The first and second are correct; the third is not.
I am assuming that there are many participants, and each has, or may have, submitted or generated a rating (or several). If that assumption is wrong, then the first is incorrect, and the third is correct.
If a word forms it plural in the usual way, adding -s (or similar), the possessive (or genitive) that would normally be formed with -'s is instead just formed with -'. Thus, if there are multiple participants who each gave rise to one or more ratings, there are many ratings originating in many participants, so both rating and participant should be plural, and participant should be possessive:
Figures showing the distribution of the participants' ratings on the math tasks
On the other hand, if you consider the fact a rating came from a participant to be an attribute of the rating, participant can be an attributive noun - which should be singular:
Figures showing the distribution of the participant ratings on the math tasks
Whether the latter example is actually appropriate in your case depend on exactly what you are trying to say. It's grammatical and makes sense, but whether it's right depends on the actual underlying situation.
: Attributive nouns are, I believe, grammatically singular even when the noun itself is one that is normally always plural - for example "trouser press".