# Why round to even integers?

According to the Mathematica help:

Round rounds numbers of the form x.5 toward the nearest even integer.

For example:

Round[{0.5, 1.5, 2.5, 3.5, 4.5}]


gives

{0, 2, 2, 4, 4}


What's the rationale behind this? Why not the usual x.5 always rounds up?

It is called bankers' rounding. The rationale is that the rounding behaves "nicely" even if you have negative numbers, i.e. rounding commutes with negation, which is nice.

• Besides that rounding to even ensures you still have an integer if you divide the rounded result by two later on. Chemist normally round to even when measurements and computations are rounded to the nearest 0.001 and stuff like that. – Ted Ersek Feb 22 '12 at 0:19
• That's a nice property, but rounding away-from-zero does has the same property, and is more intuitive. I'm pretty sure the true reason is Mr. Wizard's answer. – BlueRaja - Danny Pflughoeft Feb 22 '12 at 4:36
• I've often thought Mathematica should have an option, e.g. Direction $\rightarrow$ {"HalfDown","HalfUp","HalfEven","HalfOdd","Stochastic"} etc. – David G Jul 26 '15 at 16:30

I think the reason is to prevent biasing numbers on average upward or downward.

For example if you have a list of numbers that include a lot of x.5 and you were to round all these upward, the average magnitude of the list would also shift upward. Likewise if you round downward, downward.

By rounding to the nearest even number these would balance out, given enough samples.

SetAttributes[RoundUp, Listable]
RoundUp[a_] := If[FractionalPart[a] >= 1/2, Ceiling[a], Floor[a]]

d = Table[i, {i, 0, 100, 1/10}];

Mean[Round[d]] // N

50.

Mean[RoundUp[d]] // N

50.05

Mean[d]

50


According to Wikipedia this is also known as:

unbiased rounding, convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, odd-even rounding3 or bankers' rounding, and is widely used in bookkeeping.

Searching for some of those terms returns many good results including:

stackoverflow.com

It does not suffer from negative or positive bias as much as the round half away from zero method over most reasonable distributions. ... But the question was why [] use Banker's actual rounding as default - and the answer is that Microsoft has followed the IEEE 754 standard.

mathforum.org

Where statistics matter, and where numbers that END with the 5 are common, the method you referred to is preferred, because it avoids a bias toward rounding up. On the other hand, there may be some situations where a bias toward numbers that end in an even digit might be bad. Each method has its place.

blogs.msdn.com

Ideally when you are taking an average you want to take an average of the raw data with as much precision as you can get. But in the real world we often have to take averages of data which has lost some precision. In such a situation the Banker's Rounding algorithm produces better results because it does not bias half-quantities consistently down or consistently up. It assumes that on average, an equal number of half-quantities will be rounded up as down, and the errors will cancel out.

cplusplus.com

Given a number exactly halfway between two values, round to the even value (zero is considered even here). ... For random data this is very convenient. Bankers like it because money deposited and withdrawn is random. (There are trends, mind you, but you cannot predict exactly how much will be deposited and withdrawn.) The important point is that the bankers rounding is still biased if the data is biased. It is only unbiased with random data.

www.russinoff.com

One consequence of this result is that a midpoint is sometimes rounded up and sometimes down, and therefore, over the course of a long series of computations and approximations, rounding error is less likely to accumulate to a significant degree in one particular direction than it would be if the the choice were made more consistently. The cost of this feature is a more complicated definition, requiring a more expensive implementation.

• @Brett: nice example!! – Thomas Feb 22 '12 at 9:15
• Just as an aside, Floor[a + 1/2] is the traditional way to round positive numbers in C++ (which has no built in round function). – wxffles Feb 22 '12 at 20:17
• The methods discussed are simple deterministic algorithms, which of course people like. Another method might involve randomizing the rounding of a 5. But I think bookkeepers would really hate the idea of random bank transactions. – Ralph Dratman Mar 27 '13 at 2:34
• I liked the mention of Microsoft. NET following the IEEE standard, is this the same for Mathematica? – Mercer Jan 14 '19 at 17:54

The answer is very nicely illustrated in following link at the topics "Round-Half-Even (Banker's Rounding)" and "Round-Half-Odd":

Clive (Max) Maxfield and Alvin Brown,

Rounding Algorithms 101

http://www.clivemaxfield.com/diycalculator/popup-m-round.shtml#A5

Round to even rounds to a value with less expected followup error in further rounding operations.

It makes a weakened form of associativity hold: ((a+b)-b)+b = a+b . Basically, once you have gotten the unavoidable rounding of a+b over with, no further rounding error will occur from repeatedly subtracting and adding b.

These kind of guarantees reduce the expected maximum error of cumulative operations.