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Thread: Brain Burp Rounding??
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face BY : Wendy J Olend email (remove spam text)



It seems that we have broken into two camps, the "Always Round Up-ers"
(ARU)  and the "Round to Even Crew" (REC).  Being a member of the latter
camp, I'm going to try my hand at explaining our stance.

I've been watching everyone go back and forth, arguing if 5 is the
mid-point between 0 and 9 (REC) or if it's part of the upper half of the
set (ARU).  Of course it is part of the upper half of the set
{0,1,2,3,4,5,6,7,8,9}.  The confusion comes in at this point:  The
round-to-even method is meant to minimize rounding error.  When 0 is the
digit under consideration, there IS no rounding error introduced.  Before
the flames head my way, let me elaborate.

We all agree on this point:  when you decide to round a number you first
decide how many digits you want in the final answer, then you throw away
all but one plus that number of digits:

Example:   13.1032
If I decide that I want to report my answer with only one decimal place.  I
truncate, leaving 13.10  Now we have to ask, "What does this value mean?"
Putting this value into words, it means "I have exactly 13.1, and it what
appears to be 0 in the hundredths place."  When I "round" this number to
13.1, I have not introduced any further error, merely reduced the
precision.

Example:   13.1132 to the tenths place, I start with 13.11, which means "I
have exactly 13.1, and what appears to be 1 in the hundredths place."  When
I truncate this value, I have reported the value as less than I have
observed it to be.  Hence the "rounding error".  The same happens when you
round up.

Looking at the error inducing numbers: {1,2,3,4,5,6,7,8,9}, it becomes
obvious why we call 5 the mid-point of this set.  If we always round up at
five, we are putting a positive biasing on the data, because we are
rounding up more than half of the time.

The confusion occurs because for 0-4 it looks like you are doing the same
operation; truncation.  But for 0, you are actually just recasting the
value at a lower precision.

There's my $.02 worth,
Wendy

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