# Compare two small groups (population vs sample) over time

I am studying banks behaviour according to 6 financial rations throughout a 3-years period. I have 32 observations separated into two groups: large (6) and medium-sized banks (26). However, since in my country there are exactly 6 large banks, it does not constitute a sample, but a population, where as there are around 32 medium-sized banks (so my sample is quite large compared to the population).

To compare these two groups each year, I am using the Wilcoxon Signed Rank Test to test:

$$H_0: \mu_{i,medium} = \mu_{i,large}$$ where $\mu_{i,large}$ is the mean of financial ration $i$ of the $large$ group, which is a known parameter, which means I should not be comparing two samples. Is the Signed Rank Test adequate here?

However, I figure that since I have data over 3 years, it'd be better to compare these groups throughout the 3-years period as whole (or not?). What technique should I use to do so? Are there any test I should do prior to selecting a test?

Thanks

Bernardo

I don't think you should consider $\mu_{i,large}$ to be the population mean. Even though it consists of observations from all large banks in your country. Do you care about the characteristics of the ratios outside the 3-year period? Do you care about potential measurement error in the ratios? (i.e. your observations $y_{it}$ come from $y_{it} = r_{it} + \epsilon_{it}$, where $r_{it}$ is the true value).
• Hi, David, thanks for your thoughts! Answering your questions: 1) No, I do not care about the ratios outside the 3-year period. I am trying only to analyse whether medium banks show a signifficant different ratio when compared to large banks. 2) About the potential measurement error, I am not sure if it applies in this case. The financial ratios are based on financial statements, which we expect to be free of errors. Back to your first sentence, I am having a hard time whether or not to consider it the population mean. I'd like to "hear" why you wouldn't. Aug 17, 2014 at 21:07