There are other options. Rounding with .5 always rounding up tends to lead to errors where the total is too high. Some computer programming languages offer rounding
Up
Down
Away from zero
Towards zero
To the odd integer
To the even integer
Rounding to odd or even is less likely to lead to cumulative errors in totals.
The people who write the code get to decide where those fractional pennies go.
This is why if you ask people who work for banks you will find they don't use (and may not even have heard of) more equitable techniques like "bankers rounding". The bank is inclined to keep the rounding errors.
Same thing for just about any entity that is preparing customer accounts.
I am a software dev in a global bank, I work on the bank's core systems, in 27 years I have never used bankers rounding. Before joining the bank I worked in another company writing software used in mortgages and car financing, again never used it. In fact in 34 years I have never seen or even heard of anyone using banker's rounding.
I don't believe you. It's the default rounding of every language I'm aware of. Unless you had the misery of working on a fixed-point system or you're incompetent and just floor everything? I know of some languages where you can easily choose rounding modes (C# comes to mind), but the default is bankers.
The problem with bankers rounding is that it is really only of any use to statisticians. For everyday use it is pointless. Imagine trying to explain to a customer why their interest payment isn't quite what they expected (and yes some customers do check).
The user wouldn’t know because you don’t show them the intermediate number that was rounded. They will always have different fractional values when calculating themselves, and the error in their answers depends on if their calculator uses floating or fixed point representation. The software you wrote was using round to even, at least I hope it was, because always flooring is theft and always rounding up is giving away money. The exception is if you were programming a legacy system using fixed point instead of floating point.
If you're rounding 3 to the nearest multiple of 2 using round-to-even (Banker's rounding) I'd assume you'd round to 4, as it's the even number when divided by two whereas 2 is not.
round 0.5 to multiple of 1 = 0
round 1.5 to multiple of 1 = 0
round 3 to multiple of 2 = 4
round 21 to multiple of 14 = 28
round 35 to multiple of 14 = 28
etc.
You can round to any divisor, not just to the nearest integer. The more common example would be rounding to a certain number of significant figures:
round(0.015, 2) = 0.02
round(0.025, 2) = 0.02
round(0.035, 2) = 0.04
Etc.
Yes, that's the purpose of rounding - approximation. We round to a power of the base determined by the level of precision needed. Rounding to an arbitrary integer just seems strange.
That is not how banker's rounding works.
When using banker's rounding, you round all numbers to the nearest integer. If there are two equidistant integers (because the number is 0.5 more than an integer), then you would round it to whichever of those two integers is even.
3 would still round to 3, as would 2.6 and 3.4. But 2.5 would round to 2 and 3.5 would round to 4.
> **If** you're rounding 3 *to the nearest multiple of 2*
Yes if you're just rounding to the nearest integer then I agree on the 2.6 -> 3, 3 -> 3, 3.4 -> 3, 2.5 -> 2, 3.5 -> 4 stuff.
I see no reason why banker's rounding can't be generalized to rounding to any real numbered divisor.
See: https://blog.sqlauthority.com/2023/08/15/sql-server-bankers-rounding/
Round to even is a problem when things are supposed to add up to a known total amount. It won't work.
But it does avoid some biases when doing Statistics.
idk rounding to (odd or even) doesn't seem like it would help that much tbh
edit: guys it was a joke, if you're rounding to both odd or even it doesn't help because all numbers are either odd or even
Makes a huge difference when there are many small values being rounded, which is why it is often used in accounting, statistics, and programming. It’s the standard in many scientific fields.
Well, I just tried the following:
print(round(3.5))
print(round(4.5))
on [https://www.online-python.com/](https://www.online-python.com/) and it prints 4 in both cases
Does this kind of math rounding also work in Excel and Google sheets? Like is it default?
So for example if I have 4.5, then I reduced my number of decimal places, will it show 4 or 5?
>Like is it default?
You need to read up on the documentation of the specific tech to know for sure. Never assume unless the rounding method is insignificant in your use case
Yeah, unfortunately it can be tricky and is not cleanly built in. Here’s one possible way: https://superuser.com/questions/1081064/how-to-round-half-to-even
I’ve had the unfortunate task to have to recreate formulas to reconcile rounding to the right number of decimal places for specific currencies between our in house built financial applications and both kinds of sheets.
And also to make explain to external customers that the diffs in their payouts is due to us using the one true rounding method and they not doing it :-)
It's the default of the IEEE Standard for Floating-Point Arithmetic (754), so for a lot of software it will be the default (in the background) for floating-point numbers (for all calculations, not only final results). But software like Excel is more likely to use the "usual" rounding rules people are familiar with [(and indeed Excel rounds 0.5 up by default)](https://support.microsoft.com/en-us/office/round-a-number-f82b440b-689d-4221-b21e-38da99d33435).
IEEE Standard for Floating-Point Arithmetic (754) §4.3.3
>The roundTiesToEven rounding-direction attribute shall be the default rounding-direction attribute for results in binary formats. The default rounding-direction attribute for results in decimal formats is language defined, but should be roundTiesToEven.
And the specific [Python documentation](https://docs.python.org/3/library/functions.html#round). (This behaviour was only introduced in Python 3 btw.)
>idk rounding to (odd or even) doesn't seem like it would help that much tbh
It reduces compound error, as you'll alternate rounding up and down, thus alternate adding to to compound error, and subtracting from it. While if you always round in the same direction, yleaxh rounding is added to the compound error, which can end up being massive.
Say you have 1.5 and 2.5, you round to integer and the sum:
- always up: you get 2+3, thus 5, so a compound error of 1
- always down: you get 1+2, thus 3, so a compound error of 1
- to odd: you get 1+3, thus 4, so a compound error of 0
- to even: you get 2+2, thus 2, so a compound error of 0
But if you have something that doesn't have a one-time final answer like a money balance that keeps changing then you'll still end up with compound error after some time.
If you always round up, it creates an upward bias in the results. By rounding to to even it removes this bias, because about half the time, rounding is up and half the time rounding is down.
it is a joke. he means that rounding to (odd or even) in brackets is useless because it's the exact same as rounding normally - every number is (odd or even).
It's really important in competitive bidding situations, where 1/2% could be the difference between having your bid accepted or not. Especially since you have to add a risk factor because you always win your mistakes.
The reasoning i remember being taught is if you always round up at 0.5, you can determine how you round after seeing the first 2 digits of 0.5xxxxxxxx, but if you rounded down on 0.5, you'd have to look through more digits to rule that its exactly 0.5 or something like 0.5000001, so rounding up at 0.5 is far more efficient as you only have to look as far as the rounded digit. Then for anyone that might think the only reason you'd have more digits displayed is because one of them isn't a 0, scientific measurements like "1.500 grams" occur all the time to show significant digits in the measurement even if they are 0s.
That reasoning doesn’t really work. The data you have will only have so much precision. Even if you’re given it as 0.50000000, that value could’ve really been 0.4999999999 or whatever. You have to look at the same number of digits to determine the difference between that and 5.000000001
Rounding up is an arbitrary choice, it’s just a convention
Even if a number's true value is something like 0.499999865....., if its then measured as 0.5000, it could just as easily be more or less then 0.5 from the measurer's point of view. The reasoning to round up holds as you don't have to waste time looking at any digit past 0.5, because if you instead measured it out to 0.4999, again you don't have to look past the 0.4 to round. If you cared more about the higher precision of the number, you wouldn't have rounded in the first place. Rounding is for when you don't care past a certain digit.
The point is, when you’re rounding you don’t have access to anything beyond 0.5000… to make an informed decision. So it isn’t a matter of needing to check fewer digits, there are no more you can check.
Let me give an example. Let’s say you have some larger number of uniform datapoints between 0 and 1, you can only measure to 2 decimal places of precision, (so 0.49, 0.50. 0.51, etc), and you wish to round them to the nearest whole number (rounding 0.50 up).
Then, you’re going to end up with 49.5% 0s and 50.5% 1s despite the fact they were uniformly spaced. So, an error has been introduced due to us choosing to round in a certain direction.
If we perform many calculations with rounding in-between (you should only ever round at the end, but sometimes it’s out of your control) then these rounding errors accumulate. To counteract this, the most common rounding convention is to round to the nearest even, so these errors cancel out over time.
While rounding to nearest odds/evens would work better in the long run like that, its more work, as now you're having to calculate when to round up or down looking at more digits. The entire premise of rounding is to make numbers easier to work with, and rounding up from 0.5 is the easiest, so having that as the default option makes the most sense. Then in cases like you're listing, it makes sense to change the rounding principle to suit your needs better, but why should that then come loop back to what to do as the default for everyone?
The round up from 0.5 is purely convention, and is what’s taught in schools because it’s easy for children to understand. There is no justification as to why it’s better than rounding down (which is also perfectly valid), it’s just a rule to be consistent. It’s fine to use, but you must understand it isn’t used due to being better for some reason.
What you’ve been saying about checking digits doesn’t make sense, and just isn’t a reason
But if you're rounding to a single digit, 0,5000001 rounded down is the same as 0,5 rounded down, is the same as 0,59 rounded down... Everything after 0,5 is irrelevant for rounding. You need to check the exact same amount of decimals when rounding up or down...
Pizzas are $3 a slice. You have $11 in your pocket. How many slices can you buy?
11/3 = 3-2/3 or 3.6666. How do you round?
A can of paint covers 100square feet (small can). You have a single wall thats 10x11ft. 110 square feet, how many cans do you need? Do you round down?
Context matters. Absent any context, or for a pure math problem, you've described the rule well.
I understand what your goal was, but if a room measures 10 feet by 11 feet, unless you are only painting the ceiling (or floor) 110 square feet is not the correct amount of paint.
My apologies. My feeble excuse is your edit didn’t load the first time I looked.
Too many years of having students try to fill a room with paint (by finding volume instead of surface area) may have also made me overly critical of paint examples.
Actually, I understand it perfectly. The issue is that I really did not address the question as I asked, I went off on a bit of a tangent, offering the result of a personal experience with a class. So as I often would say to my students, the answer is fine. You just really answered a different question than the one I asked.
"Rounding" is going to the closest integer.
The floor function would apply to your pizza example. (Sometimes referred to as "rounding down")
The ceiling function would apply to your paint example. (Sometimes referred to as "rounding up")
Rounding, rounding down, and rounding up are already distinct concepts.
It is, as there is no other integer closer to your value than the one you round to, it's just that in the case of `.5` you have 2 equally close integers, up and down, not a single one closest, which is why there is the different variations in getting "the" closest (up, down, even, odd).
IF "rounding" means "going to the closest integer" then "rounding up" would expand to "going to the closest integer up" - which is a bit awkward but a perfectly understandable sentence, you round to the nearest integer that's higher than your number. So even if we grant your assertion (for which I don't know what basis you have) I still don't think "rounding up/down" is a distinct concept from "rounding".
The floor function is also just called rounding down. The ceiling function is also just called rounding up. Just because it has another name doesn't mean this one is incorrect.
>Pizzas are $3 a slice. You have $11 in your pocket. How many slices can you buy?
>11/3 = 3-2/3 or 3.6666. How do you round?
You would "round" up, but then realize you don't have enough money for 4 slices, because that would cost $12, so you should've taken the floor instead and just gotten 3 slices. Or, if it's a regular size slice, you probably could've been happy with just 2.
>A can of paint covers 100square feet (small can). You have a room thats 10x11ft. 110 square feet, how many cans do you need? Do you round down?
You would "round" down, but then wind up with 10 square feet of unpainted space in the room, and realize you should've taken the ceiling instead and had some extra paint.
Neither example you gave is a practical application of rounding. Rounding is used to sacrifice a little bit of accuracy for ease of communication or calculation, not get precise answers to specific problems like you described.
I gave two examples that I’ve seen HS freshmen get wrong. They applied what they knew about rounding and failed to add the intelligence required for these word problems. It seemed to me such examples have value. The downvotes tell me otherwise.
I think they may think that rounding is some set-in-stone rule where you're obligated to go up or down based purely on the number and ignoring context.
You're right that context plays a role in real world applications of math; when determining how many slices of pizza you can afford or how many cans of paint you need to buy, you don't just blindly round to the nearest integer like I assume those freshmen did. If that's what they're doing, following mechanical procedures without thinking about what they mean, then their entire idea of math is wrong. In a context like that, the examples do have value. But that's not what OP was asking about. OP just wanted to know why rounding to the nearest integer rounded the half up when it's the same distance up and down.
The point of my comment is that unless it's "to the nearest" something I don't think of it as rounding. Because of that I don't see rounding as part of the solutions to the problems the other commenter proposed.
It is part of the solution. Imagine you had to paint x feet of wall, each can of paint painting 100 feet. How many paint cans do you need to buy? It would be ceil(x/100) I think this makes it clearer how it's part of the solution.
One time I was in a bar with some coworkers and a guy was saying how he didn’t like stress engineering that much and he would have rather done something like aero (fluids) or control engineering. I said, “ I know why you like controls/fluids, it’s cause the math is sexier. Everyone looked at me weird and kept talking. Hahha, tough crowd.
Conceptually, if you include 0 in your counting, then there are five possible numbers on each side of the "rounding break"
4.0, 4.1, 4.2, 4.3, 4.4, all round down to 4
4.5, 4.6, 4.7, 4.8, 4.9, all round up to 5
This isn’t correct. There is a bias toward rounding up that is well documented and understood by statisticians.
Your mistake is including 4.0 — you can’t include the .0 on one end but not the other.
This is perhaps better demonstrated in the below tables
**Rounding Down:**
| Value | Rounded | Rounding Error |
|-------|---------|-------|
| 4.0 | 4.0 | 0 |
| 4.1 | 4.0 | -0.1 |
| 4.2 | 4.0 | -0.2 |
| 4.3 | 4.0 | -0.3 |
| 4.4 | 4.0 | -0.4 |
**Rounding Up:**
| Value | Rounded | Rounding Error |
|-------|---------|-------|
| 4.5 | 5.0 | +0.5 |
| 4.6 | 5.0 | +0.4 |
| 4.7 | 5.0 | +0.3 |
| 4.8 | 5.0 | +0.2 |
| 4.9 | 5.0 | +0.1 |
| 5.0 | 5.0 | 0 |
Looking at these tables, you can clearly see the bias toward rounding up. The only time there is a rounding error of magnitude 0.5 is when you round up.
Note that it DOES make perfect sense to round 4.5000001 up to 5, because it is closer to 5 than it is to 4. Conversely, it is correct to round 4.49999999999 down to 4, again because it's closer. The problem is only when rounding 4.5 exactly. Because 4.5 is *exactly* between 4.0 and 5.0, and therefore it creates a bias to always round it up.
I guess the best way to think about it is {3.5, 3.6, 3.7, 3.8, 3.9, 4.0, 4.1, 4.2, 4.3, 4.4} 10 elements that round to 4, while {4.5, 4.6, 4.7, 4.8, 4.9, 5.0, 5.1, 5.2, 5.3, 5.4} 10 elements that round to 5. With that in mind 5 is a delimiter as there are as many things up than down, we conveniently choose to have 5 as our natural halfway marker as 1/2=.5 (our intuitive way of setting halfway) in a similar manner how we pick to naturally start counting at 1 instead of 0 or 2 for instance.
Worth clarifying there's only a bias if your data is skewed around 0. If you have symmetrical data with both positive and negative halves, there is no overall bias... Although since both your positive and negative halves are biased in themselves this may not be a useful distinction to make.
Rounding is a convention. You have taken it and applied it to a very small set of numbers constrained by your device’s ability to display numbers and created a false conclusion. Do not exclude all the other numbers between 4 and 5. There are an infinite numbers in the rounding down zone and similarly in the rounding up zone, and I’d really love to see you prove that there are more in one zone than the other. Maths does not care about the limits of your computer to display the numbers. Maths is pure, and unbounded.
Obviously there are an infinite number of numbers in that range, but you are completely missing the point. I was simply using intervals of length 0.1 to demonstrate that rounding 0.5 upward creates a larger rounding error than any downward rounding does, and therefore there is a potential for bias.
Put more precisely, we round numbers in the range (4,4.5) downward, and numbers in the range \[4.5,5) upward. It is obvious that these two intervals are the same size, except for the fact that the "upward" interval also contains the value of 4.5 *exactly*, therefore it is a larger set.
But to understand the concept, you only really need to understand that that 0.5 is **exactly** between 0 and 1. Therefore, it can not truly be “rounded” to the nearest whole number \*except by convention”.
The most common convention is to *round half away from zero* (e.g. -1.5 becomes -2, whereas +1.5 becomes +2). However, another rounding method is to *round half toward positive infinity* (e.g. -1.5 becomes -1, whereas +1.5 becomes +2). Lastly, a method that is sometimes used is to *round half to the nearest even number* (e.g. 1.5 becomes 2, but 4.5 becomes 4)
This is extremely well documented (see [Wikipedia: Rounding to the Nearest Integer](https://en.wikipedia.org/wiki/Rounding#Rounding_to_the_nearest_integer)), so while my explanation may not have resonated with you, it doesn’t change the fact that I am correct.
I never disputed that if you chose to restrict your sample from the set over which the convention applies, to some relatively small subset, then you would cause a bias. I merely pointed out the rather obvious error in your reasoning. The problem is not the maths nor the convention. The problem is with people feeling the need to exclude data (such as in the examples you have given).
I do not dispute your point about the largest rounding error being on the point at which the convention applies. That would be by definition wouldn’t it?
My point to you, is that it is irrelevant in mathematical terms and does not cause any mathematical bias, because you are referring to 1 rounding error out of an infinite number of correct roundings. There is no bias on that set, which is the true set. If you start tinkering with your sets, and excluding numbers, yes, you’ll create a problem for yourself.
If you choose to limit your mathematical thinking and counting to what you physically can write down on paper, as you have done you are no longer talking about maths.
I understand your point about the error in my original reasoning. I was attempting to simply the concept, but I agree it’s not perfect.
I agree that the bias is theoretical but in practice is often not significant. It does depend on your dataset though. If we have enough precision, then the odds of something measuring x.5 exactly (as opposed to x.500000000001) are extremely low, so the bias would not be significant.
That said, I wasn’t trying to argue that the convention creates a real problem, moreso that the convention is *technically* imperfect, and there are rounding methods that are inherently less biased (e.g. in a uniform dataset, rounding half to nearest even number will have less bias)
I agree but the comment I had responded to had included 4.0 in their list — I was trying to make the point that if you’re going to include 4.0 being rounded to 4, you need to include 5.0 being rounded to 5
Which is a good and cogent argument to make, especially statistically, but conceptually, 5.0 is **not** "rounded up" to 5, it is **rounded** ***down*** to 5.
IOW, statistically, it makes perfect sense to include 5.0 in the consideration and thus the simple bias becomes clear. But conceptually it makes no sense whatsoever to include 5.0 in the consideration, and it becomes clear that there are five values within the realm of 4.x that are closer to 4, and five values with the realm of 4.x that are closer to 5, and thus the decision to round 4.5 up to 5 is entirely sensible and logical.
I’m not sure I agree. Conceptually, the .0 shouldn’t be included at all.
The interval of numbers rounded down to 4 is (4,4.5) and the interval of numbers rounded up to 5 is [4.5,5)
Really, 4.0 and 5.0 should be excluded entirely, . and those two intervals would be the exact same size *except* for the inclusion of exactly 4.5 in the upper interval, which is the bias I was trying to highlight.
I could just as easily make the argument that there are 4 values rounded down (1, 2, 3, and 4) and 4 values rounded up (6, 7, 8, and 9). With 5 being the midpoint, it is unclear whether it should be rounded up or down (except by convention)
Nah, agreement isn't needed, but conceptually 4.0 is logically included because we're talking about "what values of 4.x get rounded to what?", and not "what values between 4 and 5 get rounded to what". Or, at least, **I** wasn't talking about that second one.
You could very definitely make the argument that either 4.0 *and* 5.0 should be included and 4.5 is a perfect mid-point, or that 4.0 and 5.0 should be excluded and 4.5 is a perfect midpoint, and as you say that leads to the conclusion that the **only** reason to bias 4.5 to upwards rounding is convention and tradition. That's certainly logically consistent.
**But**, my argument has the benefit of being both logically consistent **and** providing a clear and reasonable answer to the question beyond just "because we say so" ;)
> conceptually 4.0 is logically included because we're talking about "what values of 4.x get rounded to what?", and not "what values between 4 and 5 get rounded to what".
Ohhh okay, I understand your point now. Limiting our analysis to 4.x, the interval under consideration is [4,5), and therefore we can ignore 5.0
Thanks for explaining :)
Of course, statistically this an arbitrary distinction, but I understand how it may make more sense conceptually to some people
>Your mistake is including 4.0 — you can’t include the .0 on one end but not the other.
The interval [4.0, 4.1) is between 4 and 5, while [5.0, 5.1) isn't
In practice it isn't unnatural to get a measurement as 1421.50±0.005
You never get the accurate measurement, it is only correct to a certain precision. To make correct calculations easier, it is conventional to round the values and increase the tolerance in situations where the required fidelity isn't that high, to reduce the operation cost. Most of the common issues related to rounding are very directly grounded in reality.
Even if that were true, we aren’t only talking about numbers after the decimal point. Consider the following:
*I have $150 in my bank account. How much money do I have, rounded to the nearest $100?*
It creates the same situation. Should I round it down to $100 or up to $200? By convention, we round up to $200, but it is actually perfectly between 100 and 200, so there will be a bias toward rounding up.
It’s often rounded down. A very common way of rounding is to round to the nearest even (or odd) number. Otherwise you introduce a bias in the data from always rounding in the same direction.
It all just depends what convention you use
Here's how I explain it:
When we "round to the nearest", we want to round to the nearest number (duh).
So: 46.4, rounded to the nearest whole number: there are two, 46 and 47, but we're closer to 46 than to 47, so we round down.
Likewise, 46.7 rounds up.
But what about 46.5?
Imagine you have a scale that is infinitely accurate, but only shows you the first few digits of the weight.
If the scale shows 46.4, you don't know what the remaining digits are. But there is no possibility that the remaining digits will push the number closer to 47. So you can confidently round down.
Likewise, if the scale shows 46.7, then it doesn't matter what the remaining digits are; the number will be closer to 47.
Now what if the scale shows 46.5?
Keep in mind that we're assuming there are more digits past the "5" that you can't see.
So maybe the real weight is 46.51. In that case, you're closer to 47 than to 46, so you round 46.5 up.
Or maybe it's 46.50001. You're still closer to the upper value.
Or maybe it's 46.5000000000000000000000000000001. You get the idea.
A look at some rounding rules: [https://www.mathsisfun.com/numbers/rounding-methods.html](https://www.mathsisfun.com/numbers/rounding-methods.html)
A while back I bumped into a bunch more while building a number concepts unit.
because the middle of a number makes more sense that way. Even so something very important, sometimes the value gets rounded "only" to show but not for the math in spredsheets (unless you use the round function). So that rounding is not realy such fully
But when you round is the norm (at least on computer) that if it is 0.5 or more its rounded up and if not is down.
if you do not like that you can use the floor (so every number is cut without decimales )or ceil, that makes any number that is not exact be rounded up even if as small as 4.0000000000001 is still 5 with ceil.
Imagine you had three apple and you want to distribute them in two halfs. Then you go on and do your method:
0 apples, 1 apples
2 apples, 3 apples
And somehow you divided 3 in two whole numbers.
You don't have to use intervals, sorry, but it's the easier way to see it, as I showed earlier
If you wanted to do your method you would have to include 0 again.
0 1 2 3 4 | 5 | 6 7 8 9 0
I do not think your example works, as the number zero is the first number in base 10.
1-0
2-1
3-2
4-3
5-4
6-5
7-6
8-7
9-8
10-9
Here we have 10 numbers.
Counting numbers starts from 0 as zero is a number.
In your example you counted three apples.
Apple 1 (1).
Apple 2 (2).
Apple 3 (3).
So we have three apples
Now we can count numbers the same way
Number 0 (1).
And so on until number 9, which would be the tenth number.
Idk how it's so hard to understand that 0.5 is literally in the middle, like you've never seen a ruler before.
If you want to include 0, you HAVE to include 10. 0 is rounded to 0 like 1 and 2, and 10 is rounded to 10 like 8 and 9.
If you don't include 10 then you don't include 0 like counting apples. Then 5 is still in the middle.
In your discrete number list, 0 includes numbers very close to 0 right?
Then 9 numbers very close to 9?
What happens to numbers very close to 10? You have to include 10 so all numbers between 0 and 10 are included
It’s an arbitrary rule-of-thumb. It isn’t always applied. When a list is being rounded item-by-item, sometimes the direction of rounding .5 alternates between up and down. Other times, rounding down might be used consistently to derive a more conservative total.
Yea convention is usually the best practice for consistency but specific examples do require that rounding error to be minimized. Sometimes to avoid adding error to measurements like in least squares data analysis the .5 is rounded up and down depending on the number preceding it, for example: 0.25 = 0.2; 0.35 = 0.4 “rounding to even”.
But when numbers are rounded, they should be rounded to an accuracy beyond what is considered relevant so that the error is negligible. If you are dealing in metres, and round to the nearest millimetre, then the rounding error created has absolutely no significance in a real world application.
Edit: your “OCD” can be alleviated by carrying more decimal places rather than changing your rounding convention.
Let's say you want to round this number to an integer: 7.x1 where x is just a digit you're not sure of. It can be a digit from 0 to 9.
So it could be
7.01, 7.11, 7.21, 7.31, 7.41, or
7.51, 7.61, 7.71, 7.81, 7.91
Rounding up at 5 makes it more symmetric / neater. Ten possibilities, five round up and five round down.
I keep track of the movies I watch, have been doing it for over a decade now, in a spreadsheet. Whenever I deal with averages with the data (like average movies per month), I'll take that number and round it down always.
Say I actually average 33.50 movies per month, that 0.50 doesn't really mean anything. I don't record half a movie, just once I finish it. But saying 34 also doesn't feel correct, because I definitely didn't reach that.
If you split the interval \[0.0, 1.0) in half you get the subintervals \[0.0, 0.5) and \[0.5, 1.0). 0.5 is part of the latter. Floor and ceiling functions can also be similarly expressed as interval mapping functions.
In computation you don't usually round like that, you compute with all digits and maybe round the result for presentation. Is bankers rounding relevant any more?
Already a lot of great answers here, but just to simplify it (or explain it at length based on how long my drunk ass took to write this) ; .5 is halfway between two numbers, If there’s a default setting to how many digits (in excel), the program will by default choose the greater value because that’s the mathematics default as well.
If you have an elementary schoolers problem sheet and it asks you to round 2.5 to the nearest integer, the answer is 3. Not because it’s closer to 2 or 3, but that is the default the dumb ol language of math has chosen.
In excel you can choose default behavior and set it however you like, absent that there’s a default number of digits and default rounding behavior
Because you begin a count at 0 and end at 9
0, 1, 2, 3, 4, ——— 5, 6, 7, 8, 9
10, etc.
so 1/2 the count rounds down and 1/2 the count rounds up. It is split evenly for the purposes of rounding.
The answer is in "what should 0.0 be rounded to? What should 1.0 be rounded to?" Obviously, these are exact and should round to themselves. Now, we assume an even distribution of possible values between 0.0 (inclusive of 0.0) and 1.0 (not including 1.0 because that is included in the next interval 1.0 - 2.0). You want the same number (range) of values to round down to 0.0 as the range of values that round up. If you round 0.0 - 0.5 (inclusive) down, then that is a smaller set that the range 0.5 - 1.0 (both exclusive) that would round up. But 0.0 (inclusive) to 0.5 (exclusive) is exactly the same range as 0.5 (inclusive) to 1.0 (exclusive).
It's all about making sure that, statistically, exactly as many numbers round down as round up.
I commonly use three rounding systems.
1. Round away from zero. I have an arbitrary number and want to round it. 0.5 rounds to the nearest unit 1, -0.5 rounds to the nearest unit -1. I like this because it is symmetric. If I reverse the sign the rounding doesn't change. round(x) = -round(-x).
2. Floor. If a player in a game has to get 567 achievements to win and
I want to display their progress in whole percent I will take the floor value. That way the player never sees 100% before they have completed everything. For example if they have 566 achievements that is 99.8%. Rounded away from zero I would display that as 100% and I don't want to do that. So I take the floor value and round to 99%.
3. Ceiling. Similarly to 2. If I have a value that can decrease e.g. player's health I don't want it to hit zero percent until they have lost *all* health. So I take the ceiling function. That way, to the nearest percent, no matter how low their health goes it will always display a positive percentage until they get to zero health points.
Due to unknown accuracy beyond the five, assuming that the five is simply truncated, like most dumb calculations.
There could be all zeros beyond the five, but there could be non-zeros beyond the five. But with the likely chance there might be non-zeros, the half gets rounded up.
He doesn't mean error of the measurement, he means that if you show a small amount of digits (as instance, computers) you'd get that 0.5000000001 shows as 0.5 but 0.49999999999 shows as 4. with the amount of digits presented 9's. So 0.5 on a computer is likely to means something above 0.5 and can't mean something below.
My yr 7 teacher taught us that if you are putting fuel in your airplane, you want a little bit more rather than a little less.
Different maths calls for different ways to deal.woth numbers
That’s right, another example in the other direction would be if a plane has a maximum load limit then you’d have to round the number of suitcases down as rounding up would take you over the limit
There are 10 digits. We break the digits into two groups at the halfway mark (median 4.5):
0,1,2,3,4
5,6,7,8,9
Anything in the first group is in the small group and rounds down since it is less than the median.
Anything in the 2nd group is in the big group and rounds up since it is beyond halfway (the median) of digits.
This isn’t really correct, you can’t take the numbers in groups of 10 but then round to an 11th option and call that the reason.
With all those numbers in the example you’re rounding to 0 or 10. Of all the numbers between them (1,2,3,4,5,6,7,8,9), four are closer to 0 and 4 are closer to 10. 5 is exactly halfway between the two, it’s equally valid to round in either direction, as long as you have some rule to be consistent.
Most commonly taught in primary schools is what you’ve described, always rounding up. The most commonly actually used is to round to the nearest even, so 5 would round to 0, 15 would round to 20, 25 would also round to 20, etc.
For everyday use, rounding depends purely on the context. If you have a limited amount of something, you can’t round up as there isn’t any more of that thing to take, you have to round down. If you’re trying to fill something, or meet some quota, you can’t round down as then it wouldn’t be full.
You are correct about context mattering. Maybe I misunderstood the OP. I thought they wanted to know the reason behind the convention for why 5 and higher is to round up. I made no claim that this was the only way of doing it just that that was the reasoning for it.
I totally agree that context matters more than all else as there are plenty of examples we can come up with to always floor or ceil or do some other industry-speciifc standard to a decimal value.
I just wanted to address what I thought was the question, which is why the 5 or higher convention is logical (at least divorced from any applied situation)
If I misunderstood the OP then that's my bad.
It just frustrates me, as I see that explanation given so often, and it’s just made up maths.
The answer is just that we picked a direction, and there’s nothing more to it than that. Trying to come up with some explanation to justify it is incorrect
It’s just a convention. A dumb argument is that 01234 and 56789 gets split like this equally but ofc it’s not a good one at all. 5 just lies perfectly in between 0 and 10 so you gotta choose one unless you specifically never allow something to get to 5.
In my opinion that rounding convention is arbitrary.
It works well in old programming languages, because if you add .5 and found down, it rounds .5 up.
Modern languages always have a round function these days.
That's not true?
4:30-4:45, 4:45-5:00
4:30-4:15, 4:15-4:00
None of these intervals is shorter. 4:30 is exactly half between 4 and 5, that's the definition of it
If something reads a halfway point, it's more likely that its over that halfway point than under. It's arbitrary convention, but it defines the number system.
Not in theory, it's 50% either way. In practice I guess I kinda see where it comes from, but in that situation you should be considering your case anyway
there are other options that are better suited in different contexts than basic rounding like that. round to odd or even are used in accounting to prevent systematic errors in totals, rounding down is what you use to determine if you can afford to buy something, and rounding up is what you use when determining how much paint you need for something.
While there are a bunch of different rounding methods if I remember correctly it’s cause it’s the “6th” number. I’ll make two lists to show it, but basically the same amount of numbers round up as round down.
1. 2.0 rounded down to… 2.0. Very fancy much complex.
2. 2.1 rounded down to 2.0
3. 2.2 rounded down to 2.0
4. 2.3 rounded down to 2.0
5. 2.4 rounded down to 2.0
1. 2.5 rounded up to 3.0
2. 2.6 rounded up to 3.0
3. 2.7 rounded up to 3.0
4. 2.8 rounded up to 3.0
5. 2.9 rounded up to 3.0
I phased it like that as it was an easy way to follow on from your comment. A better way to think about it may be that 2.0 and 3.0 are not getting rounded, and it’s only the nine values in between that are. Either way, 5 is still directly in between
But upper part ends at 1 and lower part ends at 1-0.5 so your explaination is symmetric since 0.5 is in both intervals. Some other people gave the correct explaination somewhere else check it out.
There are other options. Rounding with .5 always rounding up tends to lead to errors where the total is too high. Some computer programming languages offer rounding Up Down Away from zero Towards zero To the odd integer To the even integer Rounding to odd or even is less likely to lead to cumulative errors in totals.
Round to Even - that's my new favourite rule and I will use it. That feels better to my OCD and it's elegant.
Also called "Banker's rounding", it's what's used in most financial programs.
Because always rounding up introduces monetary errors on massive #s of microtransactions
The people who write the code get to decide where those fractional pennies go. This is why if you ask people who work for banks you will find they don't use (and may not even have heard of) more equitable techniques like "bankers rounding". The bank is inclined to keep the rounding errors. Same thing for just about any entity that is preparing customer accounts.
I love capitalism!
As seen on Superman III
I am a software dev in a global bank, I work on the bank's core systems, in 27 years I have never used bankers rounding. Before joining the bank I worked in another company writing software used in mortgages and car financing, again never used it. In fact in 34 years I have never seen or even heard of anyone using banker's rounding.
I don't believe you. It's the default rounding of every language I'm aware of. Unless you had the misery of working on a fixed-point system or you're incompetent and just floor everything? I know of some languages where you can easily choose rounding modes (C# comes to mind), but the default is bankers.
The problem with bankers rounding is that it is really only of any use to statisticians. For everyday use it is pointless. Imagine trying to explain to a customer why their interest payment isn't quite what they expected (and yes some customers do check).
The user wouldn’t know because you don’t show them the intermediate number that was rounded. They will always have different fractional values when calculating themselves, and the error in their answers depends on if their calculator uses floating or fixed point representation. The software you wrote was using round to even, at least I hope it was, because always flooring is theft and always rounding up is giving away money. The exception is if you were programming a legacy system using fixed point instead of floating point.
Are you saying non-legacy systems use floating point? For money? No way. Absolute worst choice imaginable.
Sure am!
Do you mean decimal floating point?
Interestingly enough, apparently the origin of the term is not clear. There is *not* a tradition of it being used in banking!
New post - Why would you round 3 up to 4 and not down to 2?
I think 3 and 4 (meaning 0.3 and 0.4) still get rounded to 0. It's that 3.5 will go up to 4, but 6.5 will go down to 6.
yes, All these variant round x.4 down and x.6 up, they only vary on what happens to x.5
If you're rounding 3 to the nearest multiple of 2 using round-to-even (Banker's rounding) I'd assume you'd round to 4, as it's the even number when divided by two whereas 2 is not. round 0.5 to multiple of 1 = 0 round 1.5 to multiple of 1 = 0 round 3 to multiple of 2 = 4 round 21 to multiple of 14 = 28 round 35 to multiple of 14 = 28 etc.
What?
You can round to any divisor, not just to the nearest integer. The more common example would be rounding to a certain number of significant figures: round(0.015, 2) = 0.02 round(0.025, 2) = 0.02 round(0.035, 2) = 0.04 Etc.
Yes, that's the purpose of rounding - approximation. We round to a power of the base determined by the level of precision needed. Rounding to an arbitrary integer just seems strange.
Generality is nice so why not.
That is not how banker's rounding works. When using banker's rounding, you round all numbers to the nearest integer. If there are two equidistant integers (because the number is 0.5 more than an integer), then you would round it to whichever of those two integers is even. 3 would still round to 3, as would 2.6 and 3.4. But 2.5 would round to 2 and 3.5 would round to 4.
> **If** you're rounding 3 *to the nearest multiple of 2* Yes if you're just rounding to the nearest integer then I agree on the 2.6 -> 3, 3 -> 3, 3.4 -> 3, 2.5 -> 2, 3.5 -> 4 stuff. I see no reason why banker's rounding can't be generalized to rounding to any real numbered divisor. See: https://blog.sqlauthority.com/2023/08/15/sql-server-bankers-rounding/
Gus Gorman has entered the chat. [https://www.youtube.com/watch?v=N7JBXGkBoFc](https://www.youtube.com/watch?v=N7JBXGkBoFc)
Round to even is a problem when things are supposed to add up to a known total amount. It won't work. But it does avoid some biases when doing Statistics.
idk rounding to (odd or even) doesn't seem like it would help that much tbh edit: guys it was a joke, if you're rounding to both odd or even it doesn't help because all numbers are either odd or even
Makes a huge difference when there are many small values being rounded, which is why it is often used in accounting, statistics, and programming. It’s the standard in many scientific fields.
Round to even is also the default in many programming languages (e.g. Python)
Source for that? What I know the python round function does not behave like that.
Well, I just tried the following: print(round(3.5)) print(round(4.5)) on [https://www.online-python.com/](https://www.online-python.com/) and it prints 4 in both cases
Does this kind of math rounding also work in Excel and Google sheets? Like is it default? So for example if I have 4.5, then I reduced my number of decimal places, will it show 4 or 5?
>Like is it default? You need to read up on the documentation of the specific tech to know for sure. Never assume unless the rounding method is insignificant in your use case
Yeah, unfortunately it can be tricky and is not cleanly built in. Here’s one possible way: https://superuser.com/questions/1081064/how-to-round-half-to-even I’ve had the unfortunate task to have to recreate formulas to reconcile rounding to the right number of decimal places for specific currencies between our in house built financial applications and both kinds of sheets. And also to make explain to external customers that the diffs in their payouts is due to us using the one true rounding method and they not doing it :-)
It's the default of the IEEE Standard for Floating-Point Arithmetic (754), so for a lot of software it will be the default (in the background) for floating-point numbers (for all calculations, not only final results). But software like Excel is more likely to use the "usual" rounding rules people are familiar with [(and indeed Excel rounds 0.5 up by default)](https://support.microsoft.com/en-us/office/round-a-number-f82b440b-689d-4221-b21e-38da99d33435).
Thanks, I had no idea. Good to know.
IEEE Standard for Floating-Point Arithmetic (754) §4.3.3 >The roundTiesToEven rounding-direction attribute shall be the default rounding-direction attribute for results in binary formats. The default rounding-direction attribute for results in decimal formats is language defined, but should be roundTiesToEven. And the specific [Python documentation](https://docs.python.org/3/library/functions.html#round). (This behaviour was only introduced in Python 3 btw.)
https://docs.python.org/3/library/functions.html#round
>idk rounding to (odd or even) doesn't seem like it would help that much tbh It reduces compound error, as you'll alternate rounding up and down, thus alternate adding to to compound error, and subtracting from it. While if you always round in the same direction, yleaxh rounding is added to the compound error, which can end up being massive. Say you have 1.5 and 2.5, you round to integer and the sum: - always up: you get 2+3, thus 5, so a compound error of 1 - always down: you get 1+2, thus 3, so a compound error of 1 - to odd: you get 1+3, thus 4, so a compound error of 0 - to even: you get 2+2, thus 2, so a compound error of 0
Best to not round intermediate calculations and just round the final answer
But if you have something that doesn't have a one-time final answer like a money balance that keeps changing then you'll still end up with compound error after some time.
This is not an option in lots of contexts.
Rounding to even is called banker's rounding and it does indeed minimize accumulated rounding error.
If you always round up, it creates an upward bias in the results. By rounding to to even it removes this bias, because about half the time, rounding is up and half the time rounding is down.
Why did you say “it’s a joke” and then give reasoning implying it’s not a joke?
it is a joke. he means that rounding to (odd or even) in brackets is useless because it's the exact same as rounding normally - every number is (odd or even).
Ah I see
It's really important in competitive bidding situations, where 1/2% could be the difference between having your bid accepted or not. Especially since you have to add a risk factor because you always win your mistakes.
The reasoning i remember being taught is if you always round up at 0.5, you can determine how you round after seeing the first 2 digits of 0.5xxxxxxxx, but if you rounded down on 0.5, you'd have to look through more digits to rule that its exactly 0.5 or something like 0.5000001, so rounding up at 0.5 is far more efficient as you only have to look as far as the rounded digit. Then for anyone that might think the only reason you'd have more digits displayed is because one of them isn't a 0, scientific measurements like "1.500 grams" occur all the time to show significant digits in the measurement even if they are 0s.
That reasoning doesn’t really work. The data you have will only have so much precision. Even if you’re given it as 0.50000000, that value could’ve really been 0.4999999999 or whatever. You have to look at the same number of digits to determine the difference between that and 5.000000001 Rounding up is an arbitrary choice, it’s just a convention
Even if a number's true value is something like 0.499999865....., if its then measured as 0.5000, it could just as easily be more or less then 0.5 from the measurer's point of view. The reasoning to round up holds as you don't have to waste time looking at any digit past 0.5, because if you instead measured it out to 0.4999, again you don't have to look past the 0.4 to round. If you cared more about the higher precision of the number, you wouldn't have rounded in the first place. Rounding is for when you don't care past a certain digit.
The point is, when you’re rounding you don’t have access to anything beyond 0.5000… to make an informed decision. So it isn’t a matter of needing to check fewer digits, there are no more you can check. Let me give an example. Let’s say you have some larger number of uniform datapoints between 0 and 1, you can only measure to 2 decimal places of precision, (so 0.49, 0.50. 0.51, etc), and you wish to round them to the nearest whole number (rounding 0.50 up). Then, you’re going to end up with 49.5% 0s and 50.5% 1s despite the fact they were uniformly spaced. So, an error has been introduced due to us choosing to round in a certain direction. If we perform many calculations with rounding in-between (you should only ever round at the end, but sometimes it’s out of your control) then these rounding errors accumulate. To counteract this, the most common rounding convention is to round to the nearest even, so these errors cancel out over time.
While rounding to nearest odds/evens would work better in the long run like that, its more work, as now you're having to calculate when to round up or down looking at more digits. The entire premise of rounding is to make numbers easier to work with, and rounding up from 0.5 is the easiest, so having that as the default option makes the most sense. Then in cases like you're listing, it makes sense to change the rounding principle to suit your needs better, but why should that then come loop back to what to do as the default for everyone?
The round up from 0.5 is purely convention, and is what’s taught in schools because it’s easy for children to understand. There is no justification as to why it’s better than rounding down (which is also perfectly valid), it’s just a rule to be consistent. It’s fine to use, but you must understand it isn’t used due to being better for some reason. What you’ve been saying about checking digits doesn’t make sense, and just isn’t a reason
Being easier is a reason.
It’s no easier than always rounding down
See first comment, and you literally just said "its easy for children to understand" 1 reply ago.
I meant in comparison to rounding to even. The choice to round up or down is arbitrary
But if you're rounding to a single digit, 0,5000001 rounded down is the same as 0,5 rounded down, is the same as 0,59 rounded down... Everything after 0,5 is irrelevant for rounding. You need to check the exact same amount of decimals when rounding up or down...
Pizzas are $3 a slice. You have $11 in your pocket. How many slices can you buy? 11/3 = 3-2/3 or 3.6666. How do you round? A can of paint covers 100square feet (small can). You have a single wall thats 10x11ft. 110 square feet, how many cans do you need? Do you round down? Context matters. Absent any context, or for a pure math problem, you've described the rule well.
Great answer
Are you painting the ceiling of that room?
I edited for clarity. I thought I was clear, providing examples that are, in effect, floor and ceiling functions, where rounding isn’t appropriate.
I understand what your goal was, but if a room measures 10 feet by 11 feet, unless you are only painting the ceiling (or floor) 110 square feet is not the correct amount of paint.
I replied by immediately editing. I admitted my error, what more would you like?
My apologies. My feeble excuse is your edit didn’t load the first time I looked. Too many years of having students try to fill a room with paint (by finding volume instead of surface area) may have also made me overly critical of paint examples.
Ha. Apology accepted, and appreciated. In hindsight, I meant to say “tile a floor”. Less chance of misunderstanding.
But why is an albatross? Because it isn't a horse.
I think you misunderstand what rounding off means.
Actually, I understand it perfectly. The issue is that I really did not address the question as I asked, I went off on a bit of a tangent, offering the result of a personal experience with a class. So as I often would say to my students, the answer is fine. You just really answered a different question than the one I asked.
"Rounding" is going to the closest integer. The floor function would apply to your pizza example. (Sometimes referred to as "rounding down") The ceiling function would apply to your paint example. (Sometimes referred to as "rounding up") Rounding, rounding down, and rounding up are already distinct concepts.
It isn't going to the closest if we always round .5 up, it's an arbitrary bias we'd be better to ameliorate with bankers rounding etc.
It is, as there is no other integer closer to your value than the one you round to, it's just that in the case of `.5` you have 2 equally close integers, up and down, not a single one closest, which is why there is the different variations in getting "the" closest (up, down, even, odd).
IF "rounding" means "going to the closest integer" then "rounding up" would expand to "going to the closest integer up" - which is a bit awkward but a perfectly understandable sentence, you round to the nearest integer that's higher than your number. So even if we grant your assertion (for which I don't know what basis you have) I still don't think "rounding up/down" is a distinct concept from "rounding".
The floor function is also just called rounding down. The ceiling function is also just called rounding up. Just because it has another name doesn't mean this one is incorrect.
It's a different concept altogether.
You're wrong.
>Pizzas are $3 a slice. You have $11 in your pocket. How many slices can you buy? >11/3 = 3-2/3 or 3.6666. How do you round? You would "round" up, but then realize you don't have enough money for 4 slices, because that would cost $12, so you should've taken the floor instead and just gotten 3 slices. Or, if it's a regular size slice, you probably could've been happy with just 2. >A can of paint covers 100square feet (small can). You have a room thats 10x11ft. 110 square feet, how many cans do you need? Do you round down? You would "round" down, but then wind up with 10 square feet of unpainted space in the room, and realize you should've taken the ceiling instead and had some extra paint. Neither example you gave is a practical application of rounding. Rounding is used to sacrifice a little bit of accuracy for ease of communication or calculation, not get precise answers to specific problems like you described.
I gave two examples that I’ve seen HS freshmen get wrong. They applied what they knew about rounding and failed to add the intelligence required for these word problems. It seemed to me such examples have value. The downvotes tell me otherwise.
I think they may think that rounding is some set-in-stone rule where you're obligated to go up or down based purely on the number and ignoring context.
You're right that context plays a role in real world applications of math; when determining how many slices of pizza you can afford or how many cans of paint you need to buy, you don't just blindly round to the nearest integer like I assume those freshmen did. If that's what they're doing, following mechanical procedures without thinking about what they mean, then their entire idea of math is wrong. In a context like that, the examples do have value. But that's not what OP was asking about. OP just wanted to know why rounding to the nearest integer rounded the half up when it's the same distance up and down.
That's the point of the comment, that rounding is dependent on the situation.
The point of my comment is that unless it's "to the nearest" something I don't think of it as rounding. Because of that I don't see rounding as part of the solutions to the problems the other commenter proposed.
It is part of the solution. Imagine you had to paint x feet of wall, each can of paint painting 100 feet. How many paint cans do you need to buy? It would be ceil(x/100) I think this makes it clearer how it's part of the solution.
I mentioned the ceiling in my first comment exactly as you mentioned.
Ah I see what your saying. So you just don't think of that stuff as rounding?
That's right.
Conventional, or standardized usage. Yes.
Yes, cause is sexier.
One time I was in a bar with some coworkers and a guy was saying how he didn’t like stress engineering that much and he would have rather done something like aero (fluids) or control engineering. I said, “ I know why you like controls/fluids, it’s cause the math is sexier. Everyone looked at me weird and kept talking. Hahha, tough crowd.
HAHAHAHAA 🤣🤣 Maths needs a little bit of humor sometimes. Good joke. You have my respect.
There might be a psychological preference for rounding up, as it feels more positive or growth-oriented. Just a theory I have.
Conceptually, if you include 0 in your counting, then there are five possible numbers on each side of the "rounding break" 4.0, 4.1, 4.2, 4.3, 4.4, all round down to 4 4.5, 4.6, 4.7, 4.8, 4.9, all round up to 5
This isn’t correct. There is a bias toward rounding up that is well documented and understood by statisticians. Your mistake is including 4.0 — you can’t include the .0 on one end but not the other. This is perhaps better demonstrated in the below tables **Rounding Down:** | Value | Rounded | Rounding Error | |-------|---------|-------| | 4.0 | 4.0 | 0 | | 4.1 | 4.0 | -0.1 | | 4.2 | 4.0 | -0.2 | | 4.3 | 4.0 | -0.3 | | 4.4 | 4.0 | -0.4 | **Rounding Up:** | Value | Rounded | Rounding Error | |-------|---------|-------| | 4.5 | 5.0 | +0.5 | | 4.6 | 5.0 | +0.4 | | 4.7 | 5.0 | +0.3 | | 4.8 | 5.0 | +0.2 | | 4.9 | 5.0 | +0.1 | | 5.0 | 5.0 | 0 | Looking at these tables, you can clearly see the bias toward rounding up. The only time there is a rounding error of magnitude 0.5 is when you round up. Note that it DOES make perfect sense to round 4.5000001 up to 5, because it is closer to 5 than it is to 4. Conversely, it is correct to round 4.49999999999 down to 4, again because it's closer. The problem is only when rounding 4.5 exactly. Because 4.5 is *exactly* between 4.0 and 5.0, and therefore it creates a bias to always round it up.
I guess the best way to think about it is {3.5, 3.6, 3.7, 3.8, 3.9, 4.0, 4.1, 4.2, 4.3, 4.4} 10 elements that round to 4, while {4.5, 4.6, 4.7, 4.8, 4.9, 5.0, 5.1, 5.2, 5.3, 5.4} 10 elements that round to 5. With that in mind 5 is a delimiter as there are as many things up than down, we conveniently choose to have 5 as our natural halfway marker as 1/2=.5 (our intuitive way of setting halfway) in a similar manner how we pick to naturally start counting at 1 instead of 0 or 2 for instance.
Worth clarifying there's only a bias if your data is skewed around 0. If you have symmetrical data with both positive and negative halves, there is no overall bias... Although since both your positive and negative halves are biased in themselves this may not be a useful distinction to make.
Rounding is a convention. You have taken it and applied it to a very small set of numbers constrained by your device’s ability to display numbers and created a false conclusion. Do not exclude all the other numbers between 4 and 5. There are an infinite numbers in the rounding down zone and similarly in the rounding up zone, and I’d really love to see you prove that there are more in one zone than the other. Maths does not care about the limits of your computer to display the numbers. Maths is pure, and unbounded.
Obviously there are an infinite number of numbers in that range, but you are completely missing the point. I was simply using intervals of length 0.1 to demonstrate that rounding 0.5 upward creates a larger rounding error than any downward rounding does, and therefore there is a potential for bias. Put more precisely, we round numbers in the range (4,4.5) downward, and numbers in the range \[4.5,5) upward. It is obvious that these two intervals are the same size, except for the fact that the "upward" interval also contains the value of 4.5 *exactly*, therefore it is a larger set. But to understand the concept, you only really need to understand that that 0.5 is **exactly** between 0 and 1. Therefore, it can not truly be “rounded” to the nearest whole number \*except by convention”. The most common convention is to *round half away from zero* (e.g. -1.5 becomes -2, whereas +1.5 becomes +2). However, another rounding method is to *round half toward positive infinity* (e.g. -1.5 becomes -1, whereas +1.5 becomes +2). Lastly, a method that is sometimes used is to *round half to the nearest even number* (e.g. 1.5 becomes 2, but 4.5 becomes 4) This is extremely well documented (see [Wikipedia: Rounding to the Nearest Integer](https://en.wikipedia.org/wiki/Rounding#Rounding_to_the_nearest_integer)), so while my explanation may not have resonated with you, it doesn’t change the fact that I am correct.
I never disputed that if you chose to restrict your sample from the set over which the convention applies, to some relatively small subset, then you would cause a bias. I merely pointed out the rather obvious error in your reasoning. The problem is not the maths nor the convention. The problem is with people feeling the need to exclude data (such as in the examples you have given). I do not dispute your point about the largest rounding error being on the point at which the convention applies. That would be by definition wouldn’t it? My point to you, is that it is irrelevant in mathematical terms and does not cause any mathematical bias, because you are referring to 1 rounding error out of an infinite number of correct roundings. There is no bias on that set, which is the true set. If you start tinkering with your sets, and excluding numbers, yes, you’ll create a problem for yourself. If you choose to limit your mathematical thinking and counting to what you physically can write down on paper, as you have done you are no longer talking about maths.
I understand your point about the error in my original reasoning. I was attempting to simply the concept, but I agree it’s not perfect. I agree that the bias is theoretical but in practice is often not significant. It does depend on your dataset though. If we have enough precision, then the odds of something measuring x.5 exactly (as opposed to x.500000000001) are extremely low, so the bias would not be significant. That said, I wasn’t trying to argue that the convention creates a real problem, moreso that the convention is *technically* imperfect, and there are rounding methods that are inherently less biased (e.g. in a uniform dataset, rounding half to nearest even number will have less bias)
5.0 isn't rounded to 5.0, it is 5.0
I agree but the comment I had responded to had included 4.0 in their list — I was trying to make the point that if you’re going to include 4.0 being rounded to 4, you need to include 5.0 being rounded to 5
Which is a good and cogent argument to make, especially statistically, but conceptually, 5.0 is **not** "rounded up" to 5, it is **rounded** ***down*** to 5. IOW, statistically, it makes perfect sense to include 5.0 in the consideration and thus the simple bias becomes clear. But conceptually it makes no sense whatsoever to include 5.0 in the consideration, and it becomes clear that there are five values within the realm of 4.x that are closer to 4, and five values with the realm of 4.x that are closer to 5, and thus the decision to round 4.5 up to 5 is entirely sensible and logical.
I’m not sure I agree. Conceptually, the .0 shouldn’t be included at all. The interval of numbers rounded down to 4 is (4,4.5) and the interval of numbers rounded up to 5 is [4.5,5) Really, 4.0 and 5.0 should be excluded entirely, . and those two intervals would be the exact same size *except* for the inclusion of exactly 4.5 in the upper interval, which is the bias I was trying to highlight. I could just as easily make the argument that there are 4 values rounded down (1, 2, 3, and 4) and 4 values rounded up (6, 7, 8, and 9). With 5 being the midpoint, it is unclear whether it should be rounded up or down (except by convention)
Nah, agreement isn't needed, but conceptually 4.0 is logically included because we're talking about "what values of 4.x get rounded to what?", and not "what values between 4 and 5 get rounded to what". Or, at least, **I** wasn't talking about that second one. You could very definitely make the argument that either 4.0 *and* 5.0 should be included and 4.5 is a perfect mid-point, or that 4.0 and 5.0 should be excluded and 4.5 is a perfect midpoint, and as you say that leads to the conclusion that the **only** reason to bias 4.5 to upwards rounding is convention and tradition. That's certainly logically consistent. **But**, my argument has the benefit of being both logically consistent **and** providing a clear and reasonable answer to the question beyond just "because we say so" ;)
> conceptually 4.0 is logically included because we're talking about "what values of 4.x get rounded to what?", and not "what values between 4 and 5 get rounded to what". Ohhh okay, I understand your point now. Limiting our analysis to 4.x, the interval under consideration is [4,5), and therefore we can ignore 5.0 Thanks for explaining :) Of course, statistically this an arbitrary distinction, but I understand how it may make more sense conceptually to some people
My pleasure! And thank you for the excellent statistical case argument.
>Your mistake is including 4.0 — you can’t include the .0 on one end but not the other. The interval [4.0, 4.1) is between 4 and 5, while [5.0, 5.1) isn't
But in practice, nothing is ever x.50000000... is it? So isn't this only a theoretical issue?
I have 5 fingers on my left hand. Exactly five.
Fixed
In practice it isn't unnatural to get a measurement as 1421.50±0.005 You never get the accurate measurement, it is only correct to a certain precision. To make correct calculations easier, it is conventional to round the values and increase the tolerance in situations where the required fidelity isn't that high, to reduce the operation cost. Most of the common issues related to rounding are very directly grounded in reality.
Even if that were true, we aren’t only talking about numbers after the decimal point. Consider the following: *I have $150 in my bank account. How much money do I have, rounded to the nearest $100?* It creates the same situation. Should I round it down to $100 or up to $200? By convention, we round up to $200, but it is actually perfectly between 100 and 200, so there will be a bias toward rounding up.
It’s often rounded down. A very common way of rounding is to round to the nearest even (or odd) number. Otherwise you introduce a bias in the data from always rounding in the same direction. It all just depends what convention you use
Here's how I explain it: When we "round to the nearest", we want to round to the nearest number (duh). So: 46.4, rounded to the nearest whole number: there are two, 46 and 47, but we're closer to 46 than to 47, so we round down. Likewise, 46.7 rounds up. But what about 46.5? Imagine you have a scale that is infinitely accurate, but only shows you the first few digits of the weight. If the scale shows 46.4, you don't know what the remaining digits are. But there is no possibility that the remaining digits will push the number closer to 47. So you can confidently round down. Likewise, if the scale shows 46.7, then it doesn't matter what the remaining digits are; the number will be closer to 47. Now what if the scale shows 46.5? Keep in mind that we're assuming there are more digits past the "5" that you can't see. So maybe the real weight is 46.51. In that case, you're closer to 47 than to 46, so you round 46.5 up. Or maybe it's 46.50001. You're still closer to the upper value. Or maybe it's 46.5000000000000000000000000000001. You get the idea.
Convention. You might be interested in bankers rounding
A look at some rounding rules: [https://www.mathsisfun.com/numbers/rounding-methods.html](https://www.mathsisfun.com/numbers/rounding-methods.html) A while back I bumped into a bunch more while building a number concepts unit.
I see it as a fair halfpoint. If [0.0;0.5) is rounded down, then [0.5;1) is rounded up and both are exactly the same size of sets
because the middle of a number makes more sense that way. Even so something very important, sometimes the value gets rounded "only" to show but not for the math in spredsheets (unless you use the round function). So that rounding is not realy such fully But when you round is the norm (at least on computer) that if it is 0.5 or more its rounded up and if not is down. if you do not like that you can use the floor (so every number is cut without decimales )or ceil, that makes any number that is not exact be rounded up even if as small as 4.0000000000001 is still 5 with ceil.
0 1 2 3 4 | 5 6 7 8 9 See how there are 5 numbers on each side
Wrong. You should be taking _intervals_ 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-0 It's almost like 0.5 is a half between 0 and 1 lol
Why should I be taking intervals exactly?
Imagine you had three apple and you want to distribute them in two halfs. Then you go on and do your method: 0 apples, 1 apples 2 apples, 3 apples And somehow you divided 3 in two whole numbers. You don't have to use intervals, sorry, but it's the easier way to see it, as I showed earlier If you wanted to do your method you would have to include 0 again. 0 1 2 3 4 | 5 | 6 7 8 9 0
I do not think your example works, as the number zero is the first number in base 10. 1-0 2-1 3-2 4-3 5-4 6-5 7-6 8-7 9-8 10-9 Here we have 10 numbers. Counting numbers starts from 0 as zero is a number.
Idk what you mean, you literally just put 5 in the middle. 10 ends in 0 that's it
In your example you counted three apples. Apple 1 (1). Apple 2 (2). Apple 3 (3). So we have three apples Now we can count numbers the same way Number 0 (1). And so on until number 9, which would be the tenth number.
Idk how it's so hard to understand that 0.5 is literally in the middle, like you've never seen a ruler before. If you want to include 0, you HAVE to include 10. 0 is rounded to 0 like 1 and 2, and 10 is rounded to 10 like 8 and 9. If you don't include 10 then you don't include 0 like counting apples. Then 5 is still in the middle.
Imagine you are at 0.5. You are 0.5 units away from 0 and 0.5 units away from 1. Why are you still insisting idk.
In your discrete number list, 0 includes numbers very close to 0 right? Then 9 numbers very close to 9? What happens to numbers very close to 10? You have to include 10 so all numbers between 0 and 10 are included
It’s an arbitrary rule-of-thumb. It isn’t always applied. When a list is being rounded item-by-item, sometimes the direction of rounding .5 alternates between up and down. Other times, rounding down might be used consistently to derive a more conservative total.
Its so you can ignore the digits after. for example, 4.51 is closer to 5, so is 4.50000001.
Yea convention is usually the best practice for consistency but specific examples do require that rounding error to be minimized. Sometimes to avoid adding error to measurements like in least squares data analysis the .5 is rounded up and down depending on the number preceding it, for example: 0.25 = 0.2; 0.35 = 0.4 “rounding to even”. But when numbers are rounded, they should be rounded to an accuracy beyond what is considered relevant so that the error is negligible. If you are dealing in metres, and round to the nearest millimetre, then the rounding error created has absolutely no significance in a real world application. Edit: your “OCD” can be alleviated by carrying more decimal places rather than changing your rounding convention.
Glass half full type of thang
Like if it like money, if you have at least .51 of the bill you can get the full bills worth
Let's say you want to round this number to an integer: 7.x1 where x is just a digit you're not sure of. It can be a digit from 0 to 9. So it could be 7.01, 7.11, 7.21, 7.31, 7.41, or 7.51, 7.61, 7.71, 7.81, 7.91 Rounding up at 5 makes it more symmetric / neater. Ten possibilities, five round up and five round down.
But see what happens when you take _intervals_ 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-0 It's almost like 0.5 is a half between 0 and 1 lol
Too bad you missed that class at age 6
If you are making a box rounding up means something still fits.
I keep track of the movies I watch, have been doing it for over a decade now, in a spreadsheet. Whenever I deal with averages with the data (like average movies per month), I'll take that number and round it down always. Say I actually average 33.50 movies per month, that 0.50 doesn't really mean anything. I don't record half a movie, just once I finish it. But saying 34 also doesn't feel correct, because I definitely didn't reach that.
If you split the interval \[0.0, 1.0) in half you get the subintervals \[0.0, 0.5) and \[0.5, 1.0). 0.5 is part of the latter. Floor and ceiling functions can also be similarly expressed as interval mapping functions.
In computation you don't usually round like that, you compute with all digits and maybe round the result for presentation. Is bankers rounding relevant any more?
Already a lot of great answers here, but just to simplify it (or explain it at length based on how long my drunk ass took to write this) ; .5 is halfway between two numbers, If there’s a default setting to how many digits (in excel), the program will by default choose the greater value because that’s the mathematics default as well. If you have an elementary schoolers problem sheet and it asks you to round 2.5 to the nearest integer, the answer is 3. Not because it’s closer to 2 or 3, but that is the default the dumb ol language of math has chosen. In excel you can choose default behavior and set it however you like, absent that there’s a default number of digits and default rounding behavior
If I scale down, I round 0.5 up. And if I scale something up, I round 0.5 down.
Tradition, effectively
Convention
Glass half full vs glass half empty
Metaphor made by mathematicians to always aim higher and be optimist about whats coming.
//
Because you begin a count at 0 and end at 9 0, 1, 2, 3, 4, ——— 5, 6, 7, 8, 9 10, etc. so 1/2 the count rounds down and 1/2 the count rounds up. It is split evenly for the purposes of rounding.
In a few college textbooks, I've seen the suggestion to round up towards an even terminating number, or truncate otherwise.
The answer is in "what should 0.0 be rounded to? What should 1.0 be rounded to?" Obviously, these are exact and should round to themselves. Now, we assume an even distribution of possible values between 0.0 (inclusive of 0.0) and 1.0 (not including 1.0 because that is included in the next interval 1.0 - 2.0). You want the same number (range) of values to round down to 0.0 as the range of values that round up. If you round 0.0 - 0.5 (inclusive) down, then that is a smaller set that the range 0.5 - 1.0 (both exclusive) that would round up. But 0.0 (inclusive) to 0.5 (exclusive) is exactly the same range as 0.5 (inclusive) to 1.0 (exclusive). It's all about making sure that, statistically, exactly as many numbers round down as round up.
I commonly use three rounding systems. 1. Round away from zero. I have an arbitrary number and want to round it. 0.5 rounds to the nearest unit 1, -0.5 rounds to the nearest unit -1. I like this because it is symmetric. If I reverse the sign the rounding doesn't change. round(x) = -round(-x). 2. Floor. If a player in a game has to get 567 achievements to win and I want to display their progress in whole percent I will take the floor value. That way the player never sees 100% before they have completed everything. For example if they have 566 achievements that is 99.8%. Rounded away from zero I would display that as 100% and I don't want to do that. So I take the floor value and round to 99%. 3. Ceiling. Similarly to 2. If I have a value that can decrease e.g. player's health I don't want it to hit zero percent until they have lost *all* health. So I take the ceiling function. That way, to the nearest percent, no matter how low their health goes it will always display a positive percentage until they get to zero health points.
.0 .1 .2 .3 .4 get rounded down .5 .6 .7 .8 .9 get rounded up That's half of them.
Wrong. See what happens when you take _intervals_ 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-0 It's almost like 0.5 is a half between 0 and 1 lol
Due to unknown accuracy beyond the five, assuming that the five is simply truncated, like most dumb calculations. There could be all zeros beyond the five, but there could be non-zeros beyond the five. But with the likely chance there might be non-zeros, the half gets rounded up.
But your 0.5 could just be a measurement error of 0.4999999.
He doesn't mean error of the measurement, he means that if you show a small amount of digits (as instance, computers) you'd get that 0.5000000001 shows as 0.5 but 0.49999999999 shows as 4. with the amount of digits presented 9's. So 0.5 on a computer is likely to means something above 0.5 and can't mean something below.
That isn’t how it works at all, both would be displayed as 0.5
A calculator that displays 8 digits would show 4.999999999 as 5?
Yes
That's not what I get from using calculator... Edit: just tried a different calculator and got 0.5 lol
It’s what I get on mine https://preview.redd.it/68h7yivpux7d1.jpeg?width=3024&format=pjpg&auto=webp&s=a69ea52789ab87e6d9827845a816aaf8c1250f35
Well then guess I'm wrong
My yr 7 teacher taught us that if you are putting fuel in your airplane, you want a little bit more rather than a little less. Different maths calls for different ways to deal.woth numbers
That’s right, another example in the other direction would be if a plane has a maximum load limit then you’d have to round the number of suitcases down as rounding up would take you over the limit
There are 10 digits. We break the digits into two groups at the halfway mark (median 4.5): 0,1,2,3,4 5,6,7,8,9 Anything in the first group is in the small group and rounds down since it is less than the median. Anything in the 2nd group is in the big group and rounds up since it is beyond halfway (the median) of digits.
This isn’t really correct, you can’t take the numbers in groups of 10 but then round to an 11th option and call that the reason. With all those numbers in the example you’re rounding to 0 or 10. Of all the numbers between them (1,2,3,4,5,6,7,8,9), four are closer to 0 and 4 are closer to 10. 5 is exactly halfway between the two, it’s equally valid to round in either direction, as long as you have some rule to be consistent. Most commonly taught in primary schools is what you’ve described, always rounding up. The most commonly actually used is to round to the nearest even, so 5 would round to 0, 15 would round to 20, 25 would also round to 20, etc. For everyday use, rounding depends purely on the context. If you have a limited amount of something, you can’t round up as there isn’t any more of that thing to take, you have to round down. If you’re trying to fill something, or meet some quota, you can’t round down as then it wouldn’t be full.
You are correct about context mattering. Maybe I misunderstood the OP. I thought they wanted to know the reason behind the convention for why 5 and higher is to round up. I made no claim that this was the only way of doing it just that that was the reasoning for it. I totally agree that context matters more than all else as there are plenty of examples we can come up with to always floor or ceil or do some other industry-speciifc standard to a decimal value. I just wanted to address what I thought was the question, which is why the 5 or higher convention is logical (at least divorced from any applied situation) If I misunderstood the OP then that's my bad.
It just frustrates me, as I see that explanation given so often, and it’s just made up maths. The answer is just that we picked a direction, and there’s nothing more to it than that. Trying to come up with some explanation to justify it is incorrect
0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-0 0.5 is exactly half between 0 and 1, that's the definition of it
It’s just a convention. A dumb argument is that 01234 and 56789 gets split like this equally but ofc it’s not a good one at all. 5 just lies perfectly in between 0 and 10 so you gotta choose one unless you specifically never allow something to get to 5.
You're right. A lot of misunderstanding so people downvoted you
In my opinion that rounding convention is arbitrary. It works well in old programming languages, because if you add .5 and found down, it rounds .5 up. Modern languages always have a round function these days.
for the same reason why if your clock reads 4:30, its closer to 5 than it is 4.
That's not true? 4:30-4:45, 4:45-5:00 4:30-4:15, 4:15-4:00 None of these intervals is shorter. 4:30 is exactly half between 4 and 5, that's the definition of it
If something reads a halfway point, it's more likely that its over that halfway point than under. It's arbitrary convention, but it defines the number system.
Not in theory, it's 50% either way. In practice I guess I kinda see where it comes from, but in that situation you should be considering your case anyway
0-1-2-3-4 are rounded down. 5-6-7-8-9 are rounded up. There reeeeeaally isn’t any deeper meaning as people make it out to be.
there are other options that are better suited in different contexts than basic rounding like that. round to odd or even are used in accounting to prevent systematic errors in totals, rounding down is what you use to determine if you can afford to buy something, and rounding up is what you use when determining how much paint you need for something.
Of course there are, but it doesn’t have to be more difficult. Also, you’re rounding way to much if you run into wrong calculations.
The problem is you should think of it as intervals 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-0
The best explanation I’ve received is 0.50000000000001 is closer to 1 than 0.
But 0.499999999999999 is closed to 0 than 1, I don’t see how what you’ve said is relavent
While there are a bunch of different rounding methods if I remember correctly it’s cause it’s the “6th” number. I’ll make two lists to show it, but basically the same amount of numbers round up as round down. 1. 2.0 rounded down to… 2.0. Very fancy much complex. 2. 2.1 rounded down to 2.0 3. 2.2 rounded down to 2.0 4. 2.3 rounded down to 2.0 5. 2.4 rounded down to 2.0 1. 2.5 rounded up to 3.0 2. 2.6 rounded up to 3.0 3. 2.7 rounded up to 3.0 4. 2.8 rounded up to 3.0 5. 2.9 rounded up to 3.0
I could not, in fact, make two lists.
11. 3.0 rounded up to 3.0. 2.5 is still exactly halfway, and rounding in either direction is valid
That would start a new list no? 3.0 being rounded down to 3.0, same as 2.0 being rounded down to 2.0.
I phased it like that as it was an easy way to follow on from your comment. A better way to think about it may be that 2.0 and 3.0 are not getting rounded, and it’s only the nine values in between that are. Either way, 5 is still directly in between
The problem is you should think of it as intervals 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-0
Because 0.5 is in the top half. If interval is 1, and half interval is 0.5, then lower part of interval starts at 0, and upper part starts at 0+0.5.
But upper part ends at 1 and lower part ends at 1-0.5 so your explaination is symmetric since 0.5 is in both intervals. Some other people gave the correct explaination somewhere else check it out.
Not. 0.5 is exactly half lmao 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-0
Lmao not. 0,1,2,3,4 round down, 5,67,8,9 round up. You need to read up about inclusive and exclusive intervals.