# Chi-Square by pengxuebo

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```									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
   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

Percentage of
IQ Range        Z-score      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.   Number of Samples: 1
2.   DV: IQ categories
3.   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
Background
1. Number of samples: 1
IV: Gender
DV: Type of sentence received
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
   Move DV into Columns
   Move IV into Rows
   Statistics
   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

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