Date:         Tue, 28 Apr 2009 23:20:40 -0700
Reply-To:     Daniel Nordlund <djnordlund@VERIZON.NET>
Sender:       "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From:         Daniel Nordlund <djnordlund@VERIZON.NET>
Subject:      Re: OT: Tuesday Puzzle
In-Reply-To:  <FE10F31634E7F34B87AA143D59608541020CAFA1@EX-CMS01.westat.com>
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> -----Original Message-----
> From: SAS(r) Discussion [mailto:SAS-L@LISTSERV.UGA.EDU] On
> Behalf Of Sigurd Hermansen
> Sent: Tuesday, April 28, 2009 7:44 PM
> To: SAS-L@LISTSERV.UGA.EDU
> Subject: Re: OT: Tuesday Puzzle
>
> Daniel:
> I understand the intent of the rounding process. Thanks for
> making it clear.
>
> Even so, my small mind that worships the hobgoblin of
> consistency has to see how the addition and subtraction of
> the magic number forces the result. If I add 1001.5 to the
> test set, I see that the odd rounded number indeed forces
> rounding up. That would appear to eliminate bias if in fact
> the real world selects numbers randomly.
>
> I suspect that the real world seldom selects numbers at
> random. Frequencies of numbers usually show that humans
> accustomed to decimal numbering prefer numbers that end in 0
> (even) over those that end in 5, and otherwise even numbers
> over odds. So if "bias" means difference between observed and
> true, then consistently rounding up may reduce bias after all.
> S
>

You are correct that human behavior comes into play when measurements are
made, and that can have more effect on bias than "rounding up" for numbers
ending in five.  In actual practice, if one is dealing with continuous
distributions, you are unlikely to get a number that is exactly
'something.5', so if you do, some loss of accuracy (truncation, measurement
error, rounding) has probably already occurred.  For 'round to even' to be
unbiased, those errors need to be uniformly (or at least symmetrically)
distributed around the measured / reported values.  But in the absence of
specific knowledge of the error distributions, rounding up is not likely to
be any better.

I might point out that the SAS function ROUND() does in fact round up for
numbers like 1.5 and 2.5, whereas you can get the 'round to even' behavior
if you want it by using the ROUNDE() function.  In most situations where "I"
might want to use some round function, it is not going to make much
difference one way or the other.  My biggest challenge with rounding is
reporting the percentage of populations that fall into various categories,
... and having to explain why they don't add to 100.  :-)

Dan

Daniel Nordlund
Bothell, WA USA