Chapter 12: The Chi-Squared Test for a Distribution

The test begins by partitioning measurement values into bins and comparing observed frequencies in each bin against those predicted by the theoretical distribution. The core of the method involves calculating the chi-squared statistic by summing the squared deviations between observed and expected counts, normalized by the expected count in each bin. This normalization ensures that deviations are evaluated relative to their expected statistical fluctuations rather than in absolute terms. The interpretation relies on comparing the computed chi-squared value to the number of bins; agreement between data and theory is indicated when chi-squared is approximately equal to the number of bins, while substantially larger values suggest the data do not follow the assumed distribution. Practical application requires careful attention to binning rules, particularly the requirement that each bin contain an expected frequency of at least five observations to ensure reliable statistical behavior. The chapter emphasizes that degrees of freedom, calculated as the number of bins minus the number of constraints imposed by parameters estimated from the data itself, provide a more precise basis for evaluation than raw bin counts. The reduced chi-squared statistic, obtained by dividing chi-squared by degrees of freedom, offers a standardized measure with an expected value of one under the null hypothesis, simplifying interpretation across studies with different numbers of bins. Finally, the chapter explains how to determine significance by calculating the probability of observing a chi-squared value as extreme or more extreme than the one obtained, using standard reference tables to decide whether to reject the proposed distribution at conventional significance levels of five percent or one percent.