Date:         Tue, 28 Apr 2009 22:44:18 -0400
Reply-To:     Sigurd Hermansen <HERMANS1@WESTAT.COM>
Sender:       "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From:         Sigurd Hermansen <HERMANS1@WESTAT.COM>
Subject:      Re: OT: Tuesday Puzzle
Comments: To: Daniel Nordlund <djnordlund@VERIZON.NET>
In-Reply-To:  <D8992A47B7EC4FD8A4A6F53DCF59C4E5@Aragorn>
Content-Type: text/plain; charset="us-ascii"

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

-----Original Message-----
From: SAS(r) Discussion [mailto:SAS-L@LISTSERV.UGA.EDU] On Behalf Of Daniel Nordlund
Sent: Tuesday, April 28, 2009 4:59 PM
To: SAS-L@LISTSERV.UGA.EDU
Subject: Re: OT: Tuesday Puzzle


> -----Original Message-----
> From: SAS(r) Discussion [mailto:SAS-L@LISTSERV.UGA.EDU] On Behalf Of
> Sigurd Hermansen
> Sent: Tuesday, April 28, 2009 1:18 PM
> To: SAS-L@LISTSERV.UGA.EDU
> Subject: Re: OT: Tuesday Puzzle
>
> Dale:
> I tried 5555555555555555 earlier, but rejected that solution because
> it "rounds" inconsistently:
>
>
> m=0.25        n=5555555600000000
>  => n=0
> m=1.25        n=5555555600000001
>  => n=1
>       |
>       v
> m=1.5         n=5555555600000002
>  => n=2
>
> m=1.75        n=5555555600000002
>  => n=2
> m=2.25        n=5555555600000002
>  => n=2
> m=2.5         n=5555555600000002
>  => n=2
> m=2.75        n=5555555600000003
>  => n=3
> m=1000.0001   n=5555555600001000
>  => n=1000
>         |
>         v
> m=1000.5      n=5555555600001000
>  => n=1000
> m=1000        n=5555555600001000
>  => n=1000
> m=2047.25     n=5555555600002047
>  => n=2047
> m=2047        n=5555555600002047
>  => n=2047
>
> I would expect to see both 1.5 and 1000.5 rounded up or both rounded
> down. Small changes in this "magic" value don't work the way I'd
> expect either. S
>

Sig,

In order to not bias the rounded results upward (or downward), when the fractional portion is exactly .5, the convention is usually to round to the odd integer value, so that you round up 50% of the time and round down 50% of the time.

Hope this is helpful,

Dan

Daniel Nordlund
Bothell, WA USA