# Determine if a series of discrete distributions are expected

*I apologize for the length of this post and I have almost no statistics experience, please keep that in mind :)

In competitive diving, a diver will perform 5 different dives and will receive scores from a panel of judges.

I have a theory that a diver is more likely to either perform the dive well and get a good score or mess up and receive a bad score, leading to a bit of a bi-modal distribution of scores. However, when I compared all divers at once, there was simply a normal(ish) distribution because the good divers' bad scores "cancel" with the bad divers' good scores(over simplification).

This lead me to separate all the data for a single person. But now I realize that because a diver's score changes with time/experience/age and also by event, this would lead to a "muddying" of the distributions. So to avoid eclipsing, what I believe will be a kind of bimodal distribution, I extracted data for a single person for each event type/round.

I have a distribution of dive scores from each dive for many different meets and rounds. (One person's data)

City  Round  Total_score  Dive_1   Dive_2  ...   Dive_5
NYC   Finals    237       60.4     59.5
NYC   Semis     199       62.7     60.1
LDN   Prelim    356       65.1     57.35


From these, for each event(NYC Finals != NYC SemiFinals), I created a histogram to determine the frequency of different dive scores. Just by eyeing some of the histograms, it seems you are more likely to do well or badly than "okay".

Note: Each graph corresponds to one event.

But I now need to use statistics to prove that I am not deluded. I was thinking of doing this by comparing the individual diver's yearly score's standard deviation to the individual event's deviation. From this I could derive an "expected" distribution and see if the individual event distributions are unlikely. However, I don't know if this is possible, or even makes sense to do. Are there any other ways of determining if a distribution is unlikely?

Once I have that(or something like it), I would need to repeat this analysis for every diver and then finally come to a conclusion of whether my hypothesis is correct.

*Edit: I am using SAS to analyze my data. Are there any built in procedures that could aid me?

• Your bar chart appears to display only ten scores. How can you hope to conclude much from that? What exactly is the hypothesis you are hoping to evaluate? How do you quantify "well," "badly," and "okay"? – whuber Sep 25 '17 at 22:12