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  "body": "\n## Definition\n\nLet $X$ and $Y$ be categorical [[variables|0PNRKMRC55]] with values in $\\set{1, \\ldots, r}$ and $\\set{1, \\ldots, c}$, observed on $n$ units.\nThe $r \\times c$ *contingency table* of the sample is the matrix $N = (n_{ij})$, where $n_{ij}$ is the number of units with $X = i$ and $Y = j$.\nThe row totals are $n_{i+} = \\sum_j n_{ij}$, the column totals are $n_{+j} = \\sum_i n_{ij}$, and $\\sum_{i,j} n_{ij} = n$.\nA contingency table of $m \\ge 3$ variables is defined in the same way as an $m$-way array of counts.\n\n## Remarks\n\nThe cells correspond to a [[partition|08QMKTWF9T]] of the sample: each unit is counted in exactly one cell.\nExample: among $100$ people, $10$ smoke and exercise, $20$ smoke and do not exercise, $40$ do not smoke and exercise, and $30$ do neither.\nThe table is $\\begin{pmatrix} 10 \u0026 20 \\\\ 40 \u0026 30 \\end{pmatrix}$, with row totals $30, 70$, column totals $50, 50$, and grand total $100$.\nNon-example: in a survey where people may tick several sports, a table of persons by sport counts some people more than once.\nThen the cells do not partition the sample, and the entries need not add up to $n$.\nTo test independence, the expected counts are $E_{ij} = n_{i+} n_{+j} / n$, and Pearson's statistic is $\\sum_{i,j} (n_{ij} - E_{ij})^2 / E_{ij}$.\nUnder independence and for large $n$, it is approximately chi-squared with $(r - 1)(c - 1)$ degrees of freedom.\n",
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