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I am familiar with the issue of 'how should one roung .5?', and I am familiar with the conventional solutions, but I don't understand why there isn't a correct answer.

When you're formulating a rounding rule, you want to (as accurately as possible) associate a number with the nearer integer (etc). Such a rule should thus produce equal amounts of each results (for evenly distributed decimal numbers). Consider perfect random distribution of single-digit decimal numbers between 0 and 1 (0.0, 0.1, 0.2 ... 0.9). There are 10 possible numbers. For the rounded results to be even, 0.5 needs to be rounded up (or away from zero if you include negative numbers). In this example, there is no choice about it. Right?

What am I missing?

Thanks

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Now try it with $\{0.1,0.2,0.3,\dots,0.9,1.0\}$; which way does $0.5$ have to go now in order to get an even split? Your $10$-element set is no more representative of the interval $[0,1]$ than mine. –  Brian M. Scott Nov 18 '12 at 22:52
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If you include 0.0 you should also include 1.0. Then there are 11 possible numbers, so the choice is arbitrary. –  Friedrich Nov 18 '12 at 22:53
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Anyway, the even distribution would be more like $(0.05, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, 0.95)$ if you divide $[0,1]$ into $10$ equal parts and pick the midpoint of each. –  Rahul Nov 18 '12 at 23:00
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1 Answer

up vote 5 down vote accepted

Consider perfect random distribution of single-digit decimal numbers between 0 and 1 (0.0, 0.1, 0.2 ... 0.9). There are 10 possible numbers.

First off, I'm not sure why you would count 0.0 but not count 1.0. You should either count both or neither. With that said, there would be either 11 or 9 possible numbers, 5 (or 4) that would round up, and 5 (or 4) that round down, leaving .5 as the only one that is of equal distance from both points.

From there we've just decided, as you mentioned, we just have decided upon the convention of rounding up.

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