# Chi-squared Test for Binned, Gaussian Samples

## TL;DR

This paper analyzes the chi-squared test for binned Gaussian data, revealing that standard assumptions underestimate the expected value and variability of the test statistic when using sample standard deviations.

## Contribution

It derives corrected formulas for the expected value and standard deviation of the chi-squared statistic accounting for sample standard deviation effects.

## Key findings

- Expected chi-squared value is larger than standard estimates.
- Standard deviation of chi-squared is also larger than previously reported.
- Results improve accuracy of goodness-of-fit assessments for binned Gaussian data.

## Abstract

We examine the $\chi^2$ 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 $\chi^2$ statistic is \begin{equation} \frac{n-1}{n-3}\pm \frac{n-1}{n-3}\sqrt{\frac{n-2}{n-5}}\sqrt{\frac{2}{N-1}}, \end{equation} where $N$ is the total number of binned values. This is strictly larger in both mean and standard deviation than the value of $1\pm (2/(N-1))^{1/2}$ reported in standard treatments, which ignore the distinction between true and sample standard deviation.