I have a memory experiment where on each trial, a 7 letter scrambled word is presented, and after a delay the participant is shown the intact word and has to type the previously seen scrambled word exactly as it appeared. They do this for 100 trials, and I collect 50 subjects worth of data. I score each trial from 0 to 7, where 7 means they got the order of letters exactly correct, 5 means they made a mistake on 2/7 letters, and 0 means they completely messed up the order of the letters.
What distribution would make the most sense for modeling this discrete and ordinal data? I'm assuming that it could be best characterized by a mixture of two underlying distributions, a random guessing distribution (randomly inputting letter orderings) and a non-guessing distribution (participant has at least some memory of the correct ordering). I am having trouble figuring out what kind of distribution (e.g., negative binomial, normal gaussian, etc.) would make the most sense to characterize these distributions.
Any thoughts are appreciated!
More details: As an example, it would be like showing COPPRON for a fraction of a second, and then showing POPCORN and asking the subject to correctly type the previously seen scrambled word. I am looking for a distribution that can characterize the 0-7 point scale data I would receive. So for the non-guessing distribution, I want a distribution that can be bounded from 0 to 7 (or 0 to 1 if I normalize the data) where there is likely a peak centered around the most common score that the subject showcased (probably 5 or 6 depending on the difficulty). For the random guessing distribution, it's the same, but it's the distribution that results if we assume a subject always randomly typing the ordering of the letters.