What is Chi-Square?• Used to examine differences in

the distributions of nominal data• A mathematical comparison

between expected frequencies and observed frequencies

• Theoretical, or Expected, Frequencies: developed on the basis of some hypothesis

• Observed Frequencies: obtained empirically through direct observation

Assumptions for Chi-Square

• The samples must have been randomly selected.

• The data must be in nominal form.

• The groups for each variable must be completely independent of each other; thus, all cell entries are independent of each other.

Chi-Square with a Single Variable

• χ2 Goodness-of-Fit Test: the fit is said to be good when the observed frequencies are within random fluctuation of the expected frequencies and the computed χ2 statistic is small and insignificant

One-Sample Hypotheses

• Null Hypothesis: There is no significant difference between the observed and expected frequencies.

• Alternative Hypothesis: There is a significant difference between the observed and expected frequencies.

Chi-Square with Multiple Variables• χ2 Test of Homogeneity: a test to determine if the

frequencies of one variable differ as a function of another variable

• The independent variable(s) in the χ2 Test of Homogeneity are called the antecedent variable(s); they are the ones which logically precede the others.

• The Chi-Square Test can accommodate multiple variables, e.g. 2 x 2 3 X 5 2 x 3 x 5

Employed Not Employed

Male

Female

O

OO

OE

E

E

E

Two-Sample Hypotheses• Null Hypothesis: The

frequency distribution of variable Y does not differ as a result of group membership in variable X.

• Non-Directional Alternative Hypothesis: The frequency distribution of variable Y does differ as a result of group membership in variable X.

The Chi-Square Distribution• There is a family of χ2 distributions, each determined by a

single degree of freedom value.• For a single variable: df = k – 1• For multiple variables: df = (r – 1)(c – 1)

Where r = the number of rows c = the number of columns

• As the degrees of freedom increase, the sampling distribution approaches the normal distribution.

Computing Chi-Square with a Single Variable

• To enter the data Create columns for each variable Each variable will have value labels The level of measurement for all variables will be nominal

• Analyze Nonparametric Chi-Square• Move the variable(s) of interest to the Test

Variable List Click OK

Output for a Single VariablePolitical Party

11 10.0 1.0

9 10.0 -1.0

20

Democrat

Republican

Total

Observed N Expected N Residual

Test Statistics

.200

1

.655

Chi-Square a

df

Asymp. Sig.

Political Party

0 cells (.0%) have expected frequencies less than5. The minimum expected cell frequency is 10.0.

Computing Chi-Square with More Than One Variable

• Analyze Descriptive Statistics Crosstabs

• Move the antecedent (independent) variable(s) to the Row(s) box Move the dependent variable(s) to the Column(s) box

• Click Statistics Check Chi-Square Click Continue Click OK

Output for a 2 X 2 Chi-SquareGender * Political Party Crosstabulation

Count

3 7 10

8 2 10

11 9 20

Male

Female

Gender

Total

Democrat Republican

Political Party

Total

Chi-Square Tests

5.051b 1 .025

3.232 1 .072

5.300 1 .021

.070 .035

4.798 1 .028

20

Pearson Chi-Square

Continuity Correctiona

Likelihood Ratio

Fisher's Exact Test

Linear-by-LinearAssociation

N of Valid Cases

Value dfAsymp. Sig.

(2-sided)Exact Sig.(2-sided)

Exact Sig.(1-sided)

Computed only for a 2x2 table