# The meaning of 0% and 100% as opposed to other percentages?

Oftentimes, percentages are rounded to the nearest whole number. A \$49.99 item may be marked 50% off, even if the price becomes \$24.99 (it should be 50.03% off). However, I have come to notice that this rounding rule does not seem to apply with 100% and 0%.

For example, certain soaps will say that they kill 99.99% of bacteria. If a (pretty bad) soap killed 39.99% of bacteria, there would be no problem with advertising as 40%. So how come soaps that kill 99.99% cannot be advertised as 100%? Why is it that the percentages of 0 and 100 are always implied to be exact, and not allowed to be rounded?

• Looks like a marketing issue rather than an ELU one.
– user66974
Aug 29 '15 at 16:19
• I think so too, but I can't seem to find the right StackExchange. I've been looking for a while now, maybe I just missed it. ELU seemed like the next best one. Aug 29 '15 at 16:22
• Because killing 99.99% of germs is very, very different from killing all of them. In any given population, there are many, many, many germs. And leaving one alive is (theoretically) sufficient for it to reproduce. So if you're not killing all (100%) of them, there's still a chance you'll get sick. Aug 29 '15 at 16:51
• It's a legal thing. If they claimed 100% and, as Dan points out, even one got through, they could be sued.
– Jim
Aug 29 '15 at 18:39
• Almost but not quite a duplicate of "In the UK, 0.77 is now legally a number between 1 and 25." independent.co.uk/news/science/… Aug 30 '15 at 1:56

"100%" is equivalent to "all". There is no rounding with "all"; either you get all of something or you don't. If a product advertised itself as "kills all bacteria" and then you found that there were 3 bacteria that it didn't kill, it doesn't matter whether that's 3 out of 10 or 3 out of 28 million; it's not all of them.

Even in ordinary conversation, if your child says "I picked up all the blocks" and you find 1 block left on the floor, you can legitimately say that they did not, in fact, pick up all the blocks. Doesn't matter if there were 10 blocks or 20,000; if there's one left on the floor, they did not pick up 100% of them.

(Similarly, "0%" = "none"; if you say "there are none left" and there's one left, you're wrong, regardless of how many there used to be.)

• Good example, if the same child picked up 39 blocks out of 80 and said "I picked up half the blocks", you wouldn't think they were blatantly lying like you would if they picked up 79 out of 80 and said "all", or 1 out of 80 and said "none". (and of couse, if those 3 missed germs in 28 million cause someone to get sick, they can't sue you if you said 99.99%...) Aug 31 '15 at 13:41

Rounding percentages is not merely a mathematical operation. Rounding highly depend on the real-life notion represented by the percentage. In your example, the complementary percentage represents the percentage of bacteria that survives after applying the soap. Lets consider the following examples without any rounding:

1. If soap A kills 40% of bacteria, and soap B kills 39.99%, then the bacteria that survives is similar in both cases (60% and 60.01%). Therefore A is slightly better.
2. If soap A kills 99.99% and soap B kills 99.98% of bacteria, the remaining amount of bacteria after applying A (0.01%) is twice smaller than the remaining amount of bacteria after applying B (0.02%). Therefore A is significantly better.
3. If soap A kills 100% and soap B kills 99.99% of bacteria, the remaining amount of bacteria after applying A (0%) is infinitely smaller than the remaining amount of bacteria after applying B (0.01%). Therefore A is much, much better.

You can see from these examples that 0.01% gap behaves differently across the percentage scale. On the edges of the scale it has much more impact. That is why when considering percentages that are close to an edge of the scale, rounding even by 0.01% can be considered as a deception.

• Who sells this soup, and how can I avoid it? Aug 29 '15 at 20:08
• No soap for you... Aug 29 '15 at 20:50
• Soup isn't deadly for being antibacterial: It's medicinal. I mean, didn't your mother ever tell you to take Chicken Soup to help cure a cold @FGreg? =P Aug 30 '15 at 2:03
• I think I liked this answer better when it still talked about soup. :D Aug 30 '15 at 3:55
• An off-topic remark: Soap that kills bacteria is actually not better than soap that does not (unless you work in a hospital), since it also kills the "good" bacteria.
– fNek
Aug 31 '15 at 12:07

The answers here are correct, but I wanted to give some statistical background on the terms.

When we think about measuring error, errors are often phrased in terms of Type I and Type II errors

• Type I errors are the "false alarm" errors or the "boy who cried wolf" errors. They occur when something is not present, but triggers detection anyway (often due to random noise sources)
• Type II errors are the "sleeping watchman" errors. These occur when the stimulus is present, but the detector doesn't detect it.

We often tune our systems to balance these two types of errors. The more sensitive they get, the fewer type II errors we get, but we pay for it by creating more type I errors by being more sensitive to noise. Likewise, we can dull sensitivity to minimize type I errors, but it increases type II errors.

With 0% and 100%, these terms fall apart. If you are looking for "all" or "nothing," there's no way to tune the detector to see none of one type without forcing yourself to deal with tons of the other type of error.

In scientific settings, more numbers are presented (such as confidence intervals) which provide a more complete picture. However, in advertisement, nobody uses those terms because they are too technical.

As such, terms like 100% are reserved for subjective situations like a "100% satisfaction guarantee," which specifically means that you can return it for any reason at all, just by claiming "you were not satisfied."

• Also known as False Positives / False Negatives. Aug 31 '15 at 13:05

This is probably a legal, not a linguistic, reason. Should even one person get a bacterial infection after using the product in question, the manufacturer might be less likely to be accountable for the resulting illness. When other claims are made (100% natural, for example) in which counter-arguments are less feasible (there may not be any hard and fast rules for what is "natural"), you will not find an decimal hedging.

• Legal issues are often linguistic ones. 100% means 'absolutely all', but 90%, 99%, 99.99% can mean varying degrees of 'almost all'. Aug 29 '15 at 16:57
• It's still a linguistic issue. A well implemented progress bar in some computer program wouldn't move from 99% to 100% until the operation was completely finished, even if it would otherwise round. Aug 29 '15 at 18:24
• And here's an example of this sort of thing happening in the other direction twitter.com/righteousaxe/status/636625315553345536 Aug 29 '15 at 18:26
• @asmeurer--Not so fast. The site baseball-reference.com compiles very detailed data about every baseball game ever played. It has a statistic, wWE, which gives the eventual winner's winning expectation as a %-age for every point in the game. If you look at the game Boston won 22-10 on Aug. 15 (baseball-reference.com/boxes/BOS/BOS201508150.shtml), you'll find that at the end of the 5th inning, it was calculated that Boston, ahead 11-2, had a 100% chance of winning, But after the first batter in the 6th walked, it became 99%, which is impossible if 100% were really 100%. Aug 30 '15 at 12:00

Note that not rounding to 100% is not a hard rule: in France, a drink that is 99.9% fruit juice can legally boast “100% de fruits”; if you actually want 100% fruit juice, you have to look for the mention “pur jus” (pure juice).

There are also many contexts where it is completely acceptable to round to 0% or 100%: if something increased by 99.9%, you could as well say that it increased by 100% (or, equivalently, that it doubled).

But in contexts where “0%” and “100%” actually mean “none at all” and “absolutely everything”, rounding makes a huge difference, which is why it’s usually avoided.

In your question you are mixing two different quantities: rounding currency is necessary because you cannot ask people to pay a fraction of cent just for mathematical precision. So you may refund all of a loan (with interests) without paying 100% of it: the rounding rule sometimes apply also with 100% (or 0%).

When rounding the amount of killed bacteria you have to remember that you are talking about numbers that exceed hundreds of billions so 1% is still a large number of bacteria that will survive and multiply themselves in a matter of hours (see zvisofer answer).