One More Example:

Say that someone describes some amount as "100 plus or minus 5." Would you literally interpret this as "100 plus 5 or 100 minus 5," which is to say "105 or 95"? Or would you recognize that it's an idiom -- referring to the mathematical ± symbol, which is itself literally not the same as its components (plus and minus) -- which clearly means "95 to 105"?

If someone says "this estimate is off by a factor of 12," would you insist that it's ambiguous, because it could mean either "the estimate is 12 times [or 1/12th of] the right result," or "this estimate is off by 1, 2, 3, 4, 6, or 12"? Or would you recognize that, in actual English usage, there's no ambiguity at all (except perhaps in extremely unusual circumstances created by unusual contexts)?

No, you should say "off by about 12 times more or less"
6.21.2006 6:18pm
Sanjay (mail):
Congratulations to Professor Volokh in this, and the below post, for rendering the often boring and pedantic science of grammar, uh.....boring and pedantic.
6.21.2006 6:22pm
jrdroll (mail):
Please. You should have said 1/10. As a writer to be succinct is to be effective in conveying your message. In a hole, stop digging.
6.21.2006 6:22pm
DrObviousSo (mail) (www):
The first says to me in english "one hundred plus or minus five". The second one says to me a value that's between estimation *11.5 and 12.5, or between the estimation *-11.5 and -12.5
6.21.2006 6:24pm
byomtov (mail):
Shouldn't you say "low (or high) by a factor of 12?" Is "off by a factor of 12" supposed to be unambiguous as to the direction of the error? Why?
6.21.2006 6:25pm
John Armstrong (mail):
The mathematical symbol does mean exactly plus or minus, and doesn't denote a range. A range would be [100-5,100+5], or more commonly [x-\epsilon,x+\epsilon].
6.21.2006 6:29pm
No, "off by a factor of twelve," when taken out of context, is not supposed to be unambihuous as to the directio of the error. I think that's why EV put the "'[or 1/12 of]'" in the quote.

Hang in there EV. I'm with you on this, comrade.

BTW, despite being an avid reader of your site for some time, today I realized I was unsure of how to pronounce your last name. I was forced to trail off in midsentence and not attribute an idea to you, because I didn't want to incorrectly pronounce your last name.
6.21.2006 6:32pm
John Armstrong (mail):
Also, "off by" is ambiguous, and should be avoided if the direction matters. As for "factor of", the one quantity contains a factor of 12 that the other doesn't. As rhetoric it's a little thin, and should probably be used as emphasis after giving the actual quantities.

"The mechanic's estimate was $497.53, but the actual bill came to slightly over $6000. The estimate was off by a factor of 12."
6.21.2006 6:34pm
Re: John Armstrong's first post: The ± can have either meaning, depending on context. When used in an equation, such as the solution to the quadratic equation, it means exactly plus or minus. In other contexts, ± may be used to denote a range or degree of precision.
6.21.2006 6:41pm
David M. Nieporent (www):
But now you're the one trying to use logic to argue about a linguistic point, Eugene. The fact that someone might not misinterpret "100 plus or minus 5" does not mean that they don't misinterpret (or have trouble interpreting) "10 times more than."

Some phrases might be understood as idioms while others may not be. You can't convincingly argue, "Logically, if one understands phrase A as an idiom, one must also understand phrase B as an idiom."

If people say that they perceive ambiguity, then they do, even if they "shouldn't".
6.21.2006 6:49pm
For the "factor" example, what is ment is a multiplicative factor (usually just called a coefficient.) A divisor os also often called a factor, so I would see only ambiguity in that, if the estimate was twelve, and the actual was off by a factor of twelve, the actual must be either 1 and 144, though in some cases, the intended meaning could be the complete set of real numbers (or maybe only integers, depending on context) from 1 through 144.
Factor is a word that has so many meanings that I would assume any reader would understand the meaning (it certainly isn't incorrect or awkward) and if there is any confusion to ask for explaination.
6.21.2006 6:52pm
For your first example, I would read MORE into what was said. I would assume that there was some sort of uneven distribution around 100 where 95 and 105 were important markers. I would assume that the +/- 5 was a parenthetical statement, probably unimportant. If it was not unimportant, I would expect more treatment of the matter which would, as an unintended consequence, eliminate question.
MikeR is absolutely correct, though. In some contexts the phrase may refer to two precise, different numbers.
6.21.2006 6:57pm
Q the Enchanter (mail) (www):
Wait a minute--are we talking octal or denary?
6.21.2006 7:01pm
lucia (mail) (www):
"this estimate is off by a factor of 12,"

Totally unambiguous. I would check the calculation to see where I forgot to convert feet to inches.

As to intrepretation of "an amount is 100 plus or minus 5", I would almost always take that to mean 95 to 105. There are exceptions where it means either 95 or 105 but the rest of the sentence or paragraph would make it clear.

In my experience, people in the process of solving quadratic equations tend to elaborate. They might say something like "The two solutions are 100 plus or minus 5." In contrast, people giving uncertainty intervals tend to only say 100 plus or minus 5.
6.21.2006 7:30pm
Christopher Cooke (mail):
I think the meaning of all of your examples is unclear, given the contexts that you posit, but may be clearer in other contexts.

So, in John Armstrong's second example, I understand the use of "off by a factor of twelve" because of the first sentence, but if he had just said "The mechanic's estimate was off by a factor of twelve as compared to the final bill," I might assume he means that the final bill was twelve times the orignal estimate, or 1/12th the estimate, but would likely assume he meant that the final bill was twelve times the original estimate (perhaps that is due to my cynicism about mechanics' bills and belief that most people over, not under, charge).

As for "100 plus or minus 5," I understand that to be stating a range of numbers between 95 and 105, not "either 95, 100, or 105," but a mathematician or scientist might have a different understanding.
6.21.2006 7:38pm
"Off by a factor of 12" is ambiguous in the sense of having two possible meanings (12 times or 1/12 of) but not unclear. Those are its only two possible meanings and without more context neither meaning is eliminated.
6.21.2006 8:01pm
Michael Lopez (mail):
Oh for Christ's sake... this is NOTHING AT ALL like "ten times less than" because now we're talking about interpretation, whereas before we're talking about structure.

If we want to say that "plus or minus" actually means "within a range from plus or minus" then that's fine. We can have words mean different things... but that's a question of first meaning.

The "Ten times lower than" thing is a question of grammar and structure not of meaning.

Different type of cat.

6.21.2006 8:20pm
Sasha (mail):
DeezRightWingNutz: "Volokh" is pronounced to rhyme with "frolic". Eugene will tend to sometimes render the "kh" in the proper Russian way, as a hardish "h", so he would say what I might try to spell "Volohh", while I render it as a "k" for convenience.

Kudos to you if you can pronounce it in the Russian way (the first "o" is pronounced roughly the way the host of Coffee Talk pronounces the "o" in "coffee"), but we won't insist.

Some people pronounce the first "o" to sound like "owe", so we rhyme with our friend Clint Bolick; others say do that and also pronounce the last syllable to sound like "lock", so the name becomes a spondee, like our friend Virginia Postrel. I don't correct people when they do that, though I do correct them in any pronunciation where they stress the second syllable.
6.21.2006 8:25pm
John Armstrong (mail):
Christopher Cooke:

My example of a more useful usage has the following thought behind it: "I want to point out the vast discrepancy between the quantity A and quantity B. The average reader isn't so innumerate that he can't tell which is bigger, but is very possibly innumerate enough that he either doesn't know that the quotient is roughly 12, or doesn't even know to consider the quotient in the first place. Thus I will first state the quantities involved and then indicate their ratio so that my point (the large ratio) isn't lost."
6.21.2006 8:35pm
Count me with the anti-prescriptivists who think "ten times less" is just fine, but these are completely different phenomena, both from each other and from "ten t I'm not sure whether "one hundred plus or minus five" meaning "x s.t. 95 <= x <= 105" is an idiomatic or pragmatic phenomenon. But as others have mentioned, "X is an idiom" doesn't entail "Y is an idiom," and the same holds for pragmatics.

The other example is simple lexical ambiguity, where one word has multiple meanings. Like 'derivitive' meaning one thing in finance and another in one-dimensional calculus.
6.21.2006 10:04pm

I understand that to be stating a range of numbers between 95 and 105, not "either 95, 100, or 105," but a mathematician or scientist might have a different understanding.

Having been trained in one of the hard sciences, 100± 5 is rendered in standard English as 100 plus or minus five, which means that the measured result, or calculated number, is 100 and the range in which the measurements all fell, or the "true" number is expected to be, is 100± 5. True, if it is intended as a statistical measure, the probability likelihood, e.g. 95% probability, should probably also be used. Nevertheless, 100± 5 is a range.
6.22.2006 12:29am

I was forced to trail off in midsentence

You might want to be careful of that. For an illustration of the consequences, see: Army of Darkness (video).
6.22.2006 12:30am
1) "Formal" and "correct" may not be one and the same thing, but I think the "more formal" communication is going to be the "more correct" as grammar and as the most precise, unambiguous expression. So, while people may say things in more than one way, the question to ask is how would you express it in written or oral argument meant to inform or persuade. And I think native speakers of English usually know what will be clearly understood by the listener/reader and what may lead to confusion on their part.
2) It always causes me pause when I hear/read "b is 10 times slower than a," whereas "a is 10 times faster than b" gives me no problem. Ought one be preferred over the other, when both are equally likely to be understood? "Faster" means something happens at a greater rate, while "slower" means something happens at a lesser rate. Consistent with the greater/lesser implication of "faster"/"slower", I expect to hear "a is 10 times faster than b" or "b is 1/10 as fast as a," and must think for a moment before concluding that the speaker/writer means the same thing by "b is 10 times slower than a." Is that just me, or do others hang up on this too?
6.22.2006 11:08am
Ken Summers (mail):
From Stryker: "In some contexts the phrase may refer to two precise, different numbers."

Not just different contexts; sometimes the same context can mean two different things in two different disciplines. The phrase "1 to 10 dilution" is interpreted differently by microbiologists and chemists (usually; there are exceptions in each group). This is not a trivial issue. It causes interpretation problems between groups and leads to incorrectly written procedures.

Microbiologists tend to read it as "one part active substance plus 10 parts diluent". Chemists (and biologists who do a lot of spectroscopy) tend to read it as "one part active substance in 10 parts final solution" (i.e., one part active plus nine parts diluent).

Both sides have good arguments for their interpretation, and either interpretation works as long as everyone involved agrees on what it means; sadly, they often don't. We have had to rewrite procedures specifically to address this issue.
6.22.2006 8:56pm
Ken Summers (mail):
Still, I don't see any ambiguity at all in "100 plus or minus 5", nor in "off by a factor of 12", except whether it's off high or low.

I don't even see any ambiguity in "ten times more"; I think everyone knows what is meant, even if it's not strictly logical.

And I am heels over head in love with the English language.
6.22.2006 9:01pm