Which average would make the most sense (mean, mode, median)? I have musical data, for tuning information for notes of an instrument. For each note played, I have the cents away from in tune (-50.0 - 50.0). I'd like to display the average each note was out of tune. I'm not sure which would make the most sense (eg. mean, mode, median) for this scenario. Any tips for determining the best solution? The goal is to inform the user what their tuning tendencies are. Is there something else I should be using instead of the above options?

I'm not sure that there is a mathematical answer to this question, as each statistic has their use. But, for what it is worth, here is my opinion on each three:

1) Tried and true, the mean is what people think of when they hear the word "average". Not a lot can go wrong with this statistic unless someone adds a note so severely out of tune that it isn't even the right note anymore.

2) Mode is almost certainly useless for you unless you plan on "bucketing" your results into small cent intervals, and even then you would do better to provide the entire histogram rather than just the mode.

3) The median is an excellent way of providing midpoint data, and it will help suppress outliers. I think it is fair to say, in general, that "natural" data won't have a mean and median that differ too greatly from one another, so this should also be acceptable. It is however not as well known as the mean, and so your users might not be expecting it, especially if you call it an "average".