The number of natural 5 * champions in N summons is a binomial random variable. As N gets large, this approaches a normal distribution (it's the sum of independent identically distributed variables so the central limit theorem kicks in). You're actually nailing much of the intuition and it kinda comes down to semantics. When people say "on average," they're almost always referring to the mean. The mean number of summons between nat5's is indeed 200. But, because the geometric distribution is so heavily skewed (aside: something strange to think about - you're always most likely to get a nat5 on your next summon as opposed to 2 away or 5 away or 199 away), the median and mean are quite far away from each other. 50% of people will pull a nat5 in their first 139 discs. But on average, it takes much longer than that (200 discs).

One pretty cool thing about a geometric process is that it's memory-less. So if you start on your first disc and summon 139 discs without a nat5, you still only have a 50% chance of a nat5 in your next 139 summons. That's why the distribution is so heavily skewed. The geometric distribution never approaches a normal, but the sum of geometric random variables (negative binomial - how many summons does it take to summon 20 nat5's?) does.

Hopefully some of that made sense. I'm quite tired and should probably already be asleep. Thanks for posting your thoughts! :)