This method also treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. In addition, for most reasonable distributions of "y" values, the expected (average) value of the rounded numbers is essentially the same as that of the original numbers, even if the latter are all positive (or all negative). However, this rule will still introduce a positive bias for even numbers (including zero), and a negative bias for the odd ones.
This variant of the round-to-nearest method is also called unbiased rounding (ambiguously, and a bit abusively), convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, or bankers' rounding. This is widely used in bookkeeping.
This is the default rounding mode used in IEEE 754 computing functions and operators.
Another tie-breaking rule that is very similar to round half to even, namely
Thus, for example, +22.5 becomes +23, +21.5 becomes +21, −21.5 becomes −21, and −22.5 becomes −23.
This method also treats positive and negative values symmetrically, and therefore is free of overall bias if the original numbers are positive or negative with equal probability. In addition, for most reasonable distributions of "y" values, the expected (average) value of the rounded numbers is essentially the same as that of the original numbers, even if the latter are all positive (or all negative). However, this rule will still introduce a negative bias for even numbers (including zero), and a positive bias for the odd ones.
This variant is almost never used in most computations, except in situations where one wants to avoid rounding 0.5 or −0.5 to zero, or to avoid increasing the scale of numbers represented as floating point (with limited ranges for the scaling exponent), so that a non "infinite" number would round to infinite, or that a small "denormal" value would round to a normal non-zero value (these could occur with the "round half to even" mode). Effectively, this mode prefers preserving the existing scale of tie numbers, avoiding out of range results when possible.
Another unbiased tie-breaking method is stochastic rounding:
Like round-half-to-even, this rule is essentially free of overall bias; but it is also fair among even and odd "q" values. On the other hand, it introduces a random component into the result; performing the same computation twice on the same data may yield two different results. Also, it is open to unconscious bias if humans (rather than computers or devices of chance) are "randomly" deciding in which direction to round.
One method, more obscure than most, is round half alternatingly.
This suppresses the random component of the result, if occurrences of 0.5 fractional parts can be effectively numbered. But it can still introduce a positive or negative bias according to the direction of rounding assigned to the first occurrence, if the total number of occurrences is odd.
In some contexts, all the rounding methods above may be unsatisfactory. For example, suppose that "y" is an accurate measurement of an audio signal, which is being rounded to an integer "q" in order to reduce the storage or transmission costs. If "y" changes slowly with time, any of the rounding method above will result in "q" being completely constant for long intervals, separated by sudden jumps of ±1. When the "q" signal is played back, these steps will be heard as a very disagreeable noise, and any variations of the original signal between two integer values will be completely lost.
One way to avoid this problem is to round each value "y" upwards with probability equal to its fraction, and round it downwards with the complement of that probability. For example, the number 23.17 would be rounded up to 24 with probability 0.17, and down to 23 with probability 1 - 0.17 = 0.83. (This is equivalent to rounding down "y" + "s", where "s" is a random number uniformly distributed between 0 and 1.) With this special rounding, known as dithering, the sudden steps get replaced by a less objectionable noise, and even small variations in the original signal will be preserved to some extent. Like the stochastic approach to tie-breaking, dithering has no bias: if all fraction values are equally likely, rounding up by a certain amount is as likely as rounding down by that same amount; and the same is true for the sum of several rounded numbers. On the other hand, dithering introduces a random component in the result, much greater than that of stochastic tie-breaking.
More precisely, the roundoff error for each dithered number will be a uniformly distributed random variable with mean value of zero, but with a standard deviation formula_12, which is better than the 1/2 standard deviation with the simple predictive methods, but slightly higher than with the simpler stochastic method. However, the sum of "n" rounded numbers will be a random variable with expected error zero, but with standard deviation formula_13 (the total remaining noise) which diverges semi-quadratically and may become easily perceptible, even if the standard deviation of the roundoff error per sample will be formula_14 which slowly converges semi-quadratically to zero. So, this random distribution may still be too high for some applications that are rounding a lot of data.
This variant of the simple dithering method still rounds values with probability equal to its fraction. However, instead of using a random distribution for rounding isolated samples, the roundoff error occurring at each rounded sample is totalled for the next surrounding elements to sample or compute; this accumulated value is then added to the value of these next sampled or computed values to round, so that the modified values will take into account this difference using a predictive model (such as Floyd–Steinberg dithering).
The modified values are then rounded with any one of the above rounding methods, the best ones being with stochastic or dithering methods: in this last case, the sum of "n" rounded numbers will still be a random variable with expected error zero but with an excellent constant standard deviation of formula_15, instead of diverging semi-quadratically when dithering isolated samples; and the overall average roundoff error deviation per rounded sample will be formula_16 that will converge hyperbolically to zero, faster than with the semi-hyperbolic convergence when dithering isolated samples.
In practice, when rounding large sets of sampled data (such as audio, image and video rendering), the accumulation of roundoff errors is most frequently used with a simple predictive rounding of the modified values (such as rounding towards zero), because it will still preserve the hyperbolic convergence towards zero of the overall mean roundoff error bias and of its standard deviation. This enhancement is frequently used in image and audio processing (notably for accurate rescaling and antialiasing operations, where the simple probabilistic dithering of isolated values may still produce perceptible noise, sometimes even worse than the moiré effects occurring with simple non-probabilistic rounding methods applied to isolated samples).