Chi-Square Test for Distribution Goodness-of-Fit Assessment

The Chi-Square test serves as a fundamental method for assessing distribution goodness-of-fit in statistical analysis. It enables validation of how well empirical data conforms to various theoretical distributions including Normal, Log-Normal, Gaussian, Rayleigh, Weibull, and other probability distribution models. The core principle involves comparing observed frequencies against expected theoretical frequencies, where the magnitude of discrepancy determines whether the data adequately fits the chosen distribution model. In computational implementations, this typically involves creating frequency bins and calculating the test statistic using the formula: χ² = Σ[(Observed - Expected)²/Expected].

During Chi-Square test implementation, comprehensive data verification and analysis are essential. Standard procedure includes performing descriptive statistics (mean, variance calculations), normality tests (such as Shapiro-Wilk or Kolmogorov-Smirnov tests), and assessing skewness and kurtosis coefficients to characterize distribution properties. Understanding statistical fundamentals is crucial - particularly degrees of freedom calculation (typically n-1-k where k represents estimated parameters) and chi-square statistic computation. In Python, this can be implemented using scipy.stats.chisquare() function, while MATLAB provides chi2gof() for distribution fitting verification.

In summary, the Chi-Square test represents a powerful statistical tool for evaluating distribution conformity across diverse data types. Proper application requires mastery of statistical concepts and meticulous data examination techniques. For accurate results, practitioners should ensure adequate sample sizes, proper binning strategies, and account for parameter estimation effects on test sensitivity.