ChiQuadratStatistik

ChiQuadratStatistik is a measure used in statistics to quantify the discrepancy between observed frequencies and frequencies expected under a null hypothesis. It is widely applied in tests of goodness-of-fit, independence in contingency tables, and homogeneity across populations.

The statistic is defined by X^2 = sum_i (O_i - E_i)^2 / E_i, where O_i are observed counts and E_i are expected counts under the null. Under the null, X^2 approximately follows a chi-square distribution with a number of degrees of freedom that depends on the test: for goodness-of-fit with k categories, df = k - 1; for a test of independence in an r-by-c contingency table, df = (r - 1)(c - 1). In 2×2 tables, a continuity-correction (Yates) may be applied to improve accuracy in small samples.

Assumptions include that data are counts from random samples, with expected counts sufficiently large (typically E_i ≥ 5). When expected counts are small, the chi-square approximation can be unreliable, and alternatives such as exact tests or category pooling may be preferred.

Interpretation centers on the size of the X^2 value and the associated p-value derived from the chi-square distribution. A larger value indicates greater discrepancy between observed and expected frequencies, and a small p-value leads to rejection of the null hypothesis of no difference or no association at the chosen significance level.

ChiQuadratStatistik is computed in many software packages (for example, R's chisq.test and chi2_contingency in SciPy; SPSS, SAS, and Excel provide similar functions). It remains a fundamental tool for analyzing categorical data and assessing model fit and relationships in contingency tables.