This method of rounding also is used with whole numbers that you want to round up to the nearest ten, hundred, thousand, etc. (Or at least I thought it was, though I see banker’s rounding is specifically for rounding decimals to an integer.)
My fault; I phrased that poorly. What I meant was that, for instance, 3.4995 rounds down to 3, while 3.5005 rounds up to 4. Which way 3.50000 rounds depends on what convention you use, but it doesn’t matter much, because you’ll almost never have exactly 3.50000 .
Of course, any sane rounding scheme is going to distort things if you care about Benford’s Law.
There are more 15s that round to 20 than there are 24s that round to 20.
The obvious solution is to round numbers ending in 5 based on whether SHA-512 hash of the the number, spelled out in French, is odd or even.
I heard about “banker’s rounding” on this message board I think, but despite having parents who both work in finance and going to school for accounting myself, I have never seen it anywhere else. No one else has ever mentioned it, and if I’ve brought it up, no one has heard of it.
I’m not sure if this is tangential, or just moving the goalposts, or whatever, but when I had to keep a stricter shopping budget (say, at Walmart) I’d round to the nearest $0.50 instead of $1.00. That was easy enough to keep a running estimate of my shopping cart without having to use a calculator. So anything under $5.25 was rounded to $5.00, $5.26-$5.74 became $5.50, and $5.75-$5.99 became $6.00. It suited my need for accuracy just fine.
I did a quick Google search to see if there was a name for this sort of rounding, and it seems that it is done (there are a lot of queries as to how to accomplish it in Excel or coding) but I don’t see a context.
However, when I first started investing, share prices were in eighths rather than hundredths/decimals. Sometimes this involved rounding to the nearest eighth, and sometimes eighths were discrete.
I first heard of “the even-digit convention” (what we called it) almost 60 years ago, and have applied it all my life. But Googling “even-digit convention” (with the quotes) yields a measly single hit. :eek:
I still don’t really get it, I’m afraid.
What I say about rounding is just how I understand math. Rounding in this case means the arithmetic mean, which involves only simple addition and division. Hence the output can be no more precise than the inputs.
Ideally, of course, one would not round the data until all arithmetic operations were completed, as each process can increase precision error.
But, even if I accept that you are right about the rounding, and that it can introduce some level of precision, it still doesn’t really answer my question. It just shifts it back one space.
Why does the normal method produce a bias in rounding? What it wrong with my reasoning that attempts to prove there is no such bias, and that it is actually “banker’s rounding” which introduces a bias?
This was answered in #2. When you asked in #3 for clarification, you got it in #4. I’m not sure what you’re missing.
The key point is that a number like 2.500 is equally well approximated by 2 or 3. To always round up would introduce a bias; you want to round up or down in a roughly fifty-fifty ratio. (Insisting on the even-digit rule rather than coin flips makes your work reproducible and avoids risk of your own subconscious bias. And makes some sense because even numbers are better — “rounder”! — than odd numbers.)
Here’s one more example. Suppose you have 21 data
{1.0, 1.1, 1.2, 1.3, …, 2.9, 3.0} – The average is 2.00
Round them to the nearest digit, using the round-up convention
{1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3} – The average is 2.0476
Round them to the nearest digit, using the “Banker’s” convention
{1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3} – The average is 2.00
Others point out that numbers like 2.50000 are rare in some applications. But in some applications (e.g. where the numbers have already been rounded before you see them, although there are other possibilities) they’re quite common. For example, Wikipedia often quotes percentages to one decimal point, but I want to simplify slightly (avoid distracting detail) for a post here. Enter the even-digit convention, which I’ve been applying for almost six decades!
I’m not sure what you’re trying to say, here. First of all, rounding and averaging are two very different operations, so I don’t see how you could mean one when you say the other. Second, the arithmetic mean (or any other sort of average) does indeed have more precision than any of its inputs, because it uses all of its inputs, not just one.
Exactly…
On the other hand, and news media are the usual culprits, we often see too much precision which is not even accurate to start with.
I remember, after some explosion, it was reported here that the terrorists had planted *about *1814kg of explosive. Clearly someone had estimated two tons, and the editor had converted it to kilos.
I am also reminded that when Radhanath Sikdar first measured Everest, he added 20 feet because he thought people would assume that the actual height was rounded.
Well, then it would seem that if you are rounding data that you believe may be already rounded, you should round 5’s up.
Also, I would point out that your 2.0476 and your 2.00 answers above are “identical”, as you should not be reporting this average of unit measurements to so many decimal points. IMHO, the answer should be 2 (or 2.0, at best) for both methods.
No and No.
The overriding rule is to round to nearest. A reported “2.5” is as likely to come from 2.48 as from 2.52.
And: it’s irrelevant that two different numbers are “identical” if rounded. The exercise there was not to compute a rounded average, but to demonstrate bias.
The one case where rounding 0.5 up specifically makes sense is if you suspect that your data has been previously not rounded, but truncated. If you think that some fool might have written down 2.573 as 2.5 and 2.482 as 2.4, then it makes sense to round the 2.5 to 3 and the 2.4 to 2. Of course, nobody should ever be doing this, for multiple reasons, but here in the real world, what people should do and what they actually do aren’t always the same.