The bias to even numbers would change the shape of the distribution.

I concede that rounding up does increase the mean. However, it only increases the magnitude of the mean. As numbers are shifted away from zero by rounding up, things should balance out when there are both positive and negative numbers.

The rounding to even method would seem to give a better mean where odd and even numbers are of the same sign. Rounding away from zero gives a better mean where odd and even numbers are of opposing signs. Admittedly, the former is a more likely set of data.

Either way, there are inaccuracies, so I'm sure we can all agree that, the error for any result should be given. Especially if using Apple's calculator.

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you're doing statistics. If you're doing any other branch of math you want to minimize the total error - the distribution is irrelevant.
Rounding towards even minimizes the total error and is the mechanism used internally by most math chips, IIRC.
Here's a link that refers to both methods as "equally valid".

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Everyone loves a clown, but no one will lend him money!

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