Chapter 24: Two-Way Tables and the Chi-Square Test*

STATISTICS IN SUMMARY

Chapter Specifics

• Categorical variables group individuals into classes. To display the relationship between two categorical variables, make a two-way table of counts for the classes. We describe the nature of an association between categorical variables by comparing selected percentages.
• As always, lurking variables can make an observed association misleading. In some cases, an association that holds for every level of a lurking variable disappears or changes direction when we lump all levels together. This is Simpson’s paradox.
• The chi-square test tells us whether an observed association in a two-way table is statistically significant. The chi-square statistic compares the counts in the table with the counts we would expect if there were no association between the row and column variables. The sampling distribution is not Normal. It is a new distribution, the chi-square distribution.

In Chapters 14 and 15, we considered relationships between two quantitative variables. In this chapter, we used two-way tables to describe relationships between two categorical variables. To explore relationships between two categorical variables, make a two-way table. Examine the distribution of each row (or each column). Differences in the patterns of the distributions suggest a relationship between the two variables. No change in these patterns suggests that there is no relationship.

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As in Chapters 21, 22, and 23, we used formal statistical inference to decide if any difference in the observed patterns of the distributions is simply due to chance. We compared what we would expect cell counts to be based on the distribution of each variable separately with the cell counts actually observed. The chi-square test answers the question of whether differences in these cell counts could be due to chance.

As in Chapters 14 and 15, we must be careful not to assume that the patterns we observe would continue to hold for additional data or in a broader setting. Simpson’s paradox is an example of how such an assumption could mislead us. Simpson’s paradox occurs when the association or comparison that holds for all of several groups reverses direction when these groups are combined into a single group.

CASE STUDY EVALUATED Here is the table that was presented in the Case Study at the beginning of this chapter:

	Female	Male	Total
Assistant professors	160	177	337
Associate professors	191	374	565
Professors	132	636	768
Total	483	1187	1670

Use what you have learned in this chapter to answer the following questions.

1. What percentage of the assistant professors are women?
2. What percentage of the associate professors are women?
3. What percentage of the full professors are women?
4. As rank increases from assistant to full professor, how does the percentage of women change? Are women overrepresented, underrepresented, or appropriately represented in the highest rank?
5. Do these data show that women have a harder time gaining promotion? Discuss.

Online Resources

• There are several StatTutor lessons that will help with your understanding of some of the details of the chi-square test. There is also a StatTutor lesson on Simpson’s Paradox.
•LearningCurve has good questions to check your understanding of the concepts.

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