Hi Enrique...

I'd start by plotting the data: metal concentrations in samples of 20 for each month. 

I'd also want to know more about how you caught the fish, where and when.  If all 20 are from a single net from one location on one day I'd be wondering how independent the observations are. 

If I was happy assuming independence I'd probably start with a quick ANOVA to capture month-to-month variability.  If I had several metals I'd be worrying about multiplicity issues, treating p-values with a pinch of salt and maybe thinking FDR adjusted p-values.

If I had data for multiple years, I'd be looking for YEAR and MONTH effects.  If I had data for several decades, I'd be thinking of an ARIMA time-series analysis fitting terms for YEAR and MONTH or YEAR and SEASON.

But, if I were you, I'd start by plotting the data.

Hope this helps...

I agree with Dennis: first plot the data. 

You can test the population distribution (over the 12-month period, e.g. X percent of fish in each month will have high metal concentrations) using a Kolmogorov-Smirnov goodness-of-fit test. This compares your actual distribution with a hypothesized distribution.  In your case, it sounds like the hypothesized distribution (i.e., the null hypothesis) would be that each month shows the same proportions of fish with high metal concentrations.  You would test your actual data against this hypothesized distribution.  Of course in this data setup, where each month has a proportion (p) of the fish with high metal concentrations and the remainder (1-p) without high metal concentrations (i.e., each fish either does or does not have high metal concentrations), a chi-square test could also be used.  But if your monthly measurements are continuous then use ANOVA. However, if you are not measuring the same individual fish month after month then I'm not sure if repeated-measures ANOVA would be valid.   

I'm assuming this is not a catch-and-release scheme where the same fish could be caught again and again (you probably need to kill the fish to analyze the metal in them). So you might be catching younger fish each month and they might not have as much metal in them because they are young.  Therefore fish size (as a proxy for age) might be a factor to deal with too.   

I agree with Dennis. A scatter plot of the concentrations by months will give you a general idea of the evolution of the concentrations over the year. You will also get an impression of possible differences in variances between months. You might also want to inspect the histograms of the distributions to get an idea of how well these look "normal". If so, you can run a simple repeated measures ANOVA. To get an idea of the covariance structure you can plot the correlations, e.g. with the R package corrplot

https://cran.r-project.org/web/packages/corrplot/vignettes/corrplot-intro.html

In SAS, you can use proc corr with the plots=matrix(histogram) option, see:

http://blogs.sas.com/content/iml/2011/08/26/visualizing-correlations-between-variables-in-sas.html

If the distributions mostly are far from normal, then you can rely on generalized linear models for longitudinal data, but this a rather complex matter.

If the distributions are not normal, you could use the nonparametric Friedman test, or even better the aligned Friedman test. (Instead of generalized linear models.)

In R: the scmamp package

Friedman ranks, aligned ranks en Post Hoc tests

https://github.com/b0rxa/scmamp/blob/master/R/tests.R

https://cran.r-project.org/web/packages/scmamp/scmamp.pdf

Enrique.

to begin an exploratory analysis, since there are independent variables that could explain the greater amount of heavy metals in some fish, such as Age, Size, Species, Sex, or other.

If you want to see the difference produced in the months you must analyze experimentally, since, you must measure the same fish at different times. and do a cohort study. and therefore an analysis of repeated measures may be the solution to look for "DIFFERENCES".

As the above is complex, as Jerry says, it is possible that you can do a distribution analysis, but I advise you to get an index value - a relationship between size, weight and concentration - (I do not know how they measure concentration).

then you perform monthly descriptive analyzes, and you could add, if necessary, an analysis between the monthly averages with effect size,

All of the above is relative, and must be very clear the purpose of the investigation to really analyze the useful statistic.