# Why Do You Round to the Five (in Math)? **URL:** \ **Category:** Factual Questions\ **Created:** [October 31, 2017, 7:22pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314 "2017-10-31T19:22:44Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Buck\_Godot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/buck_godot/32/6573_2.png) [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)\ **Post date:** [October 31, 2017, 8:28pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/21 "2017-10-31T20:28:44Z")
Of course depending on your application, round to even could actually increase your bias. Suppose you are interested in calculating how many hours people stay at work on average at your workplace. Many people clock their hours by coming in at 8:30 or leaving at 5:30. So using the round to even version to the nearest hour would round down such entry times, and round up such leaving times, resulting in a biased answer. The fancy way to take care of this would be to round 5’s at random. Basically flip a coin to decide whether to round up or down. ---
**Author:** ![sitchensis](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/sitchensis/32/4892_2.png) [@sitchensis](https://boards.straightdope.com/u/sitchensis)\ **Post date:** [October 31, 2017, 8:45pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/22 "2017-10-31T20:45:00Z")
> [@pulykamell](#): > > 0 is already round so it doesn’t round anywhere. There are ten possible options; 0123456789. For five of the 10 options (01234) we don’t change the preceding digit. For the other 5 (56789) we add one to the proceeding digit. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [October 31, 2017, 9:05pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/23 "2017-10-31T21:05:48Z")
As explained above almost any time you have a 5 after the digit you are rounding to there are more non-zero digits after it. So is 4.500000001 closer to 4 or 5? 5 right? So we round up to 5. But in elementary school, the teachers don’t teach THAT! Everything is boiled down to simplistic rules that teachers can parrot. So this entirely reasonable process has all of the concepts extracted out of it and is reduced to a husk of a rule to be memorized, “5 or higher” so when perfectly valid questions like, “What about 4.5?” gets raised the teacher responds, “Use the flippin’ rule you son of a bitch.” ---
**Author:** ![Kimstu](https://avatars.discourse-cdn.com/v4/letter/k/ecd19e/32.png) [@Kimstu](https://boards.straightdope.com/u/Kimstu)\ **Post date:** [October 31, 2017, 9:09pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/24 "2017-10-31T21:09:02Z")
> [@sitchensis](#): > > There are ten possible options; 0123456789. For five of the 10 options (01234) we don’t change the preceding digit. For the other 5 (56789) we add one to the proceeding digit. Yes, but only in the case of 0 is the rounded number exactly the same as the unrounded one. So, as **septimus** noted, if your rounding convention slightly decreases the value of the number when the final digit is {1,2,3,4}, slightly increases the value when the final digit is {5,6,7,8,9}, and leaves it unchanged when the final digit is zero, your rounded numbers will end up on average being slightly greater than their unrounded form. Because you’ve got 5/10 chance of an increase compared to only 4/10 chance of a decrease. (Assuming that your final digits are randomly distributed, which isn’t always the case.) ---
**Author:** ![iamthewalrus\_3](https://avatars.discourse-cdn.com/v4/letter/i/258eb7/32.png) [@iamthewalrus\_3](https://boards.straightdope.com/u/iamthewalrus_3)\ **Post date:** [October 31, 2017, 9:27pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/25 "2017-10-31T21:27:52Z")
> [@nelliebly](#): > > Why isn’t one method universal? That is, why would a banker round to half while someone in another field would round to zero? There must be a reason. A couple possibilities. 1. Path dependency: The standards for different processes came about differently. It doesn’t really matter which you use, but it really matters that you use the same thing as other people in your field. So you end up with different standards, neither of which is better than another. Analogy: drive on left vs drive on right. Some countries went one way, and some went another. There’s no real benefit to one over another, but switching is a major hassle, and it’s very important that everyone around you does the same thing. 1. There are subtle differences to each, so which you choose requires weighing the pros and cons of each, and different applications went with different standards based on their needs. Analogy: different screw heads. You can attach things with flathead or philips or torx, but each is slightly different in what it’s best suited for, so not all screws use the same heads. ---
**Author:** ![DrCube](https://avatars.discourse-cdn.com/v4/letter/d/a3d4f5/32.png) [@DrCube](https://boards.straightdope.com/u/DrCube)\ **Post date:** [October 31, 2017, 9:34pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/26 "2017-10-31T21:34:30Z")
> [@Kimstu](#): > > Yes, but only in the case of 0 is the rounded number exactly the same as the unrounded one. > > So, as **septimus** noted, if your rounding convention slightly decreases the value of the number when the final digit is {1,2,3,4}, slightly increases the value when the final digit is {5,6,7,8,9}, and leaves it unchanged when the final digit is zero, your rounded numbers will end up on average being slightly greater than their unrounded form. Because you’ve got 5/10 chance of an increase compared to only 4/10 chance of a decrease. (Assuming that your final digits are randomly distributed, which isn’t always the case.) With real world measurements there’s no such thing as a “final digit”. Numbers that aren’t measurements don’t need to be rounded, except for show (reporting on our $20 Trillion dollar debt instead of the more exact $20,454,070,952.73 debt, for example), in which case bias doesn’t matter. ---
**Author:** ![bordelond](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bordelond/32/150_2.png) [@bordelond](https://boards.straightdope.com/u/bordelond)\ **Post date:** [October 31, 2017, 9:37pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/27 "2017-10-31T21:37:15Z")
> [@RealityChuck](#): > > The general rule is to round to the even number: 3.5 rounds to 4. So does 4.5. The idea is that the rounding will even out over time. This wasn’t taught to my kids over the last several year of school. But then, in elementary school math, rounding is not done as a practical exercise. Our kids were taught to round down on 0, 1, 2, 3, 4; and round up on 5, 6, 7, 8, and 9. Just a simple one-rule algorithm. ---
**Author:** ![bordelond](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bordelond/32/150_2.png) [@bordelond](https://boards.straightdope.com/u/bordelond)\ **Post date:** [October 31, 2017, 9:41pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/28 "2017-10-31T21:41:26Z")
> [@pulykamell](#): > > 0 is already round so it doesn’t round anywhere. While true, it makes the rounding algorithm symmetrical and thus a little easier to memorize for young students. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [October 31, 2017, 9:48pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/29 "2017-10-31T21:48:00Z")
> [@bordelond](#): > > Our kids were taught to round down on 0, 1, 2, 3, 4; and round up on 5, 6, 7, 8, and 9. Just a simple one-rule algorithm. This is how I was taught through all my classes in the 80s and early 90s. (And I had a math and science heavy curriculum up until my sophomore year of college.) “Banker’s rounding” and all the other assorted forms here I didn’t learn until probably well after college. Maybe even from this board. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [October 31, 2017, 10:08pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/30 "2017-10-31T22:08:12Z")
> [@Saint\_Cad](#): > > Everything is boiled down to simplistic rules that teachers can parrot. More or less what I was going to say. “5 rounds up” does the right thing the majority of the time regardless of your rounding rules. It’s too hard to explain that 1.50000001 rounds up while 1.5 exactly may or may not depending on esoteric rules that aren’t even consistent. This is one of the least objectionable “lies to children” that comes up in math. They can learn the more advanced stuff when they start designing IEEE 754-compliant floating-point hardware units (and then there’s not even a 5 involved…). ---
**Author:** ![gazpacho](https://avatars.discourse-cdn.com/v4/letter/g/6f9a4e/32.png) [@gazpacho](https://boards.straightdope.com/u/gazpacho)\ **Post date:** [October 31, 2017, 11:18pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/31 "2017-10-31T23:18:50Z")
> [@Dr.Strangelove](#): > > More or less what I was going to say. “5 rounds up” does the right thing the majority of the time regardless of your rounding rules. It’s too hard to explain that 1.50000001 rounds up while 1.5 exactly may or may not depending on esoteric rules that aren’t even consistent. This is one of the least objectionable “lies to children” that comes up in math. They can learn the more advanced stuff when they start designing IEEE 754-compliant floating-point hardware units (and then there’s not even a 5 involved…). I have a document that explains the different type of resize (rounding and saturation) functions we use in designing our digital signal processing hardware. It takes a 4 pages. The options are: Truncation Symmetric Truncation Rounding (this is .5 rounds up) (really .1 rounds up because it is all binary) Odd Dither Rounding Symmetric Rounding then there are the saturation options Asymmetric saturation Symmetric saturation Then there is Sign Extension ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [October 31, 2017, 11:50pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/32 "2017-10-31T23:50:44Z")
Last time this topic came up, I Googled and concluded that the Even Digit Convention might be an old, nearly forgotten idea. I learned it in the early 1960’s. I’m curious: _**Are those of you who are familiar with this Rule (or Convention) old-timers like me?**_ 🙂 That it is old and seldom taught does not make the Rule wrong, of course. If we don’t want to waste students’ time learning a silly, seldom-applied rule of minimal value, then _ **Yes! The Even-Digit Rule is the first thing to scratch from the curriculum!** _ But the beauty, _and pedagogic purpose_, of this Rule is _NOT_ its actual utility. The Rule introduces the notion of bias, and presents a simple and elegant way to avoid that bias. It’s the thinking behind the Rule that might appeal to a teacher or student. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 1, 2017, 12:15am UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/33 "2017-11-01T00:15:36Z")
The reason I favor rounding 5s up is that you don’t always know where that number has been, and it’s possible that some idiot before you was truncating instead of rounding. Example: I’m told that a certain number is 5.5, but I want to round it to an integer. Maybe the person who told me 5.5 was actually given a number to two places past the decimal, and converted it already himself. If he was rounding, too, then I still don’t know which way to go: Maybe he turned 5.48 into 5.5 (in which case I should round down), or maybe he turned 5.52 into 5.5 (in which case I should round up). But it’s also possible that he was truncating. In that case, the number he had might have been 5.51 or 5.53 or even 5.59, but it was not 5.49 or 5.57 or whatever, and so I should round up. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [November 1, 2017, 12:25am UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/34 "2017-11-01T00:25:51Z")
While that approach may have practical merit … it seems sad to base an arithmetic method on the inelegant assumption that some unnamed predecessor might have screwed up your inputs! :eek: ---
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [November 1, 2017, 1:43am UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/35 "2017-11-01T01:43:50Z")
> [@iamthewalrus\_3](#): > > Ultimately, it doesn’t matter that much. The purpose of rounding is specifically to drop insignificant digits. Yes, rounding up on 5 introduces a slight amount of bias, but so does rounding to even because of Benford’s law. I thought Benford’s law related to the leading or most significant digit. Rounding affects the least significant digit. 😕 ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [November 1, 2017, 1:50am UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/36 "2017-11-01T01:50:14Z")
> [@septimus](#): > > Last time this topic came up, I Googled and concluded that the Even Digit Convention might be an old, nearly forgotten idea. Not forgotten in the computer world. It’s the default rounding behavior for IEEE 754 floating point. I only know of the principle via that, though when I learned of the idea it was obvious that it would apply to other even bases. ---
**Author:** ![Some\_Call\_Me.Tim](https://avatars.discourse-cdn.com/v4/letter/s/439d5e/32.png) [@Some\_Call\_Me.Tim](https://boards.straightdope.com/u/Some_Call_Me.Tim)\ **Post date:** [November 1, 2017, 2:15am UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/37 "2017-11-01T02:15:58Z")
Intel solved this problem years ago. Simply run the calculation on the right sort of Pentium processor, and you’ll find that 4.4999999999992304 rounds down to 4, while 4.500000000000239545 rounds up to 5. Even if you thought you entered 4.5! ---
**Author:** ![iamthewalrus\_3](https://avatars.discourse-cdn.com/v4/letter/i/258eb7/32.png) [@iamthewalrus\_3](https://boards.straightdope.com/u/iamthewalrus_3)\ **Post date:** [November 1, 2017, 5:21pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/38 "2017-11-01T17:21:45Z")
> [@standingwave](#): > > I thought Benford’s law related to the leading or most significant digit. Rounding affects the least significant digit. 😕 Benford’s law affects all digits in the sorts of natural samples it applies to for the same reason. Just like there are more 1s than 9s out there in the real measurable world, there are more 1.1s than 1.9s, and more 2.01s than 2.09s, and so on. And while rounding affects the least-significant digit, it references the second-least-significant. So if you round toward even, you’re going to end up rounding a lot more 1.5s up to 2 than 2.5s down to 2, and a lot more 3.5s up to 4 than 4.5s down to 4, and so on. > [@wikipedia](#): > > Benford’s law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on. To clarify my previous statement, I didn’t mean that rounding to even introduces _the same amount of_ bias that rounding 5s up does, but it will introduce bias. ---
**Author:** ![Desert\_Horses](https://avatars.discourse-cdn.com/v4/letter/d/f07891/32.png) [@Desert\_Horses](https://boards.straightdope.com/u/Desert_Horses)\ **Post date:** [November 1, 2017, 6:39pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/39 "2017-11-01T18:39:28Z")
Doesn’t a lot depend on just what it is we are rounding? If it’s meds I’m giving a patient, 0.24 or 0.26 mg can be very different - to the point of life or death - than 0.2 (rounded down) or 0.3 (rounded up) mg. If it’s keeping track of how much is in my cart at Costco so I don’t blow my budget, it’s both easier and safer to round up, no matter what the “cents” are, even if .49 or below. I’d probably feel better if the pilot is rounding up on the fuel needed to fly the plane from A to B, but my horses’ fuel (hay) only needs to be approximate. Seems to me that tolerances and thus rounding depends on how critical the application is and that no one rule fits all. There is much in life that is serious. But sometimes we take life too seriously. (I just made this up.) ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 1, 2017, 7:23pm UTC](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314/40 "2017-11-01T19:23:22Z")
If the precise dosage of medicine is that important, then you shouldn’t be rounding it at all. And while Benford’s Law applies to all digits, it applies most strongly to the most-significant digit. [Previous page](https://boards.straightdope.com/t/why-do-you-round-to-the-five-in-math/800314.md?page=1)