Advice to Lawyers: Avoid Talking About “Orders of Magnitude”

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.”