Statistical Analyses:
Chi-square test
Psych 250
Winter 2013
Types of Measures / Variables
• Nominal / categorical
– Gender, major, blood type, eye color
• Ordinal
– Rank-order of favorite films; Likert scales?
• Interval / scale
– Time, money, age, GPA
Main Analysis Techniques
Variable Type
Example
Commonly-used
Statistical
Method
Nominal by Nominal
blood type by gender Chi-square
Scale by Nominal
GPA by gender
t-test
GPA by major
Analysis of Variance
weight by height
GPA by SAT
Regression
Correlation
Scale by Scale
Question
Do men and women differ in
the % that choose jail time vs.
probation only?
Main Analysis Techniques
Variable Type
Nominal by Nominal
Example
Commonly-used
Statistical
Method
Chi-square
(categorical by categorical)
blood type by
gender
Scale by Nominal
GPA by gender
t-test
GPA by major
Analysis of Variance
weight by height
GPA by SAT
Regression
Correlation
Scale by Scale
Stat Analysis / Hypothesis Testing
1. Form of the relationship
2. Statistical significance
Variables:
Categorical by Categorical
• Form of the relationship:
Cross-tab
= two-way table
• Statistical Significance:
Chi Square
[ if n very small  Fisher’s exact test ]
Example: cross-tab
Males
Probation
Jail
n = 24
n = 16
16
4
8
12
n = 20
Females
n = 20
Example: cross-tab
Males
Probation
Jail
n = 24
n = 16
16
4
n = 20
Females
n = 20
80%
8
20%
12
40%
60%
Example
• Men more likely to choose probation
in the sample
• Can we infer men in general more
likely to choose probation?
 Statistical Significance
Statistical Significance
• Q: Is this a “statistically significant”
difference?
• Can the “null hypothesis” be rejected?
Null hypothesis: there are NO differences
between men and women in sentencing
sample
Sample
n = 40
inference
M: 80% probation
F: 40% probation
Universe
n=∞
M: ?% probation
F: ?% probation
sample
Sample
n = 40
inference
M: 80% probation
F: 40% probation
Universe
n=∞
Null Hypothesis:
M% = F%
Logic of Statistical Inference
1. If the Null Hypothesis is True…
… what are the expected frequencies
for Men and Women in any sample?
2. Do the frequencies in my sample
(n = 40) differ from the expected
frequencies?
Testing Null Hypothesis:
Expected Frequencies
Males
Probation
Jail
n = 24
n = 16
exp: 12
exp: 8
exp: 12
exp: 8
n = 20
Females
n = 20
Observed & Expected Frequencies
Males
n = 20
Females
n = 20
Probation
Jail
n = 24
n = 16
16
4
exp: 12
8
exp: 12
exp: 8
12
exp: 8
Logic of Statistical Inference
• What is the probability of drawing the
observed sample (M = 16 probation vs. F =
8 probation) from a universe with no
differences?
• If probability very low, then differences in
sample likely reflect differences in universe
• Then null hypothesis can be rejected;
difference in sample is statistically
significant
Statistical Significance
• If probability of obtaining my sample is
less than 5 in 100, the null hypothesis can
be rejected, and it can be concluded that
the difference also exists in the universe.
p < .05
• The finding from the sample is
statistically significant
Strategy
• Draw an infinite number of
samples of n = 40, and graph the
distribution of their male vs.
female probation %-s
Samples of n = 40
Universe n = ∞
M: 80%
F: 40%
M: 60%
F: 50%
Null Hyp:
M = 60% probation
F = 60% probation
M: 70%
F: 70%
M: 50%
F: 65%
Chi Square Distribution
M% = F%
2.5% of area
F% > M%
2.5% of area
M% > F%
Statistical Significance
• If probability of obtaining my sample is
less than 5 in 100, the null hypothesis can
be rejected, and it can be concluded that
the difference also exists in the universe.
p < .05
• The finding from the sample is
statistically significant
Testing Null Hypothesis:
Sample with small difference
Males
n = 20
Females
n = 20
Probation
Jail
n = 24
n = 16
13
7
exp: 12
11
exp: 12
exp: 8
9
exp: 8
Sample
N = 40
M = 65%
F = 55%
sample
Universe
N=∞
p < .05 ?
M = 60% probation
F = 60% probation
Males
n = 20
Females
n = 20
Probation
Jail
n = 24
n = 16
13
7
exp: 12
11
exp: 8
9
exp: 12
Chi Square p = .519
exp: 8
Small Difference
• p = .519
• Over 50% of samples drawn from null
hypothesis universe will have differences
this large (65% vs. 55%)
• Difference is not statistically significant
Testing Null Hypothesis:
Sample with large differences
Males
n = 20
Females
n = 20
Probation
Jail
n = 24
n = 16
16
4
exp: 12
8
exp: 12
exp: 8
12
exp: 8
Sample
N = 40
M = 80%
F = 40%
sample
Universe
N=∞
p < .05 ?
M = 60% probation
F = 60% probation
Males
n = 20
Females
n = 20
Probation
Jail
n = 24
n = 16
16
4
exp: 8
exp: 12
8
12
exp: 12
Chi Square p = .010
exp: 8
Report Findings
• “Men were found to choose probation
more frequently than women: 80% of the
time vs. 40% of the time (df = 1, χ2 = 6.67,
p. < ,05).”
• “Men chose probation 80% of the time,
and women only 40% of the time, a
difference which was statistically
significant (df = 1, χ2 = 6.67, p. < ,05).”