With regards to reputation on Stack Exchange, I did a very short analysis last year on the distribution of reputation on Stack Overflow. Thanks to the Stack Exchange Data Explorer, I can easily run the same analysis for MathOverflow:
x-axis: logarithm of reputation; y-axis: logarithm of number of users; logarithms are base-10, so the 2.0 on the x-axis corresponds to $10^2 = 100$ reputation and there are about $10^{4.25} \approx 18000$ users with this much reputation.
Some interesting points, caused by 'oddities' in the Stack Exchange reputation system:
- are the many no-activity users with just 1 reputation
- the sawtooth until x = 2.0 (±100 reputation) looks strange, but makes sense once you realize how hard it is to get a total reputation of 2 (1 question upvote followed by 2 downvotes).
- a peak on and just after x = 2.0, corresponding to 101 reputation; these are mainly users from other Stack Exchange sites who have the association bonus on Math Overflow plus some optional minor additional activity.
- the peak at 300 reputation is also caused by the association bonus. Users with 200-300 reputation either don't have other accounts on the network, or have another site where they have more reputation.
Feel free to fork the query to experiment yourselves with the data.
One of the things here that more closely follows Zipf's Law is the number of questions with a certain number of answers:
