# How do you round numbers? **URL:** \ **Category:** Factual Questions\ **Created:** [November 5, 2009, 5:03pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440 "2009-11-05T17:03:04Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![toadspittle](https://avatars.discourse-cdn.com/v4/letter/t/c0e974/32.png) [@toadspittle](https://boards.straightdope.com/u/toadspittle)\ **Post date:** [November 5, 2009, 5:03pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/1 "2009-11-05T17:03:04Z")
My wife was having a discussion with her manager yesterday about how they should be recording their lab measurements. One piece of equipment is really only accurate to 2 decimal places, even though they sometimes get readings to 3 decimal places. They were debating whether to record all 3 decimal places, to truncate the numbers at 2 decimal places, or to round the 3rd decimal place up or down. As they discussed the rounding, her boss said that, when rounding, one should round the numbers 1,2,3,4,5 DOWN and round 6,7,8,9 UP. For example: 3.394 = 3.39 3.395 = 3.39 3.396 = 3.40 This dumbfounded my wife. She (and I) had been taught to always round **5** and above UP, and to round **4** and below DOWN. Thus you’d get the following: 3.394 = 3.39 3.395 = 3.40 3.396 = 3.40 Have we been doing it wrong all these years? Or is her boss off base? ---
**Author:** ![Little\_Bird](https://avatars.discourse-cdn.com/v4/letter/l/3ec8ea/32.png) [@Little\_Bird](https://boards.straightdope.com/u/Little_Bird)\ **Post date:** [November 5, 2009, 5:06pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/2 "2009-11-05T17:06:48Z")
You and your wife are right, her boss is wrong. ---
**Author:** ![Bill\_Door](https://avatars.discourse-cdn.com/v4/letter/b/50afbb/32.png) [@Bill\_Door](https://boards.straightdope.com/u/Bill_Door)\ **Post date:** [November 5, 2009, 5:10pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/3 "2009-11-05T17:10:18Z")
They are both wrong. You skew the data either way. Her boss skews low, and she skews high. You round to the nearest even number. 3.395 rounds to 3.40, while 3.405 rounds to 3.40. In the long run you introduce less bias. ---
**Author:** ![Folacin](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/folacin/32/3195_2.png) [@Folacin](https://boards.straightdope.com/u/Folacin)\ **Post date:** [November 5, 2009, 5:11pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/4 "2009-11-05T17:11:43Z")
Superman 3 was based on this conundrum - which ever way you go, you’re going to tilt the data a little bit high or low. My high school math teacher championed rounding 5 to the nearest even number - 3.385 - 3.380 3.375 - 3.380 which seems like it should remove that bias. ---
**Author:** ![Quercus](https://avatars.discourse-cdn.com/v4/letter/q/7ab992/32.png) [@Quercus](https://boards.straightdope.com/u/Quercus)\ **Post date:** [November 5, 2009, 5:13pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/5 "2009-11-05T17:13:21Z")
Well, if it’s a lab, really, they should be recording all decimal places, and doing all calculations with +/- expected errors. (so 3.394 +/- 0.011). Unless the expected error is large enough that the third decimal point is essentially random [3.394 +/- 0.24 doesn’t really give any more information than 3.39 +/- 0.24], or there’s something about the device itself that makes them believe the third decimal point is random. ---
**Author:** ![BobLibDem](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/boblibdem/32/3149_2.png) [@BobLibDem](https://boards.straightdope.com/u/BobLibDem)\ **Post date:** [November 5, 2009, 5:15pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/6 "2009-11-05T17:15:44Z")
> [@Bill\_Door](#): > > They are both wrong. You skew the data either way. Her boss skews low, and she skews high. You round to the nearest even number. 3.395 rounds to 3.40, while 3.405 rounds to 3.40. In the long run you introduce less bias. That’s what I’ve always done. [ASTM Link](http://www.astm.org/SNEWS/SO_2008/datapoints_so08.html) ---
**Author:** ![toadspittle](https://avatars.discourse-cdn.com/v4/letter/t/c0e974/32.png) [@toadspittle](https://boards.straightdope.com/u/toadspittle)\ **Post date:** [November 5, 2009, 5:17pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/7 "2009-11-05T17:17:40Z")
Please expand on this concept. What would you round to in the following cases: 3.390 3.393 3.387 3.383 ---
**Author:** ![pan1](https://avatars.discourse-cdn.com/v4/letter/p/e495f1/32.png) [@pan1](https://boards.straightdope.com/u/pan1)\ **Post date:** [November 5, 2009, 5:23pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/8 "2009-11-05T17:23:14Z")
Alternately you can just stop a certain number of significant digits. If you have two significant digits after the decimal. 3.999=3.99. This would also result in less skew, as theoretically, there would be a 3.991 as often as a 3.999, etc. Rounding, in any way, diminishes accuracy. ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [November 5, 2009, 5:33pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/9 "2009-11-05T17:33:26Z")
According to my teachers in elementary school, ‘Five or more, round up. Four or less, round down.’ ---
**Author:** ![Superfluous\_Parentheses](https://avatars.discourse-cdn.com/v4/letter/s/8edcca/32.png) [@Superfluous\_Parentheses](https://boards.straightdope.com/u/Superfluous_Parentheses)\ **Post date:** [November 5, 2009, 5:38pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/10 "2009-11-05T17:38:34Z")
> [@pan1](#): > > Alternately you can just stop a certain number of significant digits. > > If you have two significant digits after the decimal. 3.999=3.99. > > This would also result in less skew, as theoretically, there would be a 3.991 as often as a 3.999, etc. Not true, that would always skew downwards, with a maximum error of 0.01 per calculation. Rounding up or down per 0.005 leads to a max error of 0.005 per calculation. Rounding to the nearest even digit will not lead to less errors in each calculation, but it will tend lead to lessen the total upward or downward skew for a list of calculations. ETA: I was tought to use 5 = round up rule, but most of my code rounds to nearest even. ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [November 5, 2009, 5:41pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/11 "2009-11-05T17:41:32Z")
> [@pan1](#): > > Rounding, in any way, diminishes accuracy. When I worked at a place that uses whole numbers, cents were truncated from amounts. This often resulted in crossfoot errors. (i.e., the five aging amounts did not add up to the balance amount within a $5 limit.) Records that exceeded the $5 crossfoot error limit were dropped. So for the data that I needed to write a program for, I wrote some code that rounded the numbers to the nearest whole dollar (.50 or higher, round up; .49 or lower, round down). This resulted in much fewer records exceeding the $5 crossfoot error limit. ---
**Author:** ![Superfluous\_Parentheses](https://avatars.discourse-cdn.com/v4/letter/s/8edcca/32.png) [@Superfluous\_Parentheses](https://boards.straightdope.com/u/Superfluous_Parentheses)\ **Post date:** [November 5, 2009, 5:45pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/12 "2009-11-05T17:45:21Z")
> [@toadspittle](#): > > Please expand on this concept. What would you round to in the following cases: > > 3.390 > 3.393 > 3.387 > 3.383 If you want me to round to 2 decimal places: 3.39 3.39 3.39 3.38 Everybody here except **pan1** would say that. ---
**Author:** ![tetranz](https://avatars.discourse-cdn.com/v4/letter/t/a587f6/32.png) [@tetranz](https://boards.straightdope.com/u/tetranz)\ **Post date:** [November 5, 2009, 5:45pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/13 "2009-11-05T17:45:45Z")
Rounding 5 up is the common way most people do it but something like “round to even”, also known as bankers’ rounding, is definitely better. ---
**Author:** ![CaveMike](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cavemike/32/16379_2.png) [@CaveMike](https://boards.straightdope.com/u/CaveMike)\ **Post date:** [November 5, 2009, 5:54pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/14 "2009-11-05T17:54:48Z")
> [@pan1](#): > > Alternately you can just stop a certain number of significant digits. I was thinking this too at first, but then I remembered we would always round before dropping the insignificant digits. I looked it up in wiki too: > **[Significant figures](https://en.wikipedia.org/wiki/Significant_digits#Rounding)** > > Significant figures, also referred to as significant digits, are specific digits within a number that is written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presenting the outcome of a measurement (such as length, pressure, volume, or mass), if the number of digits exceeds what the measurement instrument can resolve, only the digits that are determined by the resolution are dependable and therefore considered significant. > For instan... The wiki on rounding is interesting. It discusses different methods of rounding and different methods of tie-breaking (toward even): > **[Rounding](https://en.wikipedia.org/wiki/Rounding)** > > Rounding or rounding off is the process of adjusting a number to an approximate, more convenient value, often with a shorter or simpler representation. For example, replacing $23.4476 with $23.45, the fraction 312/937 with 1/3, or the expression √2 with 1.414. > Rounding is often done to obtain a value that is easier to report and communicate than the original. Rounding can also be important to avoid misleadingly precise reporting of a computed number, measurement, or estimate; for example, a qu... ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [November 5, 2009, 5:55pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/15 "2009-11-05T17:55:23Z")
As to the question of how to round, [there are many ways](http://en.wikipedia.org/wiki/Rounding). The IEEE floating point definition uses [round towards nearest, with ties to even](http://en.wikipedia.org/wiki/Rounding#Round_half_to_even), as the default. Really, though, you’re better off leaving in a guard digit. ---
**Author:** ![Ludovic](https://avatars.discourse-cdn.com/v4/letter/l/7ab992/32.png) [@Ludovic](https://boards.straightdope.com/u/Ludovic)\ **Post date:** [November 5, 2009, 6:01pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/16 "2009-11-05T18:01:08Z")
I agree that you should leave in the remaining digit. There’s no way that lopping off the extra digit whose accuracy is questionable will make the data _more_ accurate as long as the uncertainty is recorded. On the other hand there is a good chance the data will be less accurate. ---
**Author:** ![Napier](https://avatars.discourse-cdn.com/v4/letter/n/ce73a5/32.png) [@Napier](https://boards.straightdope.com/u/Napier)\ **Post date:** [November 5, 2009, 6:05pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/17 "2009-11-05T18:05:33Z")
I also learned the round-to-even rule. More specifically, if the digits that are being dropped are “50000…” with the number of zeros being 0 or more, and the last retained digit was odd, increase it by one, otherwise leave the last retained digit alone. This does not introduce a bias in the mean. It does, though, introduce other kinds of bias, including making even numbers a little more common. My preference is never to do any rounding, except perhaps in the final result if it is being published or distributed to many or shown in a slide or something else that makes it expensive or ugly not to round. The reason is that the supposedly meaningless numbers actually do have a small meaning. They don’t tell you anything more about the quantity you were measuring, but they create a distinctive sort of fingerprint. It often seems that we want to check or verify after the fact that there hasn’t been some kind of confusion or malfunction. If there are 5 insignificant digits and they keep repeating, perhaps some assumption about the recordkeeping is wrong. If there are only certain combinations of those digits that appear, there is probably a significant rounding happening someplace upstream, like losing the fractions of an ounce before it got converted to grams, or like only reading the 12 most significant bits out of an 16 bit A/D converter. If these digits repeat at the tops of two pages, it might tip you off that a page got printed twice. They let you check all sorts of things that you usually can’t anticipate needing to check. ---
**Author:** ![The\_Hamster\_King](https://avatars.discourse-cdn.com/v4/letter/t/8edcca/32.png) [@The\_Hamster\_King](https://boards.straightdope.com/u/The_Hamster_King)\ **Post date:** [November 5, 2009, 6:19pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/18 "2009-11-05T18:19:07Z")
I also learned to round fives to even. And that was in my numeric methods class in Computer Science graduate school were we had to calculate the error bounds on complicated computations. Another question is: What exactly is your wife doing with the data after it’s gathered? Are any calculations being done with it? Or is it just used raw? ---
**Author:** ![Harmonious\_Discord](https://avatars.discourse-cdn.com/v4/letter/h/74df32/32.png) [@Harmonious\_Discord](https://boards.straightdope.com/u/Harmonious_Discord)\ **Post date:** [November 5, 2009, 6:21pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/19 "2009-11-05T18:21:49Z")
> [@Ludovic](#): > > I agree that you should leave in the remaining digit. There’s no way that lopping off the extra digit whose accuracy is questionable will make the data _more_ accurate as long as the uncertainty is recorded. On the other hand there is a good chance the data will be less accurate. This is what I would do. I know that you can also analysis error to expect from different people taking the measurements. I haven’t done anything with that in 8 years, so can’t tell you what the name would be. ---
**Author:** ![ElvisL1ves](https://avatars.discourse-cdn.com/v4/letter/e/5daacb/32.png) [@ElvisL1ves](https://boards.straightdope.com/u/ElvisL1ves)\ **Post date:** [November 5, 2009, 6:53pm UTC](https://boards.straightdope.com/t/how-do-you-round-numbers/516440/20 "2009-11-05T18:53:14Z")
Think of it this way: Numbers ending in 0, 1, 2, 3, and 4 are rounded down, numbers ending in 5, 6, 7, 8, or 9 are rounded up. Five of each digit. That eliminates bias. Truncation is biased. As for the third digit on the gauge, do you think it’s just unreliable, or actually noise? If the former, round it, if the latter, truncate it. [Next page](https://boards.straightdope.com/t/how-do-you-round-numbers/516440.md?page=2)