Chi-squared Test for Binned, Gaussian Samples
Nicholas R. Hutzler


Chi-squared Test for Binned, Gaussian Samples]Chi-squared Test for Binned, Gaussian Samples



Division of Physics, Mathematics, and Astronomy 
 California Institute of Technology 
 Pasadena, CA 91125
hutzler@caltech.edu




  June 2019



We examine the χ^2 test for binned, Gaussian samples, including effects due to the fact that the experimentally available sample standard deviation and the unavailable true standard deviation have different statistical properties.  For data formed by binning Gaussian samples with bin size n, we find that the expected value and standard deviation of the reduced χ^2 statistic is

    n-1/n-3±n-1/n-3√(n-2/n-5)√(2/N-1),

where N is the total number of binned values.  This is strictly larger in both mean and standard deviation than the value of 1± (2/(N-1))^1/2 reported in standard treatments, which ignore the distinction between true and sample standard deviation.


































