Chi-Square Test for Distribution Goodness-of-Fit

The chi-square test is a classical statistical method primarily used to verify whether observed data follows a specific theoretical distribution. Whether testing for normal distribution, log-normal distribution, Weibull distribution, or other probability models, the chi-square test evaluates goodness-of-fit by calculating discrepancies between observed frequencies and expected frequencies.

The testing procedure typically involves the following steps: Hypothesis Setting: Establish the null hypothesis (data conforms to target distribution) and alternative hypothesis (data does not conform). Data Grouping: Divide continuous data into intervals and count observed frequencies for each bin. Expected Frequency Calculation: Compute expected frequencies for each interval based on the target distribution. Chi-Square Value Calculation: Compare observed and expected frequencies using the formula χ² = Σ[(O-E)²/E] to obtain the test statistic. Result Determination: Compare the chi-square value with critical values from chi-square distribution tables, or use p-values to determine whether to reject the null hypothesis.

The key advantage of this method is its broad applicability across various distributions. However, practitioners must ensure sufficient sample size and reasonable grouping to avoid test invalidation due to low frequencies. In implementation, the chi-square test commonly employs statistical functions like chi2gof in MATLAB or scipy.stats.chisquare in Python for automated calculations. It finds widespread application in quality control, medical research, and social sciences for distribution validation.