Good Analytical Approach for a problem I'm trying to figure out what kind of analysis will give me the results I'm looking for.
I have 4 shops and I'm trying to understand what is the typical (most likely) customer characteristics of those 4 shops and whether they differ. The data would look like this (only with more parameters):  
Customer Name,   Age,   Gender,    Shop
Mr. X,            20,     Male,   Shop1
Mrs. Y,           40,   Female,   Shop3
etc.... 

 A: What you want is to examine the joint distribution of your variables for each of the shops. Since you mentioned you want the most likely customer and to preserve dependencies between traits, you simply want to count the how much each combination appears, and the most likely combination will be that which appears the most.
For example, for shop 1:
M 20
F 30
M 20
F 15
F 60
M 20
F 60

Let's count how much each customer type appears:
M 20    -   3
F 30    -   1
F 15    -   1
F 60    -   2

So the most likely customer is M 20.
Note that without preserving dependencies, you do this for each trait separately:
Gender:

M       -   3
F       -   4

Age:

15      -   1
20      -   3
30      -   1
60      -   2

And then the most likely customer would be F 20.
In case you want the mean customer rather than the most likely customer, convert M/F to 0/1. So without dependencies you get:
Mean gender = ~0.57 (i.e. slight tendency towards female)
Mean age = ~32.1

A: Try training a decision tree with your data set, with the output variable shop number.
If you get any statistically significant rule, you can investigate it more closely. E.g. if the top node is "gender", this likely indicates that the shops differ with respect to the gender of the customers.
