Tests of independence in contingency tables I want to perform  a test of independence for a contingency table where the Y characteristic codes for whether an individual go to school or not, and X stands for its gender (MA, male; FE, female).
A typical contingency table would be:
+-----+-----------+-----------------+
| X/Y | Go School | No Go to school |
+-----+-----------+-----------------+
| MA  |           |                 |
+-----+-----------+-----------------+
| FE  |           |                 |
+-----+-----------+-----------------+

This kind of test is clear to me. Now suppose that my contingency is like the following one
+-----+-----------+-----------------+
| X/Y | Go School | No Go to school |
+-----+-----------+-----------------+
| MA  |           |                 |
+-----+-----------+-----------------+
| MA  |           |                 |
+-----+-----------+-----------------+

In other words the X variable refers to sample of "the same type"; that is, they are both extracted from the same population and have the same gender. 
My question is: If I discover that there is a significant difference between the proportions of these two samples, I can't conclude that these two samples are "similar"; rather, there must be another factor or characteristic (that I haven't considered) which causes this difference although the samples come from the same population. Is this correct? 
 A: Yes, if you took a sample from a population of males and then you took another sample from a population of males, and the chi-square for your cross-table is significant, then you may conclude that it is unlikely that the two samples are from the same population - in the statistical sense of the term "population". There must be some factor that makes the samples actually represent different populations of males. Alternatively, it is not impossible that you fell prey to type I error.
A: If the two samples come from the same population, then they come from the same population and there is no possibility that they did not. That is, if you draw two samples of (say) 10 males from the same population, using a random number generator to pick the samples, then there is 0 chance that they came from different populations.
Now, all you said was that the two samples were "both extracted from the same population". If they were not extracted randomly, then there might be some other factor. 
However, if they were extracted randomly then if they are significantly different it must be a type 1 error.  And, if you did the above 100 times, you would expect about 5 of those times to be significant at .05. You may find that there is some other factor that varies in those samples, but that difference was also randomly gotten. And you wouldn't want to say something about that based just on the cases where there was a significant difference.
And, of course, if they come from different populations, then it could easily be another factor. 
