“Chi-Square Statistics”
By
Namrata Khemka
Table of Contents
1.
2.
3.
4.
5.
What is Chi-Square?
When and why is Chi-Square used?
Limitations/Restrictions of Chi-Square
Examples
References
What is “Chi Square”
•
•
•
•
Invented by Pearson
Test for “Goodness of fit”
Tests for independence of variables
Non parametric test
Parametric vs. Non Parametric Data
Parametric data
Non Parametric data
1. Numerical scores
2. Manipulate the
scores
3. Example
•
Average height of
people in 10 cities
1. Nominal data
2. Scores not
manipulated
3. Example
•
How many people
are over 6ft and how
many are below in 2
cities
What is “Chi Square”
•
•
•
•
•
•
Invented by Pearson
Test for “Goodness of fit”
Tests for independence of variables
Non parametric test
Analyze categorical or
measurement data
SPSS or Excel
Goodness of the Fit
1.
2.
3.
4.
5.
6.
Null Hypothesis
Observed frequency
Expected frequencies
Good Fit
Poor Fit
Sum of observed frequencies = sum of
expected frequencies.
Computational Steps
• Scenario
Scenario:
•
A movie theater
owner would like to
know the factors
involved in movie
selection by people.
• A sample of 50
people were asked,
which of the following
were important to
them.
• They may choose one of the following:
1. Actors
2. Directors
3. Time the movies is playing
4. Genre
Question
• Do any of these factors play a
greater role than the others?
Computational Steps
• Scenario
• Threshold Value = 0.05
• Null Hypothesis
Null Hypothesis
• There is no difference in the importance of
these 4 factors in determining which movie
is selected
Computational Steps
•
•
•
•
•
•
Scenario
Threshold Value = 0.05
Null Hypothesis
Observed Frequencies
Expected Frequencies
p-value
Interpret the Results
• Since p is < 0.05, we
reject the null
hypothesis.
• There fore, some of
the factors are
mentioned more than
others in response to
movie selection
Test of Independence
• Examines the extent to which two
variables are related
• Example
Scenario:
•
•
University of Calgary is interested in
determining whether or not there is a
relationship between educational level
and the number of flights taken each
year.
150 travelers in the airport were
interviewed and the results are:
Scenario - Continued
2 or less
flights a
year
University
53
Student
High School 37
Student
More than 2
flights a year
22
38
Computational Steps
• Scenario
• Threshold Value = 0.05
• Null Hypothesis
Null Hypothesis
• The educational level of the travelers and
the number of flights are independent of
one another.
Computational Steps
•
•
•
•
•
•
Scenario
Threshold Value = 0.05
Null Hypothesis
Observed Frequencies
Expected Frequencies
p-value
Interpret the Results
• Since p is < 0.05, we
reject the null
hypothesis.
• These 2 variables are
not independent of
one another.
• Thus, the educational
level of travelers and
the number of flights
they take are related
Requirements and Limitations
•
•
•
•
•
Random sampling
Data must be in raw frequencies
Independence of observations
Size of the expected frequencies
Collapsing values
Collasping Values
Leather
Shoes
Sandals
Boots
Runners
Man
18
5
12
16
Women
20
19
6
10
Calculation - Details
•
•
•
•
•
Fo – fe
(Fo – fe)2
((Fo – fe)2)/fe
Chi-square = SUM((Fo – fe)2)/fe
Calculate the degrees of freedom = (R-1)
(C-1)
Calculation - Fo – Fe
University
Student
High School
Student
2 or less
flights a
year
8
More than 2
flights a year
-8
8
-8
Calculation – (Fo – Fe)2
University
Student
High School
Student
2 or less
flights a
year
64
More than 2
flights a year
64
64
64
Calculation – ((Fo – fe)2)/fe
University
Student
High School
Student
2 or less
flights a
year
1.42
More than 2
flights a year
1.42
2.13
2.13
Calculation – Continued
• Chi-square =
SUM((Fo – fe)2)/fe
• 7.1111
• Calculate the degrees
of freedom = (R-1)
(C-1)
• (2-1)(2-1) = 1
Distribution Table
df
0.9
0.1
0.05
0.025
0.01
1
0.016
2.706
3.841
5.024
6.635
2
0.211
4.605
5.991
7.378
9.21
3
0.584
6.251
7.815
9.348
11.345
4
1.064
7.779
9.488
11.143
13.277
Interpretation
Chi-Square
Conclusion
•
•
•
•
•
What is chi-square
When should chi-square be used
Limitations of Chi-square
Examples
Resources
References
•
•
•
www.ling.upenn.edu/courses/Summer_2
002/ling102/chisq.html
Statistical techniques in business and
economics by Lind, Marchal and Mason
Statistics for the behavioral sciences by
Federick J. Gravetter and Larry B.
Wallnau
Questions???