SOC613 Chap.5 Cross-Tabulation and Chi-Square
Cross-Tabulation and Chi-Square
Cross-tabulation describes how two categorical variables occur together. This chapter identifies cells, marginals and table dimensions, then calculates percentages within the independent variable. Expected frequencies represent independence while preserving observed marginals. Pearson's chi-square aggregates observed-to-expected discrepancies and uses degrees of freedom based on table dimensions.
Cramer's V adds a bounded strength measure, while lambda asks about proportional reduction in prediction error. Interpretation returns to conditional percentages and avoids causal claims from observational tables. The chapter begins with table reading because the statistical test cannot repair a wrong denominator.
Conditional percentages must total one hundred per cent within each category of the independent variable, and the substantive comparison then runs between those distributions. Existence asks whether they differ, strength asks how large the pattern is, and direction is available only when both variables carry meaningful order. Expected counts create the benchmark for independence by combining row and column marginals.
Cell contributions reveal where observed counts depart most from that benchmark, while the omnibus statistic summarises the discrepancies without explaining them. Assumptions about categorical counts, independent observations and expected-cell adequacy are checked before the reference distribution is used.
A p value then describes how surprising a discrepancy at least this large would be under independence; it is not the probability that the null is true. Cramer's V separates magnitude from sample-size-driven significance but remains nondirectional. Lambda answers a prediction-error question and may miss a patterned table when modal predictions do not change.
A complete report joins all levels: test evidence, conditional percentages, effect size, the cells driving the pattern and a design-based limit on causal interpretation.
What this chapter covers
- 01
Cells, marginals and table dimensions
- 02
Conditional percentage direction
- 03
Existence, strength and direction
- 04
Expected frequencies under independence
- 05
Pearson chi-square and degrees of freedom
- 06
Cramer's V, lambda and substantive importance
Worked example · free
Compute an expected frequency
- 1Write row total times column total divided by N.
- 1Substitute 180, 240 and 600.
- 1Calculate the expected count of 72.
- 1Interpret it under independence.
Key terms
- Marginal Total
- A row or column total at the edge of a contingency table.
- Statistical Independence
- A condition in which the distribution of one variable does not change across categories of another.
- Expected Frequency
- The cell count implied by independence and the observed marginals.
- Degrees of Freedom
- The number of freely varying cell quantities after marginal constraints are imposed.
- Cramer's V
- A bounded measure of categorical association strength derived from chi-square.
Cross-Tabulation and Chi-Square FAQ
Which direction should percentages run in a table?
Calculate percentages within categories of the independent variable. If it appears in columns, each column should total one hundred per cent and comparisons run across rows. State the denominator so the direction is visible to the reader.
What does the expected count represent?
It is the cell frequency predicted by statistical independence while retaining the observed row and column totals. It is calculated from the product of the relevant marginals divided by the grand total, using counts rather than percentages.
Why report Cramer's V after chi-square?
Chi-square evidence depends partly on sample size and has no fixed upper bound. Cramer's V rescales the result between zero and one to describe association strength. It remains nondirectional and must be interpreted alongside the actual conditional percentages.
How are degrees of freedom found for a table?
Multiply the number of non-marginal rows minus one by the number of non-marginal columns minus one. Marginal totals are constraints rather than extra categories, so they are never included in the table dimensions.
What if expected frequencies are too small?
Do not report the conventional approximation without qualification. Consider a justified category combination or an appropriate alternative procedure. Any merger must preserve substantive meaning rather than being chosen only to make a diagnostic pass.
Exam move
Draw each table with labels, marginals and the independent variable marked before calculating percentages. Check that every conditional distribution totals one hundred per cent, allowing for rounding. Compute at least one expected frequency by hand and verify that the complete expected table reproduces the observed marginals. Write degrees of freedom from the table dimensions, not from memory.
After reading the p value, return to the cells and identify the largest conditional differences. Add Cramer's V and describe magnitude separately from statistical evidence. Finish with a design sentence stating why the table supports association rather than causation. This sequence keeps the test connected to its sociological meaning.
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