It was one of the first things they taught me in grade school mathematics. Rounding off numbers.
Like the commercial says (at least where I live), fifth thirds rounds off roughly to 1.7, because the six (in the repeating fraction), is more than half way to ten. Makes sense.
But then comes five. Five is exactly midway between zero and ten. Yet it rounds off to the next highest number.
I’ll cite an example. One and five-ninths is 1.555… naturally. Yet for some reason it rounds to 1.6. Why?
And I know this must be the mathematical convention still, because I have a scientific calculator. And it rounds off 5 to the next highest number.
Again, I have to ask why? And while we’re at it, I must ask, **should ** this be the case? Is there a better way?
If you are rounding off to the closest number, then 1.55xyz… is closer to 1.6 than 1.5. The only exception would be 1.5500000… (= 1.54999999…) exactly, but you would encounter that with probability zero so it does not really matter which way you round it.
I didn’t know that. I was taught to always round .5 up to the next number, just like .6, etc.
Maybe it’s better to estimate a little high than a little low for most things. My plane is using 3.5 gallons of fuel per hour, I think I’ll just round that up to 4 so I don’t run out.
OK, if you are rounding, say, integers rather than random real numbers, then you have to even it out as explained above. That is a slightly different scenario.
ETA you should be aware of what you are doing and not confuse the two operations. E.g, fuel consumption of 3.5 should be rounded up to 4, 4.5 to 5, etc.
I have never heard of rounding to the even number, just rounding .5 and higher going up and below .5 down.
My wife had a physics (or maybe Chemistry) teacher in HS that said evens round up and odds round down (.2, .4, .6, .8 round up, and .1, .3, .5, .7, .9 round down). She argued that it was not correct and of course she lost the arguement and to this day is still bitter about it (just ask her, she will get spun up quick over this).
She is now a HS math teacher and I know she does .5 or higher up, below .5 down (no round to the next even number about it).
As for why, I just think if you need to round, you have to have a general rule. And as mentioned in the gas example, if it is better to round up, then do so.
Ultimately, it doesn’t matter that much. The purpose of rounding is specifically to drop insignificant digits. Yes, rounding up on 5 introduces a slight amount of bias, but so does rounding to even because of Benford’s law.
If you want to actually eliminate bias entirely, you can alternate rounding 5s up and down, or round to random, although that makes the process order-dependent and/or not repeatable, which might be worse than a small amount of bias. Or you can do your math in a base that doesn’t have this issue.
When I was being taught rounding, I was told that it was a convention. As in, 5 could go either way, really, and that’s the way that was decided. I’ve never run across anything that contradicted it.
If we are talking about measurements, there is no ambiguity or bias. How so?
My recommended procedure is to not lose precision during your calculations. That way, if your answer is 3.45678 and you know there should be two significant figures after the decimal point, you can readily see the rounded answer is 3.46 without invoking any arbitrary “rules.”
I wouldn’t say that’s the general rule; as a teacher, curriculum developer, and assessment developer, I’ve never encountered a state standard or state assessment, nor any teacher, that uses that rule. The only rule I’ve ever encountered is that you round up when the digit that determines whether you round up or down is a 5. (Unless you’re talking about real-world scenarios where rounding up or down might vary depending on the context.)
I don’t want to derail the OPs thread, though, which is about WHY this is so - I don’t have an answer for that.
There are actually a lot of different rounding rules, depending on context. This one is called banker’s rounding (or “round-half-to-even” if you want to get fancy.)
It’s by no means universal - every other possibility has been used here or there.
1.555… is between 1.5 and 1.6, but it’s closer to 1.6 than it is to 1.5. 1.55 is exactly halfway between 1.5 and 1.6, but if there’s anything after that second 5 (like 1.551 or 1.555), that makes it greater than 1.55, so if you’re rounding to the nearest tenth, you round to 1.6.
The controversy comes in when you have a number that ends in 5 exactly (like 1.55), so that it’s exactly halfway between the two other numbers you might round to. That’s when there are different methods (as noted by gazpacho).
If you want to approximate a real number with an integer (or with a multiple of 10, or with a 1-point decimal, or etc. — the same general principle will apply but at a different decimal place), then 4.51 or 4.501 rounds to 5; even 4.5001 rounds to 5. We’re rounding to the nearest integer.
The “problem” arises with a number like 4.5 or 4.50 or 4.500. Such a number is equally close to two integers. Now the “even-digit convention” is applied; these numbers round to 4 instead of 5, because 4 is even. (I thought this was common knowledge when I was a kid, but I used to devour books and libraries when I was still a rug-rat, so I’m not sure where I picked it up… I think it was about age 12 from a schoolmate whose father was a math professor.)
If you start with an arbitrary but precise real number, the probability would be zero, but neither of those conditions are likely to be met! You might be presented with numbers which have already been rounded but which you need to round again. (No, this won’t introduce bias; and yes, it’s a common situation.) Or the numbers might be simple fractions: if 20 families have 69 kids total, they have 3.45 kids on average. If your typesetter (:)) requires that you drop a decimal point, you’d round to 3.4 rather than 3.5.
Failure to observe the even-digit convention will introduce a tiny but real bias into your rounding.
While my preceding post describes normal rounding practice, other rounding policies may be appropriate, especially when the data distribution is non-uniform. I’ll offer just one example.
In the Jpeg and Mpeg lossy image compression methods a key step is to replace a transformed image value with a nearby integer; thus +0.55 would be replaced with 0 or +1. The specifications call for the nearest integer (+1 in this example) to be used, but (since the error 1-.55 is only slightly less than the error 0-.55) a better quality-per-bitrate can be obtained by using 0 instead! This is because the coding cost (bits spent in the Jpeg file) of a 0 is much less than that of +1. (Few Jpeg/Mpeg implementations actually do this, I’ll guess, but if you read of a product that claims “better compression than vanilla Jpeg/Mpeg” this is likely one of the techniques they use.)
(A Jpeg/Mpeg decompressor can similarly render a +1 as though it were +0.9: because of the highly non-uniform distribution shape near 0 this will reduce error on average. But woe to the mixed-vendor compressor/decompressor pairs which use both the compression hack in the preceding paragraph and this decompression hack!)
