If you want to take what I said about Shen out of context, be my guest. It's not like I haven't said anything stupid in this argument.
Actually, average for a mage is to have one skill shot in the range of 700-900, and/or a targeted spell under 800, and/or a skill shot/AoE that's over 900, under 1300. Whether the missile's range or due to AoE. Yes, obviously, there are exceptions; Annie, Cho'gath, Diana, Evelynn, etc., but the rule of thumb for mid lane mages falls under that design.
Ahri (she's a mage and assassin, please just leave it alone -_-)
Orb of Deception: 880 range. Charm: 975 range.
Anivia
Frostbite: 650 range. Flash Frost: 1175 range (distance + AoE)
Brand
Sear: 900 range. Conflagration: 625 range. Pillar of Flame: 1150 range (distance + AoE)
Cassiopeia
Noxious Blast: 925 (range + radius). Miasma: 1150 (range + radius)
I think I gave enough examples; if you don't, then I can continue later.
Akali and Katarina don't have anywhere near Syndra's burst potential. Even at just 4 spheres--a single, full spell rotation--it's impossible for either Akali or Katarina to match Syndra's burst. Neither of them have CC to guarantee their bursts, either. As for Zed, he's much closer, but it takes a nearly perfect rotation for him to deal as much burst as Syndra. If you'd like, we could even math-craft with builds and I can prove Syndra has matching, if not stronger burst, compared to Zed.
I find it amusing how you first mention Aatrox, then suddenly he disappears and you add Yi to your arsenal, conveniently forgetting he's Mana-based and more viable than Tryndamere or Yasuo at the moment.
Anyway, I can math-craft and prove Yasuo's DPS isn't on par with Marksmen after mid-game (it's mid-game where Yasuo gets his spike; after that, he falls off). Tryndamere is a fair point, but he has been problematic for reasons other than being manaless. Surprise, surprise.
Also, I'd seriously like to watch those manaless melees deal with a Udyr. Ever. Like, it's the funniest damn duel to watch. I've seen it enough times to know.
Katarina, no. Rumble, I'll give you that his ult is about as good as Zyra's. Kennen, on his ultimate, sure. Thank you for proving that you can provide a few examples.
I doubt I'm going to convince you of anything. Not even if I show you the math do I believe for one second that I could prove Yasuo's DPS is subpar late-game. Learning the mechanics is a lot smarter than just implying that I'm tripping, but I can't make you do the former. A fool will sit happily in his ignorance.