Replacement for “x times lower”

In scientific writing numbers are often compared and if something is twice the size of something else, let‘s say

A is 13 to 17 times the size of B

this can be written as

A is 13–17 times higher/larger than B

I often see cases where people turn this around and say

B is 13–17 times lower/smaller than A

and where turning A and B around makes little logical sense, i.e. the order of B and A is logically correct.

The problem is then:

How would one best rewrite this, since "x times" will, at least mathematically when x > 1, always be larger and not smaller?

Edit: I changed my example to highlight the problems with numbers other than those easily turned into fractions such as half, quarter etc.

• In maths, linear scale factors are used to avoid (1) the confusion where in everyday language 'ten times bigger' is used to mean 'x10' whereas 'one time(/s) bigger' (paraphrasing 100% bigger) means 'x2' (so I don't like your 'A is 13 to 17 times the size of B: this can be written as A is 13-17 times higher/larger than B'). (2) confusion with area, volume scale factors. Thus 'a scale factor of between 1/17 and 1/13' (or 5.9% - 7.7%) would be used. Jun 2, 2014 at 11:17
• right, "a factor of" is commonplace in english. Jun 2, 2014 at 12:23
• N times smaller has no clear meaning. When mass media do it, it's too "simplify" the language used, at the expense of clarity.
– user63230
Jun 2, 2014 at 12:38
• "The Dead Sea is 423 meters below sea level, two times lower than the second lowest place on Earth." Jun 2, 2014 at 14:58

One could say:

B is one seventeenth to one thirteenth the size of A.

But in modern scientific literature I think it more likely be expressed as something like:

B is 5.9-7.7% the size of A.

• Yes percentiles are the way to go; that's all I ever see used in these circumstances. Aug 22, 2016 at 20:03
• @AlbertRenshaw percentages are not percentiles. Oct 11, 2016 at 2:40
• Percentages or decimal proportions in scientific writing. Feb 25 at 12:39

B is 13-17 times lower/smaller than A

The problem is then, how would one best rewrite this since "x times" will, at least mathematically when x > 1, always be larger and not smaller?

I disagree with the premise of the question. The example is OK, although a little awkward. In "x times smaller," the word "smaller" inverts the ratio. There are of course other ways of expressing this, but that's a matter of style. Using "by a factor of" doesn't necessarily fix things:

(1) A millimeter is ten times smaller than a centimeter.

(2) A millimeter is smaller than a centimeter by a factor of 10.

(3) A millimeter is smaller than a centimeter by a factor of 0.1.

(4) My car is lighter than your car by a factor of x.

Of 1-3, I think 1 is the best style and is fine mathematically. 2 and 3 show that there's a potential ambiguity. In 4, the ambiguity is a real problem, because we can't tell if x is being defined as a number that's less than 1, or greater than 1.

Edwin Ashworth wrote:

[...] linear scale factors are used to avoid [...] the confusion where in everyday language 'ten times bigger' is used to mean 'x10' whereas 'one time(/s) bigger' (paraphrasing 100% bigger) means 'x2'

I think this is an innumeracy issue, not a language issue. Some people just don't understand how to convert back and forth between ratios and fractional changes, or don't realize that they're different things. For example, if they're told that B is 7% greater than A, and are then asked to find the ratio B/A, they may say 0.07. We're talking about scientific writing, where this kind of innumeracy isn't an issue.

• I am a native English-speaking professional scientist in the field of molecular biology and have published data in many papers. During my whole career I had never encountered the term x-times lower, which appears to me an oxymoron. I honestly could not deduce what it might mean and had to do a Google ngram search to find that it is used in some technical areas to mean "N/x, where N is the magnitude of the reference item". It is clear from an internet search that my experience is shared by many other numerate scientists. If the problem is of innumeracy, it relates to those using the expression. Feb 25 at 12:48

This is one of my pet peeves. In one of the responses above, the writer provides the example "(1) A millimeter is ten times smaller than a centimeter." The writer then goes on to state that this example is "fine mathematically" when in fact, it is incorrect. When you see "is" in a sentence, you replace it with an "=" sign. Thus, the example can be written as 1mm=(10)x(1cm) or 1mm=10cm. This is clearly incorrect. If the mathematical relationship between a millimeter and a centimeter is written this way: 1mm=(0.1)x(1cm) or 1mm=(1-0.9)x(1cm), then the acceptable English language translations are "A millimeter is one-tenth as long as a centimeter" or "A millimeter is 90 percent shorter than a centimeter," respectively.

"A is 13 to 17 times the size of B

this can be written as

A is 13–17 times higher/larger than B"

This is incorrect. Let's do an example:

2 is 2 times the size of 1. but... 2 is 1 times larger than 1. Because we can think of "1 times" to mean 100%: 1 is one times the size of 1. But 2 is one times larger (100% larger) than 1.

SO: "A is 13-17 times the size of B" can be rewritten "A is 12-16 times larger than B"

• Incorrect meaning ... ? As I mentioned years ago, everyday language and precisionist language part company here. But everyday language is used ... every day. Feb 22 at 19:30

Dude it's just one-seventeenth to one-twentieth the size of X.

{Incidentally: you are totally wrong to assert they have to be "in order"; whether for a multiple or a fraction. You can certainly say: "P is 10 to 5 times bigger than Q." No problem. "10 to 5" simply means "the numbers from 10 to 5".}

BTW another common ons is: A is smaller than B by a factor of 13 to 17.

that may be what you have in mind?

B is half the length of A

B is one-quarter the length of A

When in doubt, rewrite the sentence for clarity. It's always a little tricky to express math in words, that's why it has its own special formats.

If it must be done, remember this rule of thumb: digits tend to emphasize numbers while words tend to diminish their impact.

• True, but this works for simpler quantities and I realize my example was over-simplified. If one has e.g. "13-17 times", then the problem remains. Jun 2, 2014 at 10:22
• one thirteenth to one seventeenth. Jun 2, 2014 at 12:22
• "digits tend to emphasize numbers while words tend to emphasize numbers." Huh? Jan 4, 2021 at 20:20
• @Acccumulation haha good catch. Jan 4, 2021 at 20:21

I don't think there is a 'correct' way to use it. Say:

A=100

B=20

A is 5 times higher than B, or in other words 5 * B = A

B is 5 times 'lower' than A, or in other words 5 * A = B, WRONG, 5 * A = 500 != B

Just because the word 'lower' is used, it shouldn't somehow alter what 'times' means.

Proper statements:

B is 20% of A, in other words B * 0.2 = A.

A is 0.2 times lower than B, in other words B * 0.2 = A

1/5 could be used in this case, but short simple fractions aren't always possible.

• "Improper" might be confusing in this context. I encourage you to edit for clarity, and welcome. Feb 23 at 2:43
• A word meaning "not in accordance with accepted rules or standards" seems to fit well. Though I could remove the word, as I suppose it just repeats what the first part of the sentence is saying just in a different way. Thanks! Feb 24 at 10:13
• Totally! I meant regarding fractions. "Correct" is another ball of wax. Feb 24 at 22:23