Comparing proportions between two mega-groups Say I sit in 2 different restaurants, Cheesecake Factory and Panda Express, that has nothing in common on their menu. I randomly order items from the menu and record how much sugar it has as well as the item is an appetizer or entree or dessert. At each restaurant, I stop ordering 30 minutes after when I order my first item.
At the end of the day I have a data set that may look like the picture below...
I am wondering how to compare the % sugar in my orders from Cheesecake Factory to that in the ones from Panda Express? I understand that the items from the two restaurants are different and may be "incomparable" in a sense, but if I wish to simply compare the average sugar content from items listed on the menu of each restaurant, what should I do? 

 A: It depends on how sugar content is distributed. If your two restaurants' items have similar distributions of sugar content, you can use a mann-whitney-u-test to assess whether one restaurant's items tend to contain more sugar. If the distributions differ, you may need a bootstrap test, which is less exact.
You could also try data-transformation on your proportion data to fit a roughly normal-distribution. If sugar content for items from both restaurants can be transformed to fit normal distributions, Welch's $t$-test can produce another approximate estimate of the statistical-significance of any group-differences.
If you're more interested in effect-sizeestimation than in falsifying a null-hypothesis of zero differences, you can calculate effect size $r$ from an exact Wilcoxon–Mann–Whitney rank sum test or using Wilcox and Tian's (2011) robust, heteroscedastic generalization of Cohen's $d$.
To answer your commented questions about the Wilcoxon–Mann–Whitney U test:


*

*The U test is nonparametric, so the distributions don't matter as long as they're similar. 

*It works with unbalanced-classes, but with such small-samples, you'll have very little power. 

*Mostly close-to-zero is a distributional characteristic. Again, doesn't matter for the U test as long as it's similarly true of both restaurants. Lots of literal zeros is  zero-inflation, which is a problem.

*Transformation reduces bias due to violations of a normality assumption, which only applies to a parametric t-test...but your ability to avoid type-ii-errors will be very limited with 19 observations.


Reference
- Wilcox, R. R., & Tian, T. S. (2011). Measuring effect size: A robust heteroscedastic approach for two or more groups. Journal of Applied Statistics, 38(7), 1359–1368.
