Basic Statistics
The Chi Square Test of
Independence
Chi Square Test of Independence
• A measure of association similar to the
correlations we studied earlier.
• Pearson and Spearman are not applicable if the
data are at the nominal level of measurement.
• Chi Square is used for nominal data placed in a
contingency table.
– A contingency table is a two-way table showing the
contingency between two variables where the variables
have been classified into mutually exclusive categories
and the cell entries are frequencies.
An Example
• Suppose that the state legislature is
considering a bill to lower the legal drinking
age to 18. A political scientist is interested in
whether there is a relationship between party
affiliation and attitude toward the bill. A
random sample of 150 registered republicans
and 200 registered democrats are asked their
opinion about the proposed bill. The data are
presented on the next slide.
Political Party and
Legal Drinking Age Bill
Attitude Toward Bill
For
Undecided
Against
Total
Republican
38
17
95
150
Democrat
92
18
90
200
Total
130
35
185
350
The bold numbers are the observed frequencies (fo)
Determining the Expected
Frequencies (fe)
• First, add the columns and the rows to get
the totals as shown in the previous slide.
• To obtain the expected frequency within a
particular cell, perform the following
operation: Multiply the row total and the
column total for the cell in question and
then divide that product by the Total number
of all respondents.
Calculating the Expected Value for a
Particular Cell
Attitude Toward Bill
For
Republican
Undecided
38
Against
Total
17
95
150
18
90
200
55.7
Democrat
Total
92
130
35
1. 130*150 = 19500
185
2. 19500/350 = 55.7
350
Political Party and Attitude toward Bill
Attitude Toward Bill
For
Undecided
Against
Total
Republican
38
55.7
17
15
95
79.3
150
Democrat
92
74.3
18
20
90
105.7
200
Total
130
35
185
350
Numbers in Black are obtained (fo),
Numbers in Purple are expected (fe)
The Null Hypothesis and the
Expected Values
• The Null Hypothesis under investigation in
the Chi Square Test of Independence is that
the two variables are independent (not
related).
• In this example, the Null Hypothesis is that
there is NO relationship between political
party and attitude toward lowering the legal
drinking age.
Understanding the Expected Values
• If the Null is true, then the percentage of
those who favor lowering the drinking age
would be equal for each political party.
• Notice that the expected values for each
opinion are proportional for the number of
persons surveyed in each party.
Political Party and Attitude toward Bill
Attitude Toward Bill
For
Undecided
Against
Total
Republican
38
55.7
17
15
95
79.3
150
42.8%
Democrat
92
74.3
18
20
90
105.7
200
57.2%
Total
130
35
185
350
The numbers in Green are the percentage of the total for each Party.
The Expected Values
• The expected values for each cell are also equal
to the percentage of each party for the column
total.
• For example, Republicans were 42.8% of the
total persons surveyed
• If 130 people were in favor of the bill, then
42.8% of them should be Republican (55.7), if
there is no relationship between the variables
Calculating the Chi Square Statistic
The Chi Square statistic is obtained via this formula
( fo  fe )
x 
fe
2
2
The Chi Square statistic is
(1) the sum over all cells of
(2) the difference between the obtained value and the
expected value SQUARED, which is then
(3) divided by the expected frequency.
The numbers in Purple on the next slide illustrate this calculation
Calculating the Chi Square Statistic
(38  55.7)
55.7
2
=5.62
(92  74.3) 2
74.3
=4.22
(17  15) 2
15
=0.27
(18  20) 2
20
=0.20
(95  79.3) 2
79.3
=3.11
(90  105.7) 2
105.7
=2.33
X2 = 5.62 + 0.27 + 3.11 + 4.22 + 0.20 + 2.33 = 15.75
Interpreting the Results
The calculated value for the chi square statistic X 2 is
compared to the critical value found in Table H, page 544.
Note: The distribution of the Chi Square Statistic is not
normal and the critical values are only on one side. If the
obtained values are close to the expected value, then the chi
square statistic will approach 0. As the obtained value is
different from the expected, the value of chi square will
increase. This is reflected in the values found in Table H.
The Degrees of Freedom for the Chi Square Test of
Independence is the product of the number of rows
minus 1 times the number of columns minus 1
Interpreting Our Results
• In our study, we had two rows (Republicans and
Democrats) and three columns (For, Undecided,
Against).
• Therefore, the degrees of freedom for our study is
(2-1)(3-1) = 1(2) = 2.
• Using an a of .05, the critical value from Table H
would be 5.991
• Since our calculated chi square is 15.75, we
conclude that there IS a relationship between
political party and opinion on lowering the drinking
age, thereby rejecting the Null Hypothesis