Run a Draw Your Group Will Accept as Fair
Shuffle before dealing, remove picks that have already been made, and keep the process visible.
A draw is accepted when the process is visible and repeatable, not merely when the maths is sound. Doing the shuffle in front of the group removes most objections before they are raised.
Shuffle, then deal
Slicing a list in order — even an alphabetical one — produces groups correlated with whatever ordered the list. Shuffling the whole list first and then dealing round-robin keeps group sizes within one of each other and removes the correlation.
Remove names already drawn
For repeated picks across a session, taking each drawn name out gives everyone a turn. Leaving names in is also fair in the statistical sense, but rarely feels fair to a group.
Balanced size is not balanced ability
For competitive activities, use the shuffle as a first draft and adjust openly. A transparent adjustment is better received than a random result nobody believes.