I am a bit sceptical about medians. I agree that a mean sounds dodgy but in reality what Bob means by average may be more like 'total for a month'. You might measure total activity by actimeter for 30 days to even things out - and that would in common sense terms indicate 'what I can actually do over a month'. That seems to me to make more sense than the median count for any one day. And you might say that what matters most is not crashing, or having very bad days. Those very bad days might be the outliers with really low actimeter scores. You might then want a 50% improvement to be at least a 50% increase in your WORST day's actimeter score.
My suspicion is that the maths should follow the common sense interpretation of each particular type of score rather than statistical rules for populations of data points. I think maybe it relates back to the very valid point you made before about these not being linear measures much of the time - at least in terms of their human significance.
The problem is where you have a small number of samples along where some could be outliers. Using the mean and SD tends to let significant outliers (say a trip to the clinic on the last day) dominate the other values. Where sample sizes are bigger and the distributions right they tend to converge. The median is effectively the activity on a typical day rather then the overall activity. In the past I have written stochastic models where we have got outputs with the same mean and SD but very different data distributions and it was the shape of the distribution that changed decisions.
I like the idea of viewing the data as a time series and say measuring peaks and toughs (worst) particularly the worst. I guess my real concern is to explore and understand outlying data and not let it skew results.