Influence of highly unequal sample sizes on statistical tests and data visualization How would I go about comparing two populations (A and B) with unequal sample sizes.
Sample size of A, SA: 10,000 
Sample Size of B, SB: 2000
My goals are:


*

*To use data visualization tools to study relationships between the variables of each sample (Example: bar graph, histograms, scatter plots)

*To compare the means of both the samples.
a. By hypothesis testing
b. By simply estimating the mean of each sample such that the sample size of A and B are equal (Example: The sample size of A and B are equal (say 2000) with a mean of 40 and 35 respectively).
Since the sample sizes are very different, I do not know:


*

*a. If I must be worried about such a huge difference in sample sizes (I have not checked but seems like the variances are unequal too). 
b. Is sampling or weighting recommended? 
By sampling I mean: randomly selecting 2000 observations from SA 
By weighting I mean: Assign a constant to the data to increase emphasis of the information from SB.

*If there are statistical tests robust to such a huge difference in sample sizes

*Do data visualization tools (like scatter plots and bar graphs) render reliable results when sample sizes are highly unequal? 
 A: As you want to compare means, let's say you are about to use t-test (there is a normality assumption to test before that).

If I must be worried about such a huge difference in sample sizes (I have not checked but seems like the variances are unequal too).

Don't worry that much about unequal sample size. As far as I know this is not a condition for most of statistical tests. Unequal variances can be a problem for some tests but not for all. There are different t-test formula. The default one in r does assume unequal variance. 

Is sampling or weighting recommended?

Not needed. This question seems to imply that you can only compare equal groups, this is not the case, have a look on the welch t-test (the one implemented by default in r).
$$
t \quad = \quad {\; \overline{X}_1 - \overline{X}_2 \; \over \sqrt{ \; {s_1^2 \over N_1} \; + \; {s_2^2 \over N_2} \quad }}\,$$
You can see that :  


*

*Neither sample size or variances are assumed equal.

*If N1 or N2 is big, the statistic goes high (which is appreciable)



Do data visualization tools (like scatter plots and bar graphs) render reliable results when sample sizes are highly unequal?

No problem either.
Intuitively, I would just point out that the unequal sample size can result in a loss of statistical power compared to more equilibrated samples. 6000-6000 would allow you to see better if there is a significant difference between your groups. But anyway, unless you have a huge variance, 2000-12000 should give you very reliable results.
