How to describe the differences in skewed data with same median but statistically different distribution? I am comparing length of stay after laparoscopic and open appendectomy in over 160000 patients. LOS is typically a skewed variable so I use the median and interquartile range and ranksum test to describe the result, which is 2, IQR 2-3 and 2, IQR 2-4, p<0.0001. So there is a highly significant difference but it does not show that well in the median and IQR. 
The difference in length of stay is very small. This finding is a negative finding but still interesting as it goes against what most people think about LOS after laparoscopic appendectomy. That is why I want to report on the difference in LOS which is not evident if I only report median and IQR with p-value.
The arithmetic mean (3.1 vs 2.8 and a difference of 0.3 days) is not recommended as the distribution is skewed. I wonder if any other measure could be used like, geometric mean (2.5 vs 2.3, difference 0.2 days) or harmonic mean (2.1 vs 1.9 days, difference 0.2 days)?  When can I use geometric mean or harmonic mean? 
I am inclined to use the harmonic mean which comes closest to the median and gives a difference of 0.2 days. Is this incorrect?
 A: If you want to describe the difference in distribution, perhaps something like a Q-Q plot, or a pair of kernel densities would be common tools, though I assume you have quite discrete looking distributions, in which case that may make something like a pair of barcharts/histograms or a pair of ECDFs better (if possible on a single display)
If you do go with looking at the difference in ECDF, the two-sample Kolmogorov-Smirnov statistic would be the ideal "measure" of significance to go with that, since it's based on the biggest (vertical) difference in ECDF.
If you compare barplots(/histograms) for a discrete distribution, you might like to look at the kinds of measures of discrepancy indicated by smooth tests of goodness of fit (Neyman-Barton type tests), but where the statistic is adapted to two samples. This would in effect correspond to partitioning a chi-square into components associated with orthogonal polynomials, the first of which would correspond to location (a linear term), the second to variance (a quadratic term), and so on (typically you'd look out to order 4 or 6). Books and numerous articles by Rayner and Best (and some others) cover some of that territory.
A: The same question has been discussed on the Google-group Medstats. 
It was suggested that the arithmetic mean should be used as this is a more useful measure for planners. A difference in LOS of 0.3 days can be important for planning of health services and cost. Geometric and harmonic mean were dismissed as not correct to use in this situation. 
Different approaches was proposed for modelling using quantile or Cox-regression. 
I proposed poisson regression but ended up with negative binomial regression as the data showed evidence of overdispersion. This allowed me to compare the unadjusted difference of 0.3 days with difference after adjusting for age, sex, perforated appendicitis and period, which came down to a difference of only 0.05 days between the two surgical methods. This is important findings (although negative) as it goes against the common thinking about the advantages of laparoscopic appendectomy
A: Like @Andrea , I like the idea of quantile regression as it seems to get at what you want.
If you really need a mean for each then, rather than the geometric or harmonic mean, I would suggest a trimmed or Winsorized mean. I find it strange that these are not used more. However, I would also think that % in hospital after X days would be useful and informative. 
