I collect numbers from generators that yield different ranges of whole numbers with an unknown distribution. I want to estimate the mean of the numbers outputted by this generator. I'm convinced the distributions are symmetric (more specifically, uniform) though, so I can just average the min and max after I have a decent sample size. What would be the best way to test that the distribution of numbers is symmetric or uniform?  Things I've tried: -Checked that the mean, median, and midpoint converge to the same values, but I can't quantify how close the values would have to be. -Histogram, but my sample sizes (n&lt;100 because I must manually collect) are too low for me to tell -Chi-square test with the frequencies of each number in the range and testing them against the expected value for a uniform distribution, but I've read there are problems with this in other posts -Pearson's skewness coefficient and comparing that with the standard error of skewness, but I think there are limits to this kind of approach -Comparing kurtosis of data with that of a uniform distribution  I've also read about the Kolmogorov-Smirnov tests and Shapiro-Wilk tests but these seem too complicated for such a simple seeming task..  Thanks
We can test the symmetry of a distribution around $0$ by Wilcoxon sign rank test, based on its sample.   But if we want to test if a distribution is symmetric around its mean, based on its sample $X_1, \dots, X_n$, is it valid to first normalize $X_i$ by the sample mean as $Y_i := X_i - \bar{X}$, and then apply Wilcoxon sign rank test to $Y_i$'s?  If not, what are some ways?  Thanks and regards!
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