Chikwadrat

Chikwadrat, commonly known in statistics as the chi-square statistic, is a measure used to assess how observed categorical data compare with expected frequencies under a specified model. It is defined as the sum over all categories of the squared difference between observed (O) and expected (E) counts, divided by the expected count: χ² = Σ (O_i − E_i)² / E_i.

In a goodness-of-fit test, the expected counts are derived from a theoretical distribution or specified proportions. In a test of independence for a contingency table, expected counts are computed from the marginal totals, and the degrees of freedom are (r−1)(c−1) for an r by c table. The value of χ² is compared to the chi-square distribution to obtain a p-value, which indicates how likely it is to observe such a deviation if the model is correct. A higher χ² suggests a greater discrepancy between observed and expected data.

Common variants include Pearson’s chi-square and the likelihood-ratio chi-square (G-test). Assumptions include independent observations, data in the form of frequency counts, mutually exclusive categories, and adequately large expected counts (typically at least 5) for validity.

Applications span goodness-of-fit testing, tests of independence in contingency tables, and quality-control analyses. Limitations arise with small sample sizes or many categories, where exact methods or alternative tests may be more appropriate. The chi-square concept has a long history, named after Karl Pearson, and remains a fundamental tool in exploratory data analysis and hypothesis testing.