PREMISE
Rounding numbers will introduce a bias if done incorrectly. Round
off error is inevitable, but bias is not.
DISCUSSION
The following algorithm has been proposed for the rounding of numbers
to n significant figures. This method has the advantage of reducing
the bias that may accumulate when the discarded portion of the number
is exactly onehalf a unit in the nth place. The method is useful
in that it increases the accuracy of scientific research data.
The
algorithm would also be of use in financial transactions, where the
accumulation of error could potentially add up to big bucks. Imagine;
a penny here, a penny there, soon you have two cents.
The resulting number
can be said to be well rounded.
ALGORITHM
To round a number to n significant figures, discard all
digits to the right of the nth place. If the discarded
number is less than onehalf a unit in the nth place, leave
the nth digit unchanged. If the discarded number is greater
than onehalf a unit in the nth place, increase the
nth digit by 1. If the discarded number is exactly onehalf
a unit in the nth place, leave the nth digit
unchanged if it is an even number and add 1 to it if it is odd.
 Measurement Systems: Application and Design, Doebelin, Ernest O., page 61.
Copyright 1975 McGrawHill, Inc.
EXAMPLES
The following examples illustrate the major points of the algorithm.

Round 1.23456 to 3 significant digits.
Discard all digits to the right of the 3rd place. Since 0.00456 is less
than half of 0.01, the 3rd digit remains unchanged.
Result: 1.23.

Round 1.23678 to 3 significant figures.
Discard all digits to the right of the 3rd place. Since 0.00678 is greater
than half of 0.01, the 3rd digit is increased by 1.
Result: 1.24.

Round 1.245 to 3 significant figures.
Discard all digits to the right of the 3rd place. Since 0.005 is exactly
half of 0.01, and since the 3rd digit is even, the 3rd digit remains
unchanged.
Result: 1.24

Round 1.275 to 3 significant figures.
Discard all digits to the right of the 3rd place. Since 0.005 is exactly
half of 0.01, and since the 3rd digit is odd, the 3rd digit is increased
by 1.
Result: 1.28
CONCLUSION
I hope this is useful and not pedantic.