Chi-Square
CJ 526 Statistical Analysis in
Criminal Justice
Parametric vs Nonparametric


Parametric DV: Interval/Ratio
Nonparametric DV: nominal/ordinal
Chi-Square Test for Goodness
of Fit


One sample, DV is at Nominal/Ordinal
Level of Measurement
This test , the chi-square good of fit,
determines whether the sample
distribution fits some theoretical
distribution
Null Hypothesis
1.
Population is evenly distributed the
uniform distribution



Or
Some other distribution, such as the normal
distribution
The sample distribution is not different from
the theoretical distribution (such as the
uniform distribution or the normal
distribution)
Observed and expected
frequency


Observed: number of individuals from
the sample who are classified in a
particular category
Expected frequency: the frequency for
a particular category that is predicted
from the null hypothesis
Chi-Square Statistic


Sum of
(Observed - Expected)2


divided by
Expected
Degrees of Freedom



df = C - 1
where C is the number of categories
The degrees of freedom are the number
of categories that are free to vary
Interpretation


If the null hypothesis is retained, the
sample distribution is like that of the
theoretical distribution
If H0 is rejected, distribution is different
from what is expected
Report Writing: Results
Section



Null hypothesis retained: The results of
the chi-square goodness of fit test were
not statistically significant
Null hypothesis rejected
The results of the Chi-Square Test for
Goodness of Fit involving <DV> were
statistically significant, 2 (df) =
<value>, p < .05.
Report Writing: Discussion
Section


It appears as if the <sample> is (or is
not) distributed as expected.
Depends on the result
Example


Concerned about health, neither
concerned or not concerned, not
concerned about health
Could assume that a sample would be
equally split among these three
categories i.e., 120 subjects, 40 would
say concerned, 40 neither, 40 not
concerned (uniform distribution)
Example
O
E
O-E
(O-E)^2 /E
60
40
20
400
10
40
40
0
0
0
20
40
20
400
10
Chi square





Chi square = 20
D.f. = 2
See p. 726
Chi square = 20, p < .01
The distribution is significantly different
from the expected distribution
Example

Dr. Zelda, a correctional psychologist, is
interested in determining whether the
intelligence of delinquents enrolled in a
state training school is normally
distributed
Distribution of Intelligence in the
General Population
IQ Range
Z-score
Percentage of
General
Population
Below 60
-3
.0228 (23)
60-85
-2
.1359 (136)
86-100
-1
.3413 (341)
101-115
+1
.3413 (341)
116-130
+2
.1359 (136)
131+
+3
.0228 (23)
Distribution of Intelligence in
Dr. Zelda’s School
Below 60
119
60-85
150
86-100
687
101-115
32
116-130
12
131+
0
1.
2.
3.
Number of Samples: 1
DV: IQ categories
Target Population: delinquents
enrolled in the state training school
Inferential Test: Chi-Square Test for
Goodness of Fit
H0: The distribution of frequencies of the
IQ categories for the sample will not
be different from the population
distribution of frequencies of the IQ
categories
H1: The distribution of frequencies of the
IQ categories for the sample will be
different from the population
distribution of frequencies of the IQ
categories
If the p-value of the obtained test statistic
is less than .05, reject the null
hypothesis
Calculations
O
E
O-E
(O-E)^2 /E
119
23
96
9216
401
150
136
14
196
1
687
341
346
119716
351
32
341
309
95481
280
12
136
124
15376
113
0
23
23
529
23
X2 (5) = 1169, p < .001
Reject H0
SPSS: Chi-Square Goodness
of Fit Test

Weight Cases

Data, Weight Cases



Check Weight Cases by
Move weighted variable over to Frequency Variable
Analysis

Analyze, Nonparametric Statistics, Chi-Square


Move DV to Test Variable List
Enter Expected Values
Results Section

The results of the Chi-Square Test for
Goodness of Fit involving the
distribution of IQ categories for the
state training school were statistically
significant, X2 (5) = 1169, p < . 001.
Discussion Section

It appears as if the distribution of
frequencies of the IQ categories for
students enrolled in the state training
school is different from the population
distribution of frequencies of the IQ
categories.
Chi-Square Test for
Independence

Used to assess the relationship between
two or more variables
Null Hypothesis

No relationship between the two
variables (independent of one another)
Or
 Alternative: the two variables are related
to one another

Degrees of Freedom


df = (R - 1)(C - 1),
Where R is the number of rows and C is
the number of columns in a bivariate
table (review bivariate table)
Example

Dr. Cyrus, a forensic psychologist, is
interested in determining whether
gender has an effect on the type of
sentence that convicted burglars
receive
Background
Number of samples: 1
IV: Gender
DV: Type of sentence received
1.
1.
Nominal
Target Population: convicted burglars
Inferential Test: Chi-Square Test for
Independence
H0: There is no relationship between
gender and type of sentence received
H1: There is a relationship between
gender and type of sentence received
Create a bivariate table
probation
jail
total
male
14
80
94
female
46
20
66
60
100
160
Calculate expected values



For each cell, row total times column
total, divided by the total number of
subject
i.e., for the first cell, (94 x 60)/160 =
35
(66x60)/160 = 25, (94x100)/160 = 59,
(66x100)/160 = 41
O
E
(O-E)
(O-E)^2 /E
14
35
21
441
12.6
80
59
21
441
7.5
46
25
21
441
17.6
20
41
21
441
10.6
X2 (1) = 48.3, p < .001
Reject H0
Probation
Jail
Total
Male
14 (35)
80 (59)
94
Female
46 (25)
20 (41)
66
60
100
160
SPSS: Chi-Square Test of
Independence

Analyze

Descriptive Statistics

Crosstabs



Statistics


Move DV into Columns
Move IV into Rows
Chi-Square
Cells

Percentage
 Rows
 Columns
Results Section

The results of the Chi-Square Test for
Independence involving gender as the
independent variable and type of
sentence received as the dependent
variable were statistically significant, X2
(1) = 48.3, p < .001.
Discussion Section

It appears as if gender has an effect on
the type of sentence received.
Assumptions

Independence of Observations