WoW uses "realms" which is actually a cluster of servers.
There are 222 realms according to the WoW site. There are 9 million players according to
World of Warcraft - Wikipedia, the free encyclopedia.
There probably isn't an even distribution among realms as people like to choose the ones with the best/lowest ping/lag and also the gameplay they like (pvp/non-pvp servers). For the argument's sake, if we assume that there is an even distribution, 9,000,000/222 = ~40540 players per realm.
However, that's a worldwide number, and all those players will not be online at once. And I would assume their player count includes alternate characters as well, so only one character per real-life player would be on at one time.
If we assume that the > 12hr/day player is somewhat less common, you can expect that only half the world's players would be on at any point in time. From there, if you assume that 3/4 of those players have alternates that would have to be considered offline, you have about 15000 actual characters per realm, and I would think this number would be pretty high compared to real values.
I bet more than 3/4 of players have alternates, and I'm sure quite a few have more than one alternate, so that number is probably several times what it should be, which if you start dividing, it would get down to 5000 fairly easily.
There are probably other approaches to solving this problem -- it's similar to the "How many gas stations are in the US?" interview question.