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<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..


  • $\begingroup$ stat.fsu.edu/techreports/M599.pdf ? $\endgroup$ – russellpierce Feb 22 '13 at 5:42
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    $\begingroup$ The K-S test might be perfect for you if your data really are from a uniform distribution. It will give you a test that will treat H0 as 'my distribution matches the uniform distribution'. As with most hypothesis testing you won't be able to directly test whether it /is/ symmetric, you can only estimate the possibility of getting data as extreme as yours if it really were symmetric. I'd hate to see anyone assert the null. $\endgroup$ – russellpierce Feb 22 '13 at 5:46
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    $\begingroup$ John Tukey advocated a useful exploratory graphic for evaluating symmetry. It essentially plots $(\log_2(i),(x_{[i]}+x_{[n+1-i]})/2)$ for $i=n/2,n/4,n/8,\ldots,1$ where $x_{[i]}$ are order statistics. This is more powerful than any single test because a quick visual evaluation--looking for vertical deviations from the ideal horizontal plot achieved by a symmetric distribution--not only identifies asymmetry but can characterize its nature and localize its position within the distribution. Since your question sounds exploratory in spirit, this is something you might want to contemplate doing. $\endgroup$ – whuber Feb 22 '13 at 15:36

if you mean symmetry about a specified location, you can adapt a sign test, a signed rank test or a runs test or even a wilcoxon two-sample test for that. If you mean about an unspecified location, try this thesis:

Yoshizawa (1984) Some Tests of symmetry
http://www.stat.ncsu.edu/information/library/mimeo.archive/isms_1984_1460.pdf https://citeseerx.ist.psu.edu/viewdoc/download?doi=

If you're interested in diagnostic checks instead of tests, see this thread.

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