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Regarding Calculator and low precision values
Authored by: JonathanBoyd on Dec 12, '05 09:29:08AM

Having spent several years studying physics, I can reliably inform you that the proper scientific practice is to round 5s up every time. Always rounding to even numbers produces a bias towards them and brings in an additional inaccuracy.



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Regarding Calculator and low precision values
Authored by: fstengel on Dec 12, '05 09:40:24AM

I wouldn't use "proper scientific practice", I would rather use "force of habit" or more bluntly "inertia". None of the various research/lecturing colleages (in Physic, Maths, Biology) whom, ages ago, I asked why one rounds up on a 5 has been able to give any other answer than "because everybody else does so"...



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The "round to even" does NOT produce a bias
Authored by: Krioni on Dec 12, '05 10:06:07AM
"Having spent several years studying physics..."

Well, I have a degree in physics (Bachelor, and some graduate school work towards Physics PhD). Also, I know basic statistics.

"Always rounding to even numbers produces a bias towards them and brings in an additional inaccuracy."

The only way you can say that is if you know that your original numbers (to some chosen significance) tend to be odd. Do they? No, they do not. On average, you are just as likely to get even numbers as you are to get odd numbers for a given range of possibilities. Thus, if you then follow the rule to "round to nearest even for numbers EXACTLY between two digits" you will NOT have a bias. If you instead do as suggested here (and in elementary school) and round upwards, you will absolutely introduce a bias to larger values in your data.

For more, read the Wikipedia entry (look under Statistician's Method, which is what scientists who wish to have non-biased data would use): http://en.wikipedia.org/wiki/Rounding

(note the phrase "This method is sometimes known as "round to even" and is used in order to eliminate the bias that would come from always rounding a number ending in five up every time." in particular.)

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The "round to even" does NOT produce a bias
Authored by: mikemccallum on Dec 12, '05 02:45:33PM

0 is a digit, as is 1,2,3,4,5,6,7,8,9. Count them. Yup, there is 10 of them. Rounding up on 5,6,7,8,9 gives 5 chances to round up. Truncating on 0,1,2,3,4 gives 5 chances to truncate. It is a lazy analysis which states that "rounding up on 5 introduces a bias".

Zero can be a certain, significant, or uncertain digit just as any of the other nine.

This is, of course, totally to the side of the issue with the calculator function.



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The "round to even" does NOT produce a bias
Authored by: peterneillewis on Dec 12, '05 07:29:24PM

To say this is a lazy analysis while providing such a flawed analysis is impressive. If you sum the changes from 0-4 rounding down and 5-9 rounding up, you get 0 + 1 + 2 + 3 + 4 for rounding down (total 10) and 5 + 4 + 3 + 2 + 1 for rounding up (total 15). On the other hand, if you round to even, then the total changes for rounding down is 0 + 1 + 2 + 3 + 4 + (half of 5), and for rounding up is (half of 5) + 4 + 3 + 2 + 1, which you will note is 12.5 for both.

Rounding to even does generate a bias in the distribution towards even numbers, but for most purposes a slight bias towards even numbers is better than a slight bias upwards.



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Round to even changes the distribution
Authored by: JonathanBoyd on Dec 12, '05 02:50:02PM

To be honest, I don't trust Wikipedia to be terribly accurate, so I went and tried a few more trustworthy mathematical sites and they seem to say the same thing. I must confess that I'm quite surprised by that. If you take a distribution of random numbers, it should be fairly flat:
f
|
|-------------
|
+------------n
But if you then start rounding them to the nearest even, the distribution changes, so that even numbers are more prevalent:
f
|
|__---__---
|
+------------n
I would have thought that that makes it less accurate and produces a bias by clumping results around even numbers.

I guess that we're looking at it with a different intention behind the numbers. I was thinking about patterns and distributions. If you're looking at adding a long chain of numbers, or calculating a mean, as others have suggested, I concede that round to even is probably going to be better. I probably should have objected to the change in distribution, rather than saying 'bias.'

Incidentally, I don't know if it was intention or not, but saying such phrases as 'Also, I know basic statistics,' 'in elementary school' and 'which is what scientists who wish to have non-biased data would use' comes across as a little condescending, which is rather unnecessary.



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Regarding Calculator and low precision values
Authored by: dhirsch226 on Dec 12, '05 12:25:15PM

As I stated in another post for this article: Under what conditions would you conceivably care about a bias-to-even-numbers in a dataset? (I might; I deal with spatial statistics and ordering, but most people? I doubt it.)

On the other hand, there are lots of cases where you would care about the mean of a data set being artificially elevated as a result of rounding up all the time.

-Dave



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