I have two groups of values one has 13 elements and the other has 32 elements. these values are positive, continuous and not dependent. I want to know what is the best test to use to know if the values of one group are less than the other.
Attached is the boxplot of the data.
Hard to say without understanding the nature of the data/variable. Just from the information given I'd say a likelihood ratio test within a generalized linear model of the gamma family of distributions.
https://www.groups.ma.tum.de/fileadmin/w00ccg/statistics/czado/lec8.pdf
http://www.utstat.toronto.edu/burkett/sta442f15/STA442_7_LRTs.pdf
Note: a statistical test in general does not tell you "if one group of values is less than another group" (or the other way around or something similar). Instead, it just evaluates the information that is in the observed data regarding a particular model and a particular restriction of that model. If the restriction means to hold the difference between groups to zero (which is just the point that divides positive from negative differences), a low p-value indicates that there is considerable information in the data that your observations do not fit to that model and that restriction, and if the unrestricted model is appropriate, then the conclusion is that your data is very incompatible with a zero difference. If this is the case, then you can confidently interpret the sign of the (mean) observed difference. If the p value is not really small, the observed data is still not too incompatible with a zero difference and thus with hypotheses on either side of zero. So in this case you can not confidently interpret the sign of the observed difference.
Take home: tests are about *data*, not about the "truth"
Variation in the size of groups or classes never any matter at all. It is not clear what task you want to perform? your data is qualitative or quantitative? In general, for quantitative data you mostly need to use some relevant regression model, while for qualitative data you may use parametric as well as non parametric modelling techniques.
For categorical data, statistical tests may be performed as by following : https://stats.idre.ucla.edu/other/mult-pkg/whatstat/ etc.,
Jochen Wilhelm Muhammad Ali the data are for Root Mean Square Fluctuation (RMSF) of two peptides. It is a calculation of individual residue flexibility, or how much a particular residue moves during a simulation.
For me, your eyes is the best test, given sample size and indepence of of observation. They look substantively different. You then need to make an argument that such a difference makes a difference. The Iqr do not overlap. I would like to know more about the distribution of RMSF if I was doing an explicit test
Define lower. For example, do you mean if you randomly chose one from each, the probability that the one from group a is lower than the one from group b, or something like one mean being below the other?
Daniel Wright something like the mean is what I want, or any measurement (or test) showing that one group is lower than the other. I don't mean selecting one of them randomly.
Just to "formalize" Kelvyn Jones suggestions, there is IOT (inter-ocular trauma) test, if the result hits you between the eyes. Ibrahim M. Ibrahim , for something like the mean, say the trimmed mean, Rand Wilcox has several good books. Most intro textbooks cover the mean.
Daniel Wright Kelvyn Jones Muhammad Ali Jochen Wilhelm ok thank you very much.
"To know if the values of one group are less than the other" is closely related to ranks comparison. I kindly suggest you apply a Mann-Whitney test for two independent groups.
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