How to test if two sets of vectors are statistically different from each other I have a set of 2-dimensional vector observations (like to wind vectors), and I have separated them into subsamples based on whether an event that is independent of the observations occurred or not.  I know how to test whether one (scalar) sample is statistically different from another, but I haven't been able to figure out how to test wind vectors.  How do I test whether the winds in subsample 1 are statistically different from those in subsample two?  Any help is greatly appreciated.
 A: Based on the additional comment you gave to @whuber, it seems that you are looking for an extension of t-test/ANOVA type methods to situations where you have a vector of outcomes per individual (i.e. multiple outcomes per individual, and in particular in the wind vector example, you have a vector with two components for each individual observation). If interest lies in comparing whether a vector of means from one sample is different from a vector of means from another sample, you can use MANOVA which assumes multivariate normality for the outcomes. Further details on assumptions, mathematical formulation, examples and interpretation can be found here.
A: In general, it can be difficult to test for significance for high-d vectors.  Depending on your application, it may be enough that some statistic of the data is significant.  For example, you could apply labels to the subsamples (subsample A and subsample B), then apply a classifier with cross-validation to see how separable the subsamples are.  In a permutation-test-like fashion, you can then shuffle the labels across all datapoints, and re-apply the classifier, which should give you chance performance.  Do this for many shuffles to get a distribution of the cross-validated accuracies, and then see if the actual CV accuracy is statistically different from the shuffled distribution.  Other possibilities include taking the angles between the vectors, etc.
