# Social Statistics for a Diverse Society

# Chapter Summary

**Chapter 11**

The chi-square test is an inferential statistical technique designed to test for a significant relationship between nominal or ordinal variables organized in a bivariate table. The test is conducted by testing the null hypothesis that no association exists between two cross-tabulated variables in the population, and therefore, the variables are statistically independent.

The obtained chi-square (c2) statistic summarizes the differences between the observed frequencies (*f*o) and the expected frequencies (*f*e)—the frequencies we would have expected to see if the null hypothesis were true and the variables were not associated. The Yates’s correction for continuity is applied to all 2 × 2 tables.

The sampling distribution of chi-square tells the probability of getting values of chi-square, assuming no relationship exists in the population. The shape of a particular chi-square sampling distribution depends on the number of degrees of freedom.

Measures of association are single summarizing numbers that reflect the strength of the relationship between variables, indicate the usefulness of predicting the dependent from the independent variable, and often show the direction of the relationship.

Proportional reduction of error (PRE) underlies the definition and interpretation of several measures of association. PRE measures are derived by comparing the errors made in predicting the dependent variable while ignoring the independent variable with errors made when making predictions that use information about the independent variable.

Measures of association may be symmetrical or asymmetrical. When the measure is symmetrical, its value will be the same regardless of which of the two variables is considered the independent or dependent variable. In contrast, the value of asymmetrical measures of association may vary depending on which variable is considered the independent variable and which the dependent variable.

Lambda is an asymmetrical measure of association suitable for use with nominal variables. It can range from 0.0 to 1.0 and gives an indication of the strength of an association between the independent and the dependent variables.

Gamma is a symmetrical measure of association suitable for ordinal variables or for dichotomous nominal variables. It can vary from 0.0 to ±1.0 and reflects both the strength and direction of the association between two variables.

Kendall’s tau-*b *is a symmetrical measure of association suitable for use with ordinal variables. Unlike gamma, it accounts for pairs tied on the independent and dependent variable.It can vary from 0.0 to ±1.0.It provides an indication of the strength and direction of the association between two variables.

Cramer’s *V *is a measure of association for nominal variables. It is based on the value of chi-square and ranges between 0.0 to 1.0.Because it cannot take negative values, it is considered a nondirectional measure.