Compare multi-histograms for goodness of fit

I want to compare two distributions, however each of them can be shown as a group of histograms. These groups are basically has similar structures. What is the best way to compare them using goodness of fit methods?

My own idea is to compare each two histogram pairwise using known goodness of fit methods (for example chi-squared) and at the end calculate an average (or median, standard deviation etc.) of these differences.

To be more clear about the method, let me give you an example very similar to my data:

Suppose we want to compare pollution of two countries, our approach is to compare cities pairwise. For instance when comparing China with Italy, we compare four most populated cities pair-wise (numbers in parenthesis represent p-value in chi-squared test):

• Rome vs Shanghai (0.25)
• Milan vs Beijing (0.35)
• Naples vs Tianjin (0.85)
• Turin vs Guangzhou (0.50)

Because of average lower than 0.5 we may conclude Italy and China are not similar (0.25+0.35+0.85+0.50)/4=0.4875), however Italy and France that may result in 0.75 are much more similar.

Do you think this approach is correct? Is there any more accurate approach you are aware of?

Moreover, I would like to "weight" the p-value of goodlness of fit based on population.

PS:I am not a statistician! forgive my mistakes!!!

• If somebody can give me an interesting keyword related to this problem, then it worth a good answer. I can find some related articles using the keyword. Sep 22 '16 at 8:46

And let's say you have extracted the parameters including the errors for each distribution correctly. Then it is simply a matter of comparing the parameter values and their errors. E.g. if you have extracted the parameters $$x_i = 0.5 \pm .012$$ and for another sample $$x_j = 0.3 \pm .072$$ -- then you could state that there is a small probability that they match. In this context the errors are usually Gaussian, so you can actually look up how much overlap these results have.