▁to ▁the ▁closest ▁integer . ▁ ▁When ever ▁the ▁fraction al ▁part ▁is ▁ 0 . 5 , ▁alternate ▁round ing ▁up ▁or ▁down : ▁for ▁the ▁first ▁occurrence ▁of ▁a ▁ 0 . 5 ▁fraction al ▁part , ▁round ▁up ; ▁for ▁the ▁second ▁occurrence , ▁round ▁down ; ▁and ▁so ▁on . ▁( Al tern atively ▁the ▁first ▁ 0 . 5 ▁fraction al ▁part ▁round ing ▁can ▁be ▁determined ▁by ▁a ▁random ▁seed .) ▁ ▁If ▁occurr ences ▁of ▁ 0 . 5 ▁fraction al ▁parts ▁occur ▁significantly ▁more ▁than ▁a ▁restart ▁of ▁the ▁occurrence ▁" count ing ", ▁then ▁it ▁is ▁effectively ▁bias ▁free . ▁With ▁guaranteed ▁zero ▁bias , ▁it ▁is ▁useful ▁if ▁the ▁numbers ▁are ▁to ▁be ▁sum med ▁or ▁aver aged . ▁ ▁Random ▁tie - bre aking ▁ ▁If ▁the ▁fraction al ▁part ▁of ▁ ▁is ▁ 0 . 5 , ▁choose ▁ ▁randomly ▁among ▁ ▁and ▁, ▁with ▁equal ▁probability . ▁All ▁others ▁are ▁rounded ▁to ▁the ▁closest ▁integer . ▁ ▁Like ▁round - half - to - even ▁and ▁round - half - to - odd , ▁this ▁rule ▁is ▁essentially ▁free ▁of ▁overall ▁bias ; ▁but ▁it ▁is ▁also ▁fair ▁among ▁even ▁and ▁odd ▁ ▁values . ▁The ▁advantage ▁over ▁alternate ▁tie - bre aking ▁is ▁that ▁the ▁last ▁direction ▁of ▁round ing ▁on ▁ 0 . 5 ▁fraction al ▁part ▁does ▁not ▁have ▁to ▁be ▁" rem ember ed ". ▁ ▁Sto ch astic ▁round ing ▁R ounding ▁as ▁follows ▁to