rounding 5 exactly to evens

Factual Questions
1

All my life, I was taught to round up if the next digit is a 5. But, lately, I’ve been hearing that, if it’s exactly 5 (e.g. 0.5000 not 0.5001), I should round to the closest even number instead, to avoid bias.

But I can’t see how this doesn’t make things worse. Here’s my logic: no matter which place you take it to, you will always have the same number of chunks for [-0.5,+0.5).

Let me show an example. We’ll just use 25. If I stick with integers, I have 15,16,17,18,19,20,21,22,23,24, which would normally be rounded to 20, and 25,26,27,28,29,30,31,32,33,34, which would normally be rounded to 30. That’s 10 each. If I use 15,15.1,15.2,15.3, etc, I wind up with 100 each. 15.00 gets 1000, and so on.

But, if you follow the “round 5 to evens” rule then you get 15,16,17,18,19,20,21,22,23,24,25 which round to 20 and 26,27,28,29,30,31,32,33,34 which round to 30. That’s 11 and 9. If I use the tenth’s place, I get 15,15.1,15.2,…24.9,25.0, which is 101 that round to 20, and 25.1,25.2,…34.8,34.9, which gives me 99.

So, for any fixed level of precision to start with, it would seem that this “round 5 to evens” rule would introduce rather than remove bias. Yet I’m told exactly the opposite.

Can anyone, using the level of math I am used if possible, show why the normal method of always rounding up on 5 is biased?

And, probably more importantly, can someone explain why my logic here is wrong?

2

The issue with always rounding a 5 up is that it increases the mean (average) of the sample. The difficulty lies in the fact that with this rounding algorithm, one digit does not change (0), 4 digits get decreased (1-4) and 5 digit (5-9) get increased. This leads to an inherent bias that increases the mean value. By changing it so that 50% of 5 digits are rounded up and 50% are rounded down, the bias is eliminated.

For example, in your sample set of integers, the true mean of the values is 24.5. The mean of the samples after applying the traditional rounding method is 25, but the mean of the samples after applying the banker’s rounding method (roll to nearest even) is still 24.5.

3

But then 24.5 would be rounded down to 24, which is still just as wrong as 25. Your average of numbers with a certain precision can’t have a higher precision than the numbers you started with. Inherently, rounding to the nearest integer means that you will be off by 0.5 either direction.

And it still doesn’t explain what’s wrong with my reasoning.

4

You don’t round the numbers, average, then round again.

5

Having 11 cases round to evens and 9 round to odds is exactly the goal. The only way for rounding to be symmetric is if there are an odd number of cases: an equal number on both sides, plus the center element.

This doesn’t happen if you group the elements by tens. You do if you group by 11s and 9s. Sure, you end up with some bias towards the evens, but as **Rysto **said, it makes the *mean *unbiased.

6

In fact, an average will have better precision and that’s usually the reason one takes an average.

7

I usually see this called banker’s rounding. And it usually only comes up in computer systems or programming languages’ optional treatment.

8

Bigt There’s nothing wrong with your reasoning when you are looking at the sum of the absolute of the errors. Round up is then more accurate.

When its rounding to remove just decimal place, the average is worst affected and you might have to consider even rounding.

if the numbers are losing 8 decimal places, the average would have a tiny error and you’d best use round up.

ergo, ordinary folk uses round up by default - sometimes someone might use round to even for a specific use such as bankers or for where they are going to require averages and its a simple one or two digit loss just because of differences in data acquisation/storage standards.

9

There is no circumstance where the always-round-5-up rule is more accurate. At best, there are some circumstances where the two rules are equally accurate.

In the real world, of course, it makes almost no difference, because you’ll almost always have more digits after the 5, and you use those to decide which way to round.

10

I don’t follow that. If there are any digits (other than zero) after the 5 then rounding must go up; surely?

If the original data set is only accurate to ½ then it is probably not going to be a problem either way. Finance is always at least two digits after the decimal (Dollars & cents or Pounds and pence) and frequently more. The likelihood of any individual data being exactly ½ is going to be quite small.

11

I thought the rule was always 5 and above, give it a shove, 4 or below let it go. I never heard the rule about 5s.

12

Interesting stuff. I prefer round up 5, myself. The only semi-rational alternative I’ve heard is to look at the preceding digit and round up or down based on whether that’s odd or even. E.g., 1.5 goes up to 2 while 2.5 goes down 2. That creates an appearance of uniform distribution of rounding direction.

OTOH, it ignores properties like Benford’s Law for naturally occurring data. That strongly suggests rounding up is a much better idea. I.e., 5 thru 9 appear less often in a given digit’s position that 0-4 so rounding the former up counterbalances a bit of rounding the latter down.

13

In my engineering classes I was taught unbiased rounding. Greater than 5, round up, less than 5, round down. If 5, then:

  1. If the number before 5 is even, round down (1.25 becomes 1.2)
  2. If the number before 5 is odd, round up (1.35 becomes 1.4)
14

This is the way I was told: round to the even number. By doing this that way, you skew the data less than if always rounding up or down.

15

I thought rounding ***only ***considered the next digit, not those beyond? (Although I admit I follow Patx2’s rule on the basis of Chrono’s reasoning, and 'cuz that’s how I was taught.)

When a colleague challenged me on rounding rules, I looked them up. What Patx2 and I do is called “Rounding away from zero”, which is perfectly cromulent and unbiased IF your numbers are equally likely to be positive or negative.

(In fact, in my daily life, numbers are almost always positive, but I just decided that I don’t have enough time and brain cells to waste in on hairsplitting that requires me to check extra conditions to see how I will round. But my numbers are not as precise as they pretend to be, so I can see that science and engineering might ask for more.)

16

Rounding considers the entire string of digits as a whole from the one you’re rounding and onwards. The problem is that you usually only have a finite string to work with. If the readout on your three-digit scales/thermometer/speedometer/whatever is 26.5, this means that the actual value is somewhere between 26.45 and 26.55, but you have no way of knowing whether it’s in the lower or upper half of that interval.

17

so, does 4.5000005 round to 4 or 5 using “round to even” rules?

It is closer to 5 in fact.

Rounding to millionths place should give you 4.500000, which is way more precise than rounding to units.

Rounding to units gives 5 if all information (to 10 millionths) is considered, but 4 if someone has previously rounded to millionths.

There is something counterintuitive about this.

18

You don’t round twice. If rounding more than once was acceptable you could round to any arbitrary number depending on the rounding convention.

19

Really? You could round 4.50000005 to 9? How about to 1,000,000? Which rounding convention would get you there?

20

Anyway, with this in hand, I feel more justified than ever in “rounding away from zero”. Why should I ben my brain to accommodate occasional cases where trailing zeros go out (well-nigh) forever?