they were talking about” when they mentioned “orders of magnitude”.
The story sounds like a highly garbled variant of the snarky interdisciplinary jokes one sometimes hears among technically trained people, similar to “banjo player jokes”, “viola player jokes” among musicians.
Say, something along the lines of “We’re physicists. We’re building a bigger bomb. He’s a civil engineer. He only knows how to build bigger targets.”
July 2, 2010, 1:47 pmJim says:
No. Like was said earlier, order of magnitude implies approximation. 40 is an order of magnitude larger than 5.
July 2, 2010, 1:50 pmD.R.M. says:
Heh. However, at least then you can easily substitute something like, “The odds are one in ” . . . hell, what is the chance of a twelve-sigma event? We’re talking “If they held a lottery every second from the Big Bang to today, it still wouldn’t have happened”, right?
July 2, 2010, 1:53 pmPierre Corneille says:
I’m disinterested in such nitpicking, and see it all as a moot point that’s not up for discussion anymore. Hopefully in future posts, I’ll look for better instantiations of innumeracy amongst the general populace, irregardless, or not, of how impactful it is.
July 2, 2010, 1:54 pmDairyqueen says:
I think this is a general point that carries a lot further than simply “orders of magnitude.” There are multiple times when I read Supreme Court cases, where I don’t know the word and have to look it up. Sometimes I find it incredibly impressive that the justices (or their clerks) can find words or expressions that are relatively unusual yet pass perfectly with the expression they are trying to get across. On the other hand, why would you use the word when there is an easier and more effective way to get a communication across? Our justices shouldn’t be striving to write poetically – though its nice when they do – if 40% of lawyers don’t understand what they are saying… It’s not a writing contest (or a math contest for that matter), its a judgment.
Same goes with much of the contractual work I have come across. Yes, with a significant amount of time and effort, anyone can understand a 90 page SPA agreement. But, why should it be so (generally unnecessarily) complicated in the first place.
July 2, 2010, 1:56 pmDairyqueen says:
@Patent Lawyer “If the school district simply used a random lottery, couldn’t that lead to segregated schools by random chance?” Unfortunately, the simplest response, “That would be a 12 sigma event”, was unavailable to me for the reasons described in this post. :)”
but, you could have said that would be like a monkey writing Shakespeare.
July 2, 2010, 2:01 pmSeaDrive says:
I wouldn’t use percent in that example either, as in sales of x are 100000% bigger than sales of y.
July 2, 2010, 2:05 pmSmooth, like a Rhapsody says:
Thanks for making me look up the definition of a “12 sigma event”; I have become less innumerate today.
July 2, 2010, 2:05 pmDavid Welker says:
I disagree. You could not say that something is more than three orders of magnitude when it is less than 1 x 10^3 larger. At this point, I would say the statement is incorrect.
July 2, 2010, 2:06 pmRV says:
I agree that is a language, not a math, question. In my experience, lawyers, except in their own minds, are generally not masters of language. They too frequently use words improperly or
I’ll take that bet, even without mandating that we remove small children from the pool. Sure, there are plenty of incompetent people out there, but I’d bet that saying that over 40% of them can’t solve 3×3 or read a clock is just madness. I’d (more or less randomly) put the over-under at around 12% being unable to answer one or both questions, if we limit it to 10-39 year olds.
July 2, 2010, 2:07 pmL Nettles says:
Blaming humans for global warming is an orders of magnitude problem.
July 2, 2010, 2:12 pmPatent Lawyer says:
Something like that. The question was fairly absurd, and could only have been asked by an innumerate judge. But as the judges were my fellow law students, that was certainly possible.
July 2, 2010, 2:12 pmbyomtov says:
AJK,
What alternate usage would you suggest? “A lot”?
Roberts could have written,
The demand for hunting depictions is more than 100 times estimated demand for crush videos or animal fighting depictions.
Though as pointed out in the other thread, the use of “demand” in this sentence is technically incorrect. Better would be:
Sales of hunting depictions are than 100 times estimated sales of crush videos or animal fighting depictions.
July 2, 2010, 2:15 pmDairyqueen says:
Really RV?
http://www.youtube.com/watch?v=lbm4mAilZZk&feature=related
July 2, 2010, 2:17 pmgrrzzly says:
I have to admit that it drives me nuts to read this:
Especially, from Prof. Volokh himself, whom I always considered very well educated. Have you ever heard about the Galilean principle of relativity? It’s just as correct to say that the Sun orbits the Earth from the point of view of a person located on the Earth as it is to say that the Earth orbits the Sun from the point of view of the Sun. There could be good reasons to prefer the latter point of view, but to dismiss the former one as ignorant only betrays your own profound ignorance of physics.
And then the same people pretend to have an opinion on non-trivial scientific concepts, such as, say, the AGW. Oh, the horror! Get me out of here!
July 2, 2010, 2:20 pmDG says:
If we are going to insist that lawyers have a four year degree before attending law school, than shouldn’t we also insist that they learn some useful information during that period? No one gets into medical school without having taken (and done well in) organic chemistry, for example.
I don’t think lawyers need calculus. But I do think that they all need a good statistics and probability class. Stats/Prob is immensely useful to lawyers in many areas and a good understanding of the subject is quite valuable to judges.
How many lawyers take such a class during their undergrad?
July 2, 2010, 2:29 pmGuest12345 says:
There certainly is a difference. A huge difference.
1.000 x 10^3 is one order of magnitude larger than 9.990 x 10^2. However the difference between those two numbers is 1. A spoken multiplier would be “1000 is 1.001 times larger than 999.”
Consider:
1.0000 x 10^4 compared to 9.9900 x 10^2. Two orders of magnitude, 10.01001 times larger.
1.00000 x 10^5 compared to 9.99000 x 10^2. Three orders of magnitude, 100.1001 times larger.
Orders of magnitude are only a matter of exponent. Mantissa is irrelevant.
July 2, 2010, 2:35 pmDavid Welker says:
Guest12345,
To say that 1000 is one order magnitude larger than 999 I think would have a tendency to deceive. Something that is approximately one magnitude larger should be approximately 10 times as large. I am not sure if it is technically correct to say that 1000 is one order of magnitude larger than 999 or not, but I do know that it would be confusing.
Imagine you know:
X = 1000
Y = 999
And you say to someone, X is an order of magnitude larger than Y, I think you are nearly guaranteed to give that person the wrong idea about the relative sizes of X and Y. So, even if this were technically correct (and I am not sure that it is) it would be an unwise usage.
July 2, 2010, 2:44 pmhattio says:
DG says;
DG, you’re correct of course. The problem is that a basic course in English Writing, science courses, biology courses, history courses, psychology courses, sociology courses, physics, and about 100 others would be useful as well. You can’t mandate all the useful classes.
July 2, 2010, 2:44 pmPraetorius says:
I’m pretty sure we covered the concept of ‘magnitude’ in 7th or 8th grade arithmetic…not anything advanced.
And I’ve been in Burbank when Leno’s “Man on the Street” interviews were conducted: They go through a LOT of people with intelligent answers to find those morons or whores (who will act like a moron to get on TV) they publicize.
July 2, 2010, 2:45 pmRandom Wine Geek says:
(emphasis added)
Another possibility is that respondents to the poll followed Professor Volokh’s instructions when responding to the poll, which neither of your snarky conclusions take into account:
In this case, the “right” answer was the honest one, not the correct one. For respondents like me, who knew that our “right” answer was also almost certainly incorrect, the incentives were to either give the “wrong,” but correct answer, or to ignore the poll. I know the contempt I’ve felt for that nebulous group of poll respondents who can’t give the right answer to questions so basic they shouldn’t challenge the abilities of a poorly-trained Labrador retriever. My pride recoiled at the thought of voluntarily joining their ranks. And I suspect that a number of other readers shared my own irrational discomfort with this poll. But in the end, I decided to swallow my pride and provide accurate data to Professor Volokh.
Immediately thereafter I looked up the correct answer, then spent the evening reading the 704(b) regulations as penance for my ignorance.
July 2, 2010, 2:47 pmdclawyer says:
There’s also the likelihood that many of the people who got it correctly may have been guessing (or using some educated guessing framework). I didn’t know the technical meaning of the phrase, but I arrived at the correct answer apparently. And I figured that because there was no penalty for guessing, why not!
My (tortured) reasoning was fairly simple…demand for A exceeds demand for B by nearly three orders of magnitude, so 3 had to factor into the answer somehow. If it was simply that A was three times more in demand than B, I couldn’t imagine someone not just saying that. So I didn’t think that was the answer. The next most likely option I came up with was that the three was a multiplying factor. The word “magnitude” reminded me of magnifying glasses, which multiply the object by a certain amount. So, if something is going to be 3 magnifying factors bigger, how about it being 10 ^ 3 times the original?
By that logic, I probably could have talked myself into quite a few wrong answers. I doubt I was the only one using somewhat educated guesswork!
July 2, 2010, 2:57 pmSeaDrive says:
No, it’s not. The principle of Galilean invariance extends to only observation, and not all science is observation.
July 2, 2010, 2:59 pmClayton E. Cramer says:
Here’s a variant of the tale that indicates Grove did know what “order of magnitude” meant–he was just shocked at how uncertain they were.
July 2, 2010, 3:00 pmgrrizzly says:
Move to strike: non-responsive.
July 2, 2010, 3:11 pmCrunchy Frog says:
Monkeys don’t do iambic pentameter.
July 2, 2010, 3:12 pmCrunchy Frog says:
Bad example, as only 5 out of 9 SCOTUS Justices can understand what a “basic and fundamental” legal concept is.
July 2, 2010, 3:20 pmRoscoe says:
Prior to reading this post, I didn’t know the definition of the term “orders of magnitude,” but had a general understanding of it being “a really big number.” And if I used this term in a brief, I would expect the judge to know I was trying to convey “a really big number” even if he or she didn’t know the precise number.
Obviously, if I had the actual numbers on a subject, I would prefer to use them. But if I don’t, I think the term is a useful and elegant way of saying “a really big number.”
That being said, I am given a bit of pause by the fact that 18% of the survey respondents thought three orders of magnitude meant 3 or 4.
July 2, 2010, 3:22 pmsardonic_sob says:
It’s fun to say this to computer programmers:
“You know how you can tell the difference between a good random number generator and a bad one?”
(They hem and haw.)
“You can’t!”
This of course is not true at all, but somebody who understands how to call a random function (or bet on a roulette wheel) but doesn’t understand how random numbers are generated by computers or the concept of “randomness” in general will usually buy it. Hilarity ensues.
However, it should be noted that somebody who *does* understand these things can still quite easily lead themselves astray believing they understand their inputs as well as they understand their calculations. (“The map is not the territory.”) Wasn’t the sequence of events that lead to the downfall of LTCM something like a nine-sigma in their model? What would a “reasonable” economist, even one who understood statistics, have said in 2005 was the likelihood of AIG, GM and Lehman Brothers going broke, or the entire US housing market going kerblooie and forcing the Fed and the USG to spend a zillion dollars in less than two years? Or an EMU member being in danger of imminent default on its debts and throwing the global financial system into a paranoid frenzy?
In fact, I would say one of the gravest dangers of innumeracy is that people who don’t know how to think logically and follow basic math are even more easily misled by those either equally ignorant or actively malevolent than people usually are by leaders and authorities.
July 2, 2010, 3:25 pmDairyqueen says:
It can reasonably be questioned why we insist on that 4 year undergrad degree in the first place.
July 2, 2010, 3:27 pmsardonic_sob says:
It’s just as correct because, if you’re going to pick nits, they’re both incorrect. However, saying “the Earth orbits the sun” is much more accurate relative to what is actually happening than saying “the Sun orbits the Earth,” no matter where you’re standing. Funny concept, bad execution. Four out of ten.
July 2, 2010, 3:29 pmgrrizzly says:
So, what’s actually happening? Where’s the center of the universe?
I was talking about a frame of reference. In a frame of reference where I’m at the origin, the Sun orbits me every single day.
It’s not the end of the world not to have a decent education in science. But then it’s better not to embarrass yourself publicly, while pontificating on scientific ignorance.
July 2, 2010, 3:52 pmBlah says:
Reminds me of the discussion of “orthogonal” in the oral arguments earlier in the term (VC post here: http://volokh.com/2010/01/11/orthogonal-ooh/.
Given that the appendix in Barber v. Thomas in OT09 contained a rather long explanation of how to calculate prison time by solving a system of simple linear equations with 2 variables, perhaps it’s time for a math section on the bar exam?
July 2, 2010, 3:59 pmDavid P says:
I always understood it as a 10^x ratio. But I did my undergrad in engineering, and am not surprised that some people don’t really understand it.
July 2, 2010, 4:02 pmlucia says:
I knew the results weren’t going to be flattering to lawyers when I read the first comment on the post requesting the survey. It asked you to clarify your perfectly clear question.
July 2, 2010, 4:08 pmDG says:
{DG, you’re correct of course. The problem is that a basic course in English Writing, science courses, biology courses, history courses, psychology courses, sociology courses, physics, and about 100 others would be useful as well. You can’t mandate all the useful classes.}
Almost all liberal arts programs have a basic math/science requirement. Simply steer pre-law kids into taking stats instead of something both useless and more difficult like calculus.
I don’t think physics is particularly useful, nor is sociology (worthless in general). Psychology, history, and biology are so obviously useful that they should be part of pre-law by definition. College-level composition courses are useful, but literature courses are much less so. Simply by suggesting which courses that pre-laws use to meet existing requirements would help.
July 2, 2010, 4:10 pmUrso says:
It must be a terrible burden on you to have to constantly suffer the existence of fools.
July 2, 2010, 4:11 pmGuest12345 says:
I certainly agree that it can be confusing, but it’s not wrong. I didn’t answer the original post, but if I had then it would have been this:
As an aside, I find it sad that language is becoming so imprecise in general usage.
July 2, 2010, 4:11 pmerp says:
Praetorius, order of magnitude may have been discussed in 7th or 8th grade classes back when, but I don’t think it’s true today.
RV, I forgot the s/off at the end of my comment, although I’m no longer surprised by the level of ignorance displayed by people who attended our public schools since the teachers unions took them over. A recent high graduate who was doing some yard work for us couldn’t tell me how much I owed him for eight hours work at $8.00/hr. On line at the supermarket, a well-dressed attractive woman of about 40 admired my watch. When I told her I got it locally for a modest price, she said she couldn’t use one like it because she doesn’t know how to tell analog time!
I do have a question. What’s a 12 sigma event?
July 2, 2010, 4:13 pmJmaie says:
GUEST12345 says:
1.000 x 10^3 is one order of magnitude larger than 9.990 x 10^2. However the difference between those two numbers is 1. A spoken multiplier would be “1000 is 1.001 times larger than 999.”
As soon as the multiplier changes, you’re comparing apples and oranges. If your assertion was correct you could claim, for example, 1000 x 10^2 is an order of magnitude smaller than 1 x 10^3.
July 2, 2010, 4:21 pmM. Gross says:
I know there’s no point complaining of pedants on a law blog, but the commentors really outdid themselves this time.
July 2, 2010, 4:29 pmSeanF says:
And here I would say the opposite. The implication of “three orders of magnitude” is, “You’re not only wrong, you’re wronger than wrong. In fact, you’re wronger than wronger than wrong.” :)
What about 1.000×10^3 and 9.999…x10^2?
July 2, 2010, 4:36 pmKMK says:
I got the answer wrong because of my ignorance of the term. After looking it up, I quickly discovered the right answer. I think it was kind of silly of you to ask people not to look up the term. Any good lawyer would look up an unfamiliar term. Indeed, being able to effectively research (and quickly understand your research results) is an integral part of being a good lawyer. So maybe your survey showed that some VC reading lawyers did not immediately understand the term, but it hardly shows that lawyers cannot comprehend math. I think the more interesting question is whether a lawyer – with an opportunity to research the math question – could get the answer right.
July 2, 2010, 4:37 pmBob says:
Only a third? I’d have guessed many more. I’ve been deposed on a number or occasions in things environmental. The lack of scientific knowledge and even the ability to correctly pronounce chemical names is painful, especially to a chemist. I’d like to laugh but they tend to get a bit testy.
July 2, 2010, 4:39 pmcboldt says:
– What’s a 12 sigma event? –
July 2, 2010, 4:40 pmBackground: “Sigma” is a measure of variability in a normal distribution, aka “bell curve.” Sigma is also referred to as the “standard deviation.” If one takes the entire population of “things” or “events,” there will be an average, and some individuals above, others below the average. When the spread between largest and smallest is wide, sigma (for the entire population) is larger.
A twelve sigma event is the area under the bell curve. One way to view it would be starting at a point that is 12 sigma distant from the average. Another would be to take plus or minus 6 sigma from the mean, and find the area beyond those +/- 6 sigma points. The area is radically different (but small either way) depending on which way one applies 12 sigma. In a perfect normal distribution, 0 sigma covers half of the population, either to the right or to the left of the average. Use a “Z table” to look up the area for other values of sigma, but most “Z tables” don’t go as high as 6 sigma, let alone 12.
I found one source that says the probability that a given event/item is 12 sigma or more removed from the average is about 2 in 10^33.
cboldt says:
– What about 1.000×10^3 and 9.999…x10^2? –
July 2, 2010, 4:42 pmGreat question. They are exactly the same number. EXACT.
Howard says:
Robert Heinlein has a discussion in one of his books about ‘on the order of x”, vs “on the close order of x. I’d be surprised if anywhere close to 2/3 of any group (except maybe engineers) understood that distinction.
July 2, 2010, 4:43 pmbyomtov says:
Given that the appendix in Barber v. Thomas in OT09 contained a rather long explanation of how to calculate prison time by solving a system of simple linear equations with 2 variables, perhaps it’s time for a math section on the bar exam?
Actually, I once did some work on this issue – not the Barber case – and the contortions the Bureau of Prisons goes through to do the calculation are amazing. This is not just a pedantic gripe. Using BOP’s procedure it’s easy to make a mistake, which is of course much less likely with a simpler method.
July 2, 2010, 4:54 pmTTT says:
In defense of lawyers:
1) It looks like almost all of us followed directions by giving 1 integer. Lawyers are good at following directions.
2) I’d wager that almost all of us knew this was a definition question. Lawyers are good at reading questions.
3) I’d wager that of those who were wrong, most of us knew we did not know (for certain) when we answered. That fact was implied by your asking the question and by your asking us to guess. Lawyers are good at understanding implications.
July 2, 2010, 5:30 pmarch1 says:
Eugene, while you may ‘avoid’ short term, I urge you to teach long-term.
Not having ‘order of magnitude’ (and the back-of-envelope-estimation mindset that it facilitates/signifies) in one’s mental toolkit is IMO a handicap of the same order of magnitude as not knowing the difference between a necessary and a sufficient condition, or having a basic grasp of conditional probability or of evolutionary theory. These concepts may all have originated on the math/sci/engineering side of the cultural divide, but their utility is universal.
Guest12345, the recent objections to your ‘exponent only’ interpretation of ‘order of magnitude’ can be addressed by requiring that the mantissa be normalized to the range [1..10). But it’s still IMO a silly interpretation. For instance, 100.0000001 and 99.9999999 meters differ by 1 order of magnitude using your interpretation; but if you instead measure the same two distances in feet, they will suddenly be the SAME order of magnitude, though their ratio has changed not a whit. This is (gratuitously) wacky.
July 2, 2010, 5:37 pmDaniel says:
Having reconsidered, I believe an an order of magnitude is 8. That’s the point where a restaurant automatically includes the gratuity. Any smaller order is not considered is not of sufficient magnitude to enforce standard tipping protocols.
July 2, 2010, 5:43 pmarch1 says:
I would have called 8 the tipping point.
July 2, 2010, 5:53 pmAnatid says:
Psychology agrees with you. The subconscious “feel” refers to a concept known as heuristics, the largely-subconscious processes by which we determine what “seems” right.
It depends on your school – what neighborhood it’s in, who the teachers are, how much funding it receives, what state it’s in. If you sent your kid to the same primary school I went to, you’d still find excellence in education.
On the other hand, if you’ve taken a look at some of the science textbooks that Texas has issued recently …
July 2, 2010, 5:54 pmkarrde says:
I presume that there is a way to measure the entropy (as defined by Claude Shannon) of the stream of random numbers produced by the RNG.
(Caveat: I have received a Master’s in Math, and have done a great deal of work in Computer Software and Computer Engineering…but I never had to take a class in which learning how RNG’s worked was essential to understanding the class. Thus, I haven’t learned this subject yet.)
With respect to “orders of magnitude”, I knew the answer when I read the problem, but I cannot recall learning the answer. It must have seeped into my mind during pre-Algebra or Algebra, and remained firmly fixed.
I agree with the rest of your statement, that innumerate or barely-numerate people are much more easily misled by malevolent (or even well-intentioned-but-misguided) people.
July 2, 2010, 5:55 pmDaniel says:
I concede. You win.
July 2, 2010, 5:58 pmFub says:
Thanks for that. It also had some interesting details that are consistent with popularly circulated tales about the various physicists’ personalities, as well as Grove’s.
July 2, 2010, 6:21 pmCan't find a good name says:
I’m pleased to see that John Roberts does appear to know what an order of magnitude is. I had suspected that Eugene posed the original question because some lawyer had used the term incorrectly, but that turned out not to be the case.
While there is no doubt potential for confusion when the phrase “order of magnitude” is used, Chief Justice Roberts did significantly reduce the chance of confusion by providing the actual numbers he was referring to in the next sentence.
July 2, 2010, 6:23 pmCiarand Denlane says:
I would have guessed 16. I’m a lawyer, but my first childhood exposure to “magnitude” was stellar magnitude, and I guess that meaning must have stuck,
July 2, 2010, 6:27 pmElliot says:
Maybe the answer, as in so many written communications, is to use simple, clear, and accurate language. I don’t see how using “order of magnitude” rather than “a lot” or “far greater” added anything to the opinion. I’d suggest it detracted from the opinion. It just limited the size of the audience that could understand it.
If people feel a need to flex their literary muscles, they can write poetry.
July 2, 2010, 6:50 pmGuest12345 says:
Indeed. Which is EV’s point. Orders of magnitude don’t tell you much about the ratio of the values in question.
In scientific notation you won’t have 1000 x 10^2, you’d have 1.000 x 10^5. As has been mentioned, the mantissa is normalized to the range [1,10).
July 2, 2010, 7:21 pmMichelle Dulak Thomson says:
Urso,
How many mathemeticians can explain what consideration is — a “basic and fundamental” legal concept if ever there was one. But, rightly, no one cares, because that’s not the kind of thing mathemeticians are supposed to know.
IANAM (“mathematician,” as we spell it hereabouts). IANAL either, but I know what is meant by “consideration,” just as I know what is meant by “order of magnitude.” Strange that either thing should be thought of as esoteric knowledge.
July 2, 2010, 7:42 pmsardonic_sob says:
Right back at you: Which universe?
In any event, you know as well as I do that what is really happening – and can be observed readily – is that both bodies are orbiting their common center of mass. Of course, the fact that this isn’t a two-body problem makes the observation and the deduction far more complicated. But while you can make geocentric epicycles to any degree of accuracy you want, they’ll never anticipate the observed motion of the solar system as well as a heliocentric (or better yet, system barycentric) model will. Relativity is all well and good but any theory must match observable reality or… well, it’s not a good theory.
It’s kind of analogous to cultural relativity, in a certain sense. Without proof of the divine inspiration/mandate of any particular cultural practice, it’s not really possible to say that one is indisputably “better,” but there are clearly systems which work much better than other systems. It may be culturally “chauvinistic” to make such a claim, but if it matches observable reality, then the fact that it may not be “polite” doesn’t make it not true.
July 2, 2010, 8:35 pmAnatid says:
But again, this is relative. A culture that is adaptive to one environment may not be adaptive to another environment. What works in a region of low population and limited resources probably won’t work in a region of dense population and abundant resources, and neither will be suitable for a region of dense population and limited resources (just to pick two of many variables).
July 2, 2010, 8:47 pmModerate says:
This thread has gone orthogonal.
July 2, 2010, 9:47 pmPaul Allen says:
Lawyers should not say 1000 times greater as a reinterpretation of three orders of magnitude. For instance, if something is 5000 times greater, three orders of magnitude is still correct. Therefore translating three orders of magnitude into 1000 is very misleading as to the specificity of the statement.
Eugene, your advice is horribly dangerous.
July 3, 2010, 12:00 amMark Horning says:
Not. Even. Close.
1 x 10^3 = 1 x 10^3
9.990 x 10^2 = 1 x 10^2.99956…
The difference is approximately 5-thousandths of an order of magnitude.
An “order of magnitude” = “roughly a factor of 10. You will even hear scientists and engineers say “half an order of magnitude” which = “roughly a factor of 3″ (or Pi, it’s technically the square root of 10, which is close enough to Pi or 3 for most scientists to not worry about).
July 3, 2010, 12:27 amDavid-2 says:
It depends on the context – specifically, the number system in use. In a lot of computer science, orders of magnitude are binary orders of magnitude – therefore 3 orders of magnitude would be 2^3 or a factor of 8.
Since the consensus of the comments here is that lawyers are innumerate, we must assume they’re only capable of counting in unary (one and one and one and one …). Therefore 3 orders of magnitude would be 1^3, or 1. So that to a lawyer, any difference in order of magnitude is no difference at all.
July 3, 2010, 1:13 amNorCal Dave says:
But that’s not at all what the Principle of Galilean Relativity says. It says the laws of physics are the same in all inertial reference frames.
July 3, 2010, 1:20 amNorCal Dave says:
This is not really the central issue, but it is bad form to say A “exceeds” B by orders of magnitude. He should have said A “is” orders of magnitude “higher” than B.
You can say 5000 is an order of magnitude higher than 500. But 5000 exceeds 500 by 4500. 4500 is not an order of magnitude.
July 3, 2010, 1:41 amGuest12345 says:
A number’s magnitude is not the log of that number, it is the integral portion of the log. In a decimal magnitude system, the magnitude is the integral portion of log base 10. Since log10(999) = 2.99956, the magnitude of 999 is 2. Ranking the magnitudes into orders involves a fixed ratio between adjacent classes.
July 3, 2010, 2:21 amGuest12345 says:
Not so. A multiplier of 5000 is only three orders of magnitude larger than numbers in the range [1,2), [10,20), [100,200), etc.
1 = 1×10^0 & (5000 x 1) = 5×10^3
July 3, 2010, 2:33 am3 = 3×10^0 & (5000 x 3) = 1.5×10^4
bbbeard says:
I am so disappointed in you, Eugene.
As several commenters have noted, “factor of 1000″ and “three orders of magnitude” do not mean the same thing. And I have to say, despite your BS in math, I’m not convinced you understand the difference. Mark Horning actually had a better handle on it when he said
The point is that “order of magnitude” does not merely mean that an approximation is involved, such as a case where the ratio of A to B is, say, 4.7 +/- 1.8. There are times when the important quantitative fact is that the ratio of A to B is roughly 10, or perhaps 1000, or a billion. It may be the case that the factor is known to high precision, for example, the ratio of the mass of the proton to the mass of the electron is 1836.1526724718(80) — but the important thing for us is that m_p is three orders of magnitude larger than m_e, not two or four, and the world would be very different if that were not the case. It may be that some ratio is not known very precisely — the Higgs mass is believed to be a couple of orders of magnitude larger than the proton’s, but very few model-independent statements beyond that can be made. The fact that it is two orders of magnitude, probably not three or more, is significant — indeed, billions of dollars are being spent on particle accelerators by physicists and governments betting on that order of magnitude.
This is not just a ‘terminological’ question. “Order of magnitude” expresses a concept fundamentally different from “factor of”. And yes, it’s more subtle — you learn about “factors” in second grade and about “orders of magnitude” in fifth grade. The fact that 1/3 of over 1000 law student respondents did not understand the meaning of this elementary concept is (a) disappointing, and (b) not surprising. But I would suggest that, before banishing an intellectual concept from the lawyer’s meager toolkit, some more careful consideration be given to other mitigations for legal innumeracy.
And I have to add, as someone who literally designs rockets for a living, that I cannot begin to express the frustration of living under the rule of people who cannot even handle elementary mathematics.
July 3, 2010, 4:16 amGeeky language meets legal realism: Guess who wins? | The Robb Report says:
[...] Advice to Lawyers: Stop Talking About Orders of Magnitude [...]
July 3, 2010, 4:24 ambbbeard says:
That is not the sense in which Justice Roberts was using the term, nor is it the sense in which it is being discussed in this thread. In particular, in this case it is the order of magnitude of a ratio that is under consideration. “Three orders of magnitude” literally encompasses factors from 100*sqrt(10) to 1000*sqrt(10). 999 is most certainly “three orders of magnitude”. So are 500, and 2000.
July 3, 2010, 4:24 amS says:
His advice is to say 1000x greater when you are talking about 1000 times greater (5000x when you are talking about 5000 times) and avoid orders of magnitude altogether. If you don’t understand his advice, it’s a good thing you are not a lawyer or perhaps you’re dangerous.
July 3, 2010, 4:27 ambbbeard says:
Heh. Forget Galilean invariance. General relativity tells us that not only are there no preferred frames, there are no preferred coordinate systems. A coordinate system tied to the Earth’s crust can be used to solve physical equations just as legitimately as a coordinate system tied to the Sun. It is a little-remarked irony that when Einstein formulated general relativity he overthrew not just Newton, but Copernicus as well….
July 3, 2010, 4:33 amPhilC says:
Well, as you are ‘living under their rule’, it shows what an useful idiot you are.
July 3, 2010, 4:38 ambbbeard says:
Oh, right, because it would be pointless to have lawyers understand difficult and useless concepts like “rate of change” and “exponential growth”.
July 3, 2010, 4:43 ambbbeard says:
No experiment should be believed until it is confirmed by theory. ;-)
See previous comment about general relativity. The point is that we can do physics in any coordinate system, not just ones tied to the Sun.
July 3, 2010, 4:55 ambbbeard says:
I would have said it has become chaotic — because it is evidently characterized by strange attractors.
July 3, 2010, 4:57 ambbbeard says:
Actually, 5000 is closer to four orders of magnitude — it’s only a factor of 2 away from 10^4 but a factor of 5 from 10^3. That is, the log to the base 10 of 5000 is about 3.6990, which is closer to 4 than 3.
July 3, 2010, 5:08 amlonetown says:
It depends on whether your are talking dollars or earthquakes.
July 3, 2010, 8:23 amPierre Corneille says:
Yeah, it gets exponentially more frustrating everyday, by a factor of 2!
July 3, 2010, 9:14 amRoger Zimmerman says:
What is more, in computer science, where base 2 rules, an OOM corresponds to roughly a doubling (or halving). Since legal writing should be largely domain agnostic, I would tend to agree with EV’s prescription.
July 3, 2010, 10:19 ambbbeard says:
Eugene: Perhaps I could put it this way: suppose you found out that 1/4 of all lawyers do not know what an “uncertainty interval” is. Would you then counsel lawyers to avoid using uncertainties in legal arguments? E.g.: “If a particular quantity is believed to be, e.g. 800+/-200 pounds, just say ‘one thousand pounds’”?
July 3, 2010, 10:46 ambyomtov says:
How many mathemeticians can explain what consideration is — a “basic and fundamental” legal concept if ever there was one. But, rightly, no one cares, because that’s not the kind of thing mathemeticians are supposed to know.
Mathematicians don’t offer professional advice on legal matters. Lawyers do often give professional advice on matters that involve, if not higher mathematics, at least basic and sometimes involved calculations.
July 3, 2010, 12:54 pmMichelle Dulak Thomson says:
Not to mention that a fair number of mathematicians probably do know what “consideration” means. For that matter, all jokes about mathematicians’ illiteracy aside, most mathematicians can actually spell “mathematician.” Urso is likely not a mathematician.
July 3, 2010, 1:19 pmBill Woods says:
No, it is not correct to say the Sun orbits the Earth. If you want to be persnickety, you could say it’s not correct to say the Earth orbits around the Sun either; rather both orbit around the Solar System’s barycenter. But the difference would only matter if you’re an alien astronomer wondering if the Sun has planets.
July 3, 2010, 1:39 pmKen Mitchell says:
Precisely. “Three orders of magnitude larger” implies somewhere between 500 times larger and 5,000 times larger. If you meant “something close to 1000 times larger”, then you need to SAY that.
July 3, 2010, 1:51 pmKen Mitchell says:
Interesting sidenote: I used to teach celestial navigation for aircraft in the Navy. Trying to (manually!) do the math for cel nav is simple if you assume a geocentric universe, and devilishly complicated otherwise.
July 3, 2010, 1:58 pmjab says:
Guest, you are quite wrong in this, at least how mathematicians and scientists use the phrase.
Before you compare exponents, you must round off both numbers to the same number of significant digits (usually just 1 significant figure). Then you compare exponents. Order of magnitude is strictly used to illustrate relative magnitude or size. The way you have defined it obscures that fact.
So if you are comparing 1000 to 999, round them off to the same significant digit:
1000 –> 1000 –> 1 x 10^3
999 –> 1000 –> 1 x 10^3
999 and 1000 are the SAME order of magnitude.
For the record, I have a PhD in astrophysics, and an a university professor. (Not that we can’t be wrong, but on this, I am 100% certain).
July 3, 2010, 3:02 pmjab says:
Roger… true… but 2 written in binary is 10.
July 3, 2010, 3:06 pmSo order of magnitude in binary is still 10 (it’s just that 10 in binary means 2 in decmal!)
jab says:
I need to correct myself… I did not mean to say round off to the same number of significant figures… but I meant to say round off to the nearest power of ten.
The other way to to this is using base 10 logarithms…
July 3, 2010, 3:13 pmlog_10 (1000) = 3
log_10 (999) = 2.9995 –> 3
Same order of magnitude
jmaie says:
Guest12345 is working under the theory that 2.9995 truncates to 2 for the purpose of this calculation.
July 3, 2010, 4:03 pmCornellian says:
I know what “orders of magnitude” means but I wouldn’t use that term in a legal brief. The risk is too high that the reader won’t know what you mean or, worse yet, assume he knows when he really doesn’t.
July 3, 2010, 4:23 pmPaul Allen says:
You’re not thinking things through. There is the case when the source material says x orders of magnitude. Your lawyer should not translate that into 1000x.
Strictly preferring Nx can lead to misstatements of fact.
July 3, 2010, 4:45 pmbbbeard says:
I’m not ‘arguing from authority’, but since there are a significant number of professionals here who seem to be unable to convince you of your mistake in comprehension, perhaps you could do some googling and report back to us with a link supporting your point of view.
July 3, 2010, 5:08 pmbbbeard says:
Oh, quite. And also a separate issue… There are coordinate systems that can make a given problem trivial. But it is a fallacy to think that because a coordinate system makes things easy to solve, that the coordinates are “real”. And, fundamentally, the argument about whether the Earth goes around the Sun or vice versa is about a choice of coordinate systems.
July 3, 2010, 5:16 pmlucia says:
The second story makes orders of magnitude more sense. After all, engineers used slide rules back in the day; their use requires performing side calculations to keep track of the order or magnitude. So, it really makes no sense to suggest a civil engineer would not be familiar with the term order of magnitude. (My father in law– still alive at 94– is a civil engineer. He knows what an order of magnitude means.)
Also, whether or not Groves was shocked to discover how uncertain the physicists estimates were, the fact that the uncertainty was so high was going to present a very serious challenge to people whose job it was to bring into being industrial type facilities to create sufficient fissile materials to make bombs. Ordinarily, whether the engineer in charge understands nuclear physics or not, he would like to know if he needs to build a plant to manufacture 5lbs, 50lbs or 500lbs of material. More perplexing for Groves would be the fact that there could also exist large uncertainties in whether process for making material would scale up.
Were it not for the pressures of the war, it’s quite likely reasonable people would have deferred the task Groves was charged with until research had reduced the uncertainty in the amount of fissile material required for a bomb had been reduced to substantially less than an order of magnitude.
July 3, 2010, 5:26 pmbbbeard says:
That’s getting at the philosophical issue here, I think. At what points does simplification for the purpose of communication cross the line into falsehood? Eugene was careful when he wrote
But it takes only a slight tweak to this formulation to cross the afore-mentioned line. What if what you mean to say is “greater by three orders of magnitude” but not necessarily “more than 1000 times greater”? If X is 800 or 900 times Y, then it would be true to say that “X is greater than Y by three orders of magnitude” but it would be false to say “X is a thousand times greater than Y”.
And I don’t think the answer to that dilemma is to back down the factor until the statement becomes true enough for the court system. It could be that the uncertainties are poorly characterized, for example, or that fluctuation in the ratio over time forces unnecessary specificity. For example, for the last forty years, NASA’s budget has been three orders of magnitude less than the GDP, although it was higher in the 1960′s. That fact alone can be significant in certain contexts, depending on what it being discussed. Does it really clarify things to say, “Since 1970, the GDP has been more than 333 times NASA’s budget” or “Since 1970, NASA’s budget has been smaller than the GDP by a factor between 333 and 888″?
July 3, 2010, 5:49 pmcboldt says:
– After all, engineers used slide rules back in the day; their use requires performing side calculations to keep track of the order or magnitude. –
July 3, 2010, 6:57 pmBeing off by an order of magnitude (or more than one) has a definite, precise meaning in that context. Even if the significant digits are correct, the error in order of magnitude is exact, a factor of 10 to some (positive or negative) power.
Andy Skelton says:
More reasons to avoid “order of magnitude”.
July 4, 2010, 11:12 amToby says:
Unless it happened the very first time. Perhaps “it probably still wouldn’t have happened”
July 6, 2010, 4:32 pm