# How do I determine statistical differences between two independent 2 samples groups?

I have two varieties of Barley; Copeland and Synergy.

I have data on the protein levels of this barley. I have 28 data points for protein of Copeland and 24 data points for protein of Synergy.

What would be the best statistical test to determine whether the barley protein levels are statistically different?

Thanks

The Mann-Whitney U test is probably the best test to use without having enough data to determine if the protein is normally distributed or not. This has several features: 1) It is a test that ranks outcomes so that the underlying shape of the distributions becomes irrelevant, that is, it is a so-called non-parametric test. 2) It is a test for independent random variables and as such can have a different number of results for each variable. 3) It is not as powerful as a paired-sample test that can look at differences of ranking only between the same subjects (e.g., Wilcoxon signed-rank test).

The main alternative to the Mann-Whitney U test proposed by Carl would be Student`s t-test. There is some normal distribution assumption in there, but at n= 2*28 it is pretty robust against violations of that assumption. The main advantage is not power (if one of these happens to be significant and the other not significant, then it is a borderline case and one should be cautious to trust any of them). The main advantage is, that it investigates the mean of the protein content, which is a concept easier to grasp for non-statistical audiences. Your audience will be happy to hear, that the mean of one sort is significantly larger by roughly 3.5 units. Also, the t-test is favourable if you are interested in any sort of power analysis.

So depending on your audience and what else except for a p-value you want to present, the usual choices are Mann-Whitney U aka Wilcoxon rank sum or t-test for independend samples.

• @Bernard I do not trust Student's t-testing without testing for normal conditions. Testing for normality is not something that can be left out with impunity. For example, if data is non-normal, and the t-test is less significant than Mann-Whitney U testing, the results are not then 'borderline," due to the t-test result. That is precisely why I did not recommend t-testing. Sure, inexperienced users of t-testing abound, but it is the job of experienced users to avoid recommending tests whose basic assumptions do not apply to any given circumstance.
– Carl
Commented Nov 5, 2016 at 14:34
• @Bernard I do not recommend using the mean value as an estimate of location without testing of the distribution type. To use mean values one needs to know that the distribution is not so heavy tailed, e.g., Cauchy or Pareto distributed among others, that the expected value is not a measure of location.
– Carl
Commented Nov 5, 2016 at 15:07
• @Carl (response to first comment) I understand your reservation towards t-testing in non-normal conditions and yes, under certain circumstances the power of the U-test is far higher under non-normal circumstances. I do not think, that either test was not valid but just suggested caution. These are the situations where soneone with some expertise should have a closer look. I think we can agree so far. I explained where I see advantages of the t-test and we differ probably only on how we weigh the advantages and disadvantages of each test. If there was room for only one answer I'd concur with... Commented Nov 5, 2016 at 17:21
• (continued)...if there was room for only one answer I'd concur with your single recommendation. Not knowing dws nor his audience of circumstances I still think there is a chance that the better known test and more commonly used mean are advantageous in his case. Commented Nov 5, 2016 at 17:24
• @Carl (response to second comment) I assume, that the mean is the superior statistic for practical purposes. Non-technical audiences are usually aware of what a mean is of have at least some idea about it. "How much protein will I get from 1 kg of Barley" is probably asking for a mean or expectation of income. Obviously you are right if academic rigor is what counts. The protein levels of Barley are obviously never negative and a grain of Barley will not weigh more than a 1 kg and can only consist of up to 100% protein. This rules out Cauchy and Pareto distribution in the case at hand. Commented Nov 5, 2016 at 17:50