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If I've conducted an experiment with only 10 samples and the data acquired have some wide gap in values, what would I use from the two statistical method.

Here are my values sample values:

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I am assuming that you are mentioning t-tests and sign tests because you are interested in comparing some sort of measure of centrality. In which case the t-test compares the mean of the sample while the sign test compares the median. The t-test is a parametric test with more assumptions (such as normally distributed sample data for a one-sample test) but with more power. So if all assumptions hold you would use a t-test, otherwise look for alternatives. Also worth noting that with only 10 samples it will be hard to determine distribution.

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