Have been present at the taking of about 20 polar bears, while sledding with Canadian Inuit in the 1970s on a project of searching for polar bear denning areas. The Inuit present were allowed taking their legal allotted polar bear quota. The bears varied in size between 250-800 lbs. respectable but no giants.
The largest caliber brought by the Inuit on any of the trips was a .303 British, the smallest .222Rem. With in between .270, .243, 6mm rem., 25/06, .308 win, and 30-06.
They far preferred the smaller rounds, since they abhor exit holes, always used c&c bullets in factory loads brought in by the Hudson's Bay manager. Exit holes=fur damage.
They also liked to be able to carry the ammo in their pants pockets, smaller cartridges= more ammo for longer trips.
The most popular were then the .222rem, and the .243 since most game hunted was first and formost the ringed and bearded seals to be taken with head shots only to prevent them convulsing back into the seal hole. The second most hunted animal was the caribou.
Polar bears were regularly encountered on those sealing trips and therefore dispatched with the seal rifle at hand.
Now if they went out specifically for polar bear as the main target, they would bring a .303, .308, 270, 25-06 or .243. The 222 would be left at home or carried as a backup seal gun.
At one time I brought along a Remington .350 mag, the cartridge was much admired until a bear was shot with it, spraying blood and lung tissue on the piece of ice besides it.
The bear was no more then 350-400 lbs.
No Inuit after that wanted anything to do with the .350 mag.
Bears shot with the .222 were usually shot in the heart, or brain and died with heart shots within 2 minutes. Caribou were shot in the lungs or heart and died always very rapidly with the .222 rem. They admit that doing bears with a .222 needs caution and a cool disposition, since there is no margin for error. However it is done frequently.
The .243, .270 25/06, .308 win, 06, all killed the bears very quick with heart, lung or brain shots. Bears were shot well within the 50-100 yard range.
Then I only saw around 20 bears shot, so, a limited sample.
Another reason they are confident with their small bores is that the bears are chased until tired, not much steam left for charging, then shot at ranges less then 100 yards.
Caribou up there are approached easily within range, I do not believe I have seen shots exceeding 200 yards, mostly 40-100 yards.
The Inuit knew the anatomy of their animals and knew to place their shots.