CONTINGENCY TABLE

Tags: #◇Computer Science #◇STT101A #◇Conceptual #◇Contingency Table

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◇Overview

A Contingency Table (two-way cross-tabulation table) organizes bivariate categorical counts or probabilities across $r$ rows (levels of variable $A$) and $c$ columns (levels of variable $B$). It allows immediate extraction of joint, marginal, and conditional probabilities.

◇Key Definitions

• Joint Probability ($P(A_i \cap B_j)$): Probability in interior cell $(i, j)$, computed as $\frac{n_{ij}}{N}$ where $N$ is the grand total.
• Marginal Probability ($P(A_i)$ or $P(B_j)$): Row total $\frac{n_{i\cdot}}{N} = \sum_j P(A_i \cap B_j)$ or column total $\frac{n_{\cdot j}}{N} = \sum_i P(A_i \cap B_j)$ along the margins.
• Conditional Probability ($P(A_i \mid B_j)$): Interior cell divided by its corresponding column (or row) marginal total: $P(A_i \mid B_j) = \frac{n_{ij}}{n_{\cdot j}} = \frac{P(A_i \cap B_j)}{P(B_j)}$.

◇Worked Example

• Joint: $P(M \cap F) = \frac{90}{1000} = 0.09$.
• Marginal: $P(F) = \frac{150}{1000} = 0.15$.
• Conditional: $P(M \mid F) = \frac{90}{150} = 0.60$.

◇References & Navigation

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• Related Notes: ◇Total Probability Rule, ◇Generalized Multiplication