At full distance traveled (max length of your W) the damage should be:
250 + 10% bonus AD + 25% missing health.
Let's assume for argument's sake that you had built nothing more than your Doran's and your AD Marks for a total of +16 AD.
Let's say Azir was level 8: so 986 health and 36.4 armor. He may have +9 armor from armor seals, so let's go with 45.4 armor.
Let's give you 7% armor pen from the ferocity tree and 5% damage from executioner.
Let's say he was actually at 250hp out of 986hp.
So a quick damage calculation says:
Initial: 250 + 0.1x16 + 0.25x(986 - 250) = 435.6
Executioner: x1.05 = 457.38
Effective armor = 45.4 x 0.93 = 42.222
Physical damage factor = 100/(100+42.222) = 0.703126
Calculated damage = 321.6
So yea, something seems wrong here if you're certain it only did 160 damage.
If we assume it's a problem with the execute (I suspect this as I feel I've shot it a few times and it didn't do the damage I was expecting to the allies who didn't die when I executed someone with it):
Initial = 250 + 0.1x16 + 0.25 x 736/986 = 251.79 <- i.e. it's miscoded to apply 25% to percent missing health, or basically nothing
...
Calculated damage = 185.9
Seems a lot more likely. So yea, I bet the percent missing health got mucked up.