# The Chi-Square

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```					              The Chi-Square

Tests for Goodness of Fit
And
Independence

The Chi-Square
I.     Introduction
II.    Expected versus Observed Values
III.   Distribution of X 2
IV.    Interpreting SPSS printouts of Chi-Square
V.     Reporting the Results of Chi-Square
VI.    Assumptions of Chi-Square

Introduction
Often when we are testing hypotheses, we only
have frequency data. Our hypothesis concern
the distributions of the frequencies across
various categories.

Examples:
Are there an equal number of males and
females in a group?
Are Republicans more likely to be
Fundamentalist Christians than Democrats?

1
Introduction
With these data we have the number of
people of a certain type in a category.
This is qualitative, not quantitative date.
The scale of measurement is nominal.

Compare this to age as a variable. Age is
a quantitative variable, measured on a
ratio scale.

Introduction
If one were to ask are Republicans older
than Democrats, then one could
measure the age of a sample of people
in each group, calculate the means of
each sample, and test if the difference
in the sample means is statistically
significant (i.e., the sample means
represent a difference in the population
mean).

Introduction
Compare this to the question: “Are
Republicans more likely to be males
than Democrats?” Our sample would
contain a number of males and females.
We would not want to calculate a mean
gender.

2
Introduction
Age and Party Affiliation
Republican      Democrat
M = 51.2        M = 47.5
Appropriate statistical test:
Independent samples t test.

M1-M2
t=     -------------    df = are (n 1 -1) + (n2-1)
sM1-M2

Introduction
Gender and Party Affiliation
Males               Females
Republicans 58                 42

Democrats 70                   80

Appropriate statistical test:
Chi-Square

Expected versus Observed
Values
With the Chi Square, you test the
distribution of scores across the groups
against a hypothetical distribution (the
Ho, or null hypothesis).

For example, the null hypothesis might be
that males and females are equally
likely to be Republican and Democrate.

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Expected versus Observed
Values
For example, in a sample of 100
Republicans, the null hypothesis might
be that there would be 50 males and 50
females.
Expected values:
Males     Females
Republicans:     50         50

Expected versus Observed
Values
However, what if you know the
population is 60 percent female, then
the expected values should be as
follows:
Males     Females
Republicans:    40          60

Expected versus Observed
Values
In any random sample of 100 people, I will not
observe exactly 60 females and 40 males,
any more than I get exactly 50 heads in a
100 coin tosses.

Chi Square measures the difference between
the observed values and the expected values,
and compares that difference to what one
might expect by chance.

Chi-square = Χ2 =    (f o -fe)2
fe

4
Expected versus Observed
Values
Males          Females
Republicans:             58             42
Observed
40             60
Expected

Χ2 = (58-40)2 + (42-60)2
40           60

Χ2 =        8.1   +        5.4 = 13.5

Distribution of X                 2

Large values of X 2 are unlikely to be observed by chance alone
(null hypothesis).

Distribution of X                 2

Shape of the distribution depends on the degrees of freedom.

5
Distribution of X                 2

The degrees of freedom are determined by the
number of rows and columns in the table.
If there is only one row,
df = C-1
With more than one row,
df = (R-1)(C-1)
R = number of rows.
C = number of columns.

In our example, df = 1

Distribution of X                 2

With two dimensions: 2 X 2 Chi-Square

Gender and Party Affiliation (observed values)
Males       Females            Totals
Republicans 58               42                   100

Democrats         70            80              150

Totals   128            122             250

Null hypothesis: counts will be equally distributed
Across the cells.

6
With two dimensions: 2 X 2 Chi-Square

Gender and Party Affiliation (expected values)
Males      Females                                   Totals
Republicans 100*128/250 100*122/250
100
= 51.2              = 48.8

Democrats                   150*128/250         150*122/250    150
= 76.8              = 73.2
Totals         128                 122            250

Use these values to calculate Chi Square:
Χ2 =            (fo -fe)2
fe

Interpreting SPSS printouts of
Chi-Square
Data Structure:

Interpreting SPSS printouts
of Chi-Square
Case Processing Summary
Cases
Valid                Missing             Total
N        Percent     N         Percent   N         Percent
Party * Gender        250         100.0%    0         .0%       250       10

Party * Gender Crosstabulation
Gender
male      female    Total
Party        RepublicanCount                58         42       100
Expected Count        51.2      48.8      100.0

Democrat Count                  70        80        150
Expected Count         76.8      73.2      150.0

Total        Count              128       122       250
Expected Count     128.0     122

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Interpreting SPSS printouts
of Chi-Square
Compare this value to alpha (.05)
Chi-Square Tests
Value      df Asymp. Sig.       Exact Sig.     Exact Sig.
(2-sided)         (2-sided)      (1-sided)
Pearson Chi-Square           3.084 a   1        .079
Continuity Correction        2.648     1        .104
Likelihood Ratio             3.094     1        .079
Fisher's Exact Test                                            .093            .052
Linear-by-Linear             3.072     1         .080
Association
N of Valid Cases             250

a. 0 cells (.0%) have expected count less than 5. The minimum expected count is 48.80.
b. Computed only for a 2x2 table

Reporting the Results
“A Chi Square test was performed to
determine if males and females were
distributed differently across the
political parties. The test failed to
indicate a significant difference, Χ2 (1)
= 3.08, p = .079 (an alpha level of .05
was adopted for this and all subsequent
statistical tests).”

Assumptions of Chi-Square
1. Independence of Observations
Each person contributes one score.

2. Size of Expected Frequencies
Fewer than 20% of the cells should have
expected frequencies less than 5.

8

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