If X and Y are R.V.s with a certain distribution and you need to measure their similarity to answer which hypothesis is right (threshold based), you can use for example the Kullback–Leibler divergence (KLD).

Performing an exploratory data analysis is almost always a good starting step. This would allow the data to speak for itself and guide you to the "better" analysis, or to the design of better / further experiments.

Scatterplot + correlation analysis + regression analysis would tell you if, what and how good is the relationship between your two variables.

If you want to measure the level of dependency between X and Y, I would advise you to start with the Spearman's rho coefficient.

You may look at its significance to conclude if there is some sort of association between X and Y. The exact formulae can be found here

https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient

How small is the population?

In fact, on an assumption that it is not feasible to study the entire population even if it is finite, we go for probabilistic sampling to draw conclusions regarding the population. If however, the population is small enough, then there is no need to go for sampling at all. The entire population can then be studied, and no statistical tests etc. would be needed.

@Kassem

KL divergence may be a good choice if the RV X and Y are from the same distribution. The KL divergence will fail to detect a link if Y = -X for instance. Rank correlation seems more appropriate.

You first all need data explorative analysis to ascertain the distribution of the data,this will determine the test criteria.

Thoriq

The answer depends on the measurement level of your variables.

X and Y categorical, a contingency table analysis is ok.

X and Y ordinal, Spearman rank correlation

X and Y numeric, check the kind of relation (linear or non linear)

X categorical and Y numeric, anova

X numeric and Y categorical, binary or multinomial logistic,

etc.

This is not an exhaustive list of options. For example you may consider principal components analysis if X and Y are numeric and you are not interested in modeling one of them as explained by the other.

Jorge Ortiz

Thank You very much for all the answers, that were very helpful :)

Thoriq -

For continuous data, after considering scatterplots, and you decide on linear or nonlinear regression based on both subject matter considerations and the scatterplot(s), I suggest estimating the variance of the prediction errors.  The square root of that, by the way, in SAS PROC REG, is STDI.  For other data types, you'd need to use whatever kind of regression may be appropriate.  Subject matter considerations are important.

When you are looking at a small population, that can make a big difference for some applications, such as estimating totals or means and their standard errors for continuous or yes/no data from such a finite population, where, for instance, a finite population correction (fpc) factor may be important.  But often it is the superpopulation mechanism, which produces even a finite population, which is important. Then you may sometimes want to drop the fpc factor.  You have to consider your application, data, and what you are trying to learn. 

Note that for a census, the fpc would throw out any variance due to sampling.  But that should serve as a reminder that no matter what, you can have nonsampling error such as measurement error. 

Since you have a census, it would seem that descriptive statistics are particularly important.   But because you want to see how two variables relate, not knowing other details of your application, it sounds like scatterplots and regression are most important to your study. 

 In general, I do not recommend hypothesis tests:  

 https://www.researchgate.net/publication/262971440_Practical_Interpretation_of_Hypothesis_Tests_-_letter_to_the_editor_-_TAS

Best wishes - Jim 

Article Practical Interpretation of Hypothesis Tests - letter to the...

Dear Thoriq,

I agree with above: is your sample size equals the population size: no statistical test is neccesary because the population parameter is known and has not to be estimated (with the corresponding uncertainty; p-value).

(besides take into account that with small populations the sample size must be about 80 or 95 % of the population; Small Population statistics is something special.)

And depending on the measerment level you choose the correlational measure. (Pearosn spearman etc).

The interpretation of the result is not as usual because no p-value is calculated, you only look at the size of the result. If you think that the size is relevant, than it is.

Cheers, Pierre

Jorge has said it all but another thing you can try is t distribution

Remember that the purpose of hypothesis testing is to generalize the outcome based from the sample you have taken from the target population, since your research was the study population, you do not need to conduct hypothesis testing.