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  "body": "\n## Summary\n\nA chi-squared test is a [[hypothesis test|087P03NEND]] whose test statistic has approximately a [[chi-squared distribution|0RFMMBHFTV]] under the null hypothesis. The main example is Pearson's chi-squared test for counts in categories.\nFor observed counts $O_1, \\dots, O_k$ and expected counts $E_1, \\dots, E_k$ under $H_0$, Pearson's statistic is\n$$\\chi^2 = \\sum_{i=1}^{k} \\frac{(O_i - E_i)^2}{E_i}.$$\nOne rejects $H_0$ when $\\chi^2$ is large.\n\n## Remarks\n\nIn the goodness-of-fit test, $H_0$ fixes the probabilities of the $k$ categories. For a large sample, $\\chi^2$ has approximately the law $\\chi^2_{k - 1}$ under $H_0$.\nIn the test of independence for an $r \\times c$ contingency table, $H_0$ states that the two categorical variables are [[independent|0XCE070V8B]]. Then $E_{ij}$ is (row total $\\times$ column total) / $n$, and the law is approximately $\\chi^2_{(r-1)(c-1)}$.\nThe approximation is poor when expected counts are small. A common rule asks for $E_i \\ge 5$ in each cell. For small tables, Fisher's exact test is an alternative.\nKarl Pearson introduced the test in 1900.\n\n## Explanations\n\n### L1 human\n\nThe chi-squared test compares the counts that you observe with the counts that a hypothesis predicts.\nIf the squared differences, scaled by the predictions, add up to a large number, the hypothesis does not fit the data.\n",
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