Is it acceptable (accurate, valid and/or reliable) to calculate correlation with a small sample when population is also very small? Suppose the population of interest has size 15 (which is the total number of employees in a specific function at a small company). 
Suppose I randomly select 10 of them and note how long they have worked at the company (X) as well as how much they earn at the company (Y).
I want to look at the correlation between X and Y, and use it as an estimate for the total population correlation (n=15). 
Would this be acceptable to calculate? In terms of accuracy, validity and reliability? This is because the population size is also quite small, so I would expect 10/15 to be enough in this case.
 A: I think this is "acceptable" but it very much depends on what do you want to do with this figure, and what do you think about the data.


*

*essentially you are doing linear regression with a single explanatory variable, so you should still have 8 degrees of freedom in your size-10 sample, and t-values should be valid.

*however, this assumes all the nice assumptions of OLS, including homoskedastic normal independent disturbances.  If homoskedasticity is violated and you are considering using some sort of robust standard errors, there may be a substantial small sample bias (at least with some methods).  Alternatively, the more robust methods may have way too wide confidence intervals to be of any use.

*if you attempt to test any of the assumptions you probably fail to reject any H0, whatever you are testing.  So you are out in the dark in this sense (or maybe out of the dark as you can always say you tested all the violations and nothing came out significant).

*Ordinary OLS assumes your sample is drawn from an infinite population, not just of population of 15.  So you may have to adjust your standard errors for the finite population (but I don't think it matters much in practice).

*if your data misbehaves in any way (outliers, non-linear relationship etc) your options to address this may be pretty limited.


Another set of questions is related to what you want to do with this number.  


*

*Do you want to use a sample of 10 to estimate the effect on the full population of 15?  If yes, validity is not a problem (assuming your 10 people form a representative sample and the distributions behave well).  Note that there are good chances no high-wage or low-wage earner ends up in your sample here--but if it matters depends on the wage distribution.

*If you want to generalize to a wider population we obviously have to ask how representative is your data.  But this is not really sample size related.

*I don't see any issues with reliability here, at least as the word is commonly used.

*I guess with "accuracy" you mean something like confidence intervals.


Here is my proposal:


*

*see what the others have found on similar data.  Do their data behave well?  (And you probably need to take log of the wage to make it look normal).  Assume your data would be similar if it were larger.

*check if your data is comparable.  You may formally test the most crucial properties (e.g. homoskedastic normality)

*check if the standard errors for your small finite population are reasonably similar.

*if yes, just go and estimate.


You can also consider adding a Bayesian prior based on the literature to your regression, and then updating that one.
