“Chi-Square Statistics”

By Namrata Khemka

Table of Contents

1. What is Chi-Square?2. When and why is Chi-Square used?3. Limitations/Restrictions of Chi-Square4. Examples5. 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 DataParametric data

1. Numerical scores2. Manipulate the

scores3. Example

• Average height of people in 10 cities

Non Parametric data1. Nominal data2. Scores not

manipulated3. 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. Null Hypothesis2. Observed frequency3. Expected frequencies4. Good Fit5. Poor Fit6. 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.Actors2.Directors3.Time the movies is playing4.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 - Continued2 or less flights a year

More than 2 flights a year

University Student 

53  22 

High School Student

37  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

LeatherShoes

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

2 or less flights a year

More than 2 flights a year

University Student 

8 -8

High School Student

-8 8

Calculation – (Fo – Fe)2

2 or less flights a year

More than 2 flights a year

University Student 

64 64

High School Student

64 64

Calculation – ((Fo – fe)2)/fe

2 or less flights a year

More than 2 flights a year

University Student 

1.42 2.13

High School Student

1.42 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 Tabledf 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_2002/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???