I think you should have a look on Chi-square test for the representativeness of the data ! It is something currently used for poll, in order to test whether the sample is representative of the global population or not. 

See : http://www.tqmp.org/RegularArticles/vol08-3/p173/p173.pdf (French)

or 

http://uregina.ca/~gingrich/ch10.pdf

Best regards

Yes, if you have the correct sample size, then your sample mean will be representative of the the population mean. That is the basis of parametric tests such as Student's t-test, ANOVA, etc. You can reduce bias to improve the 'representativeness'. For example, creating a power curve to obtain the correct sampling level :)

Thank you for the answers.

The first populations give these results for chi2 test :

Chi squared

Rows, columns: 8, 60 Degrees freedom: 238

Chi2: 306.89 p (no assoc.): 0.001718

Monte Carlo p : -1

for the second population

Chi squared

Rows, columns: 13, 60 Degrees freedom: 504

Chi2: 572.08 p (no assoc.): 0.018964

Monte Carlo p : -1

There is negative values for variables and it seems that this kind of test doesn't like it. Any other tricks to avoid this problem?

I think you need to do one big Chi Squared test, not two separate ones, so that you can compare populations. The same goes for ANOVA, if you decided to use that :)

You say that you collected 21 samples divided into two groups. How then do you get any test with 500 degrees of freedom?

No idea. I just think I'm not doing the proper statistical test to answer to the representativeness of samples from each species. These previous quoted test confirm that we have two groups significantly different and we already know that with the PCA.

It is a critical aspect of experimental design to know what is a replicate, and to make sure that any statistical test reflects a true replicate when assigning degrees of freedom. Not being able to answer this question is a big problem.

Dear Mr Ebert from Florida, I'm doing chemistry and of course as you can see I'm not a statistic specialist. Results here are of course weired and it's the problem when you use automated function of a software. But the important thing here is that This website allow non specialist to be helped by specialist and  previous people have done that  not you. Thank you Professor for your comments but it doesn't help me. Best

Dear Sylvain, I understand your frustration. However, the correct statistical method is determined by many factors, among them the number of replicates. What I see here is an apparent disagreement between the stated experiment and the statistical output. How am I to figure out if the problem is with your description of the problem or with your use of the program.

However, I can see your point that I have not addressed your question. The only way to tell if your sample of 10 (21 samples in 2 groups = 10 samples in one group and 11 in another) is representative is to take more samples. I would suggest a few hundred. The reviewer is effectively criticizing your work as having too few replicates. The consequence is that another scientist who takes the same measurements will have a good chance of arriving at a different conclusion.

You have three choices, but I have no idea which one to choose because you cannot answer my question.

1) Collect more data.

2) Reword your description of your methods such that it is clear that you have 742 replicates, not 21. That would seem like sufficient replication (or at least a publishable start). Biological populations are variable in space and time and a sample size of 21 is not close to sufficient to properly estimate the population mean or standard deviation.

3) Run a new experiment, maybe a mark-recapture type experiment, to quantify species abundance. Using the variability in these data do the sample size calculations for estimating the appropriate sample size for your existing study. Assuming that answer is less than 21, then you have satisfied the reviewer's comments.

It might be more efficient to find a statistician and get more personal help. Option 2 will be the simplest, if it is true. However, if you have 21 replicates, and 721 pseudoreplicates, then you will be forced to deal with #1 or #3. It is possible that the reviewer's comment is a different form of rejecting the manuscript.

Chi-square tests will not help you determine if your data are representative of the population. So I generate a population of 20 million using a random number generator of my choice, I will let you see 11 values. A Chi-square test will not let you tell if your 11 values are representative of the entire population.

thank you for your help.

I totally agree that the number of samples is too low and I cannot change that. This work is about macroalga and collection is really difficult. The experiment design was made to avoid this kind of question. Temporal and geographical variation of chemical fingerprints were investigated. In parallel phenotype and DNA barcofding were also verified. So everything goes to the same conclusion chemical fingerprints helps to discriminate one species from another. Even with 1000 samples conclusion would have been the same. Different biosyntheses pathway lead to a simple quick and easy differentiation.

Maybe this is a presentation issue, and the reviewer is trying to encourage you to describe your project more clearly.

1) For macroalga, I think of going to the intertidal zone at low tide and pulling a handful of seaweed off the rocks. Doesn't sound all that difficult. So what makes your particular collection of algae difficult? Please keep in mind that your hardship in collecting these samples is irrelevant. What readers need to understand is that any researcher would have trouble getting more samples. So even Woods Hole Oceanographic would have had this trouble.

2) Novelty. What makes your samples interesting? Joe et al. (2017) was able to collect 5224 samples of diatoms on the maiden voyage of the good ship Discovery, and during this time we were able to collect 21 samples of macroalgae as we passed over the deep sea vents (latitude: longitude).

3) How do you know that the conclusions would have been the same with 1000 samples. You did not collect 1000 samples. Presumably no one else has either. Is there no variability in your samples? That would be unusual, especially if the samples were collected over many months and hundreds of miles distant from each other.

4) I don't think of chemical fingerprints as a tool to discriminate species as a novel idea. As GC and LC get cheaper and inventions like MS make identifying compounds more certain, this just becomes another character state in a species description.

Some answers you might want to include in your manuscript, but you might want to write a special letter to the editor explaining why a sample size of 21 while generally insufficient should be accepted in this case. You also need to solve the problem with degrees of freedom and replication. Even if it slips past the editors and reviewers, there are readers that will have questions. What will you do if this is a question from the audience when you present your work at the international meeting?