Why round to even integers?

Question

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?

Answer 1

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.

Answer 2

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