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The Chi Square Test
Diana Mindrila, Ph
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Phoebe Balentyne, M
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Based on Chapter 23 of The Basic Practice of Statistics (6th ed
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Describe the problem of multiple comparisons
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Describe the chi-square test statistic
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Describe uses of the chi-square test
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Perform a chi-square goodness of fit test
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S
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I, & Flinger, M
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(2013)
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New York, NY: W
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Freeman and Company
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For each question, a categorical (or nominal) variable is created
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However, these procedures would be conducted separately by education
level and then by instructional preference
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Correlation analysis is typically used to measure the association between
variables, but correlation can only be used with quantitative variables
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This is called a crosstab because two
variables are being tabulated at the same time, and the frequency, or the
percentage of individuals in each subcategory, are being counted
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The next step is to obtain the frequencies for each category, which can be
done using statistical software, especially for a very large sample
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Chi-Square Test
To determine whether the association between two qualitative variables is
statistically significant, researchers must conduct a test of significance called
the Chi-Square Test
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Step 1: Formulate the hypotheses
Null Hypothesis:
H0: There is no significant association between students’ educational level
and their preference for online or face-to-face instruction
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If there is no association between the two variables, the individuals would be
uniformly distributed across the cells of the table
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(It is
technically multi-sided because the differences may occur in both directions
in each cell of the table)
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or
Ha: There is a significant difference in the distribution of instructional
preferences between undergraduate and graduate students
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The formula for computing the expected values requires the sample size, the
row totals, and the column totals
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The observed values are the actual counts computed from the sample
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The image below shows the table that SPSS creates for the two variables
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Step 4: Compute the test statistic
The chi-square statistic compares the observed values to the expected
values
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The Chi-Square Distributions
The Chi-Square Distributions
The chi-square distributions are a family of distributions that take only positive
values and are skewed to the right
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The chi-square test for a two-way table with r rows and c columns uses critical values
from the chi-square distribution with (r – 1)(c – 1) degrees of freedom
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The image above shows that the distribution of the chi-square statistic starts
at zero and can only have positive values
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The shape of the distribution changes as the degrees of freedom increases
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If these values are entered into the formula for the chi-square tests statistic,
the value obtained is 28
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Is this value high enough to reject the null hypothesis?
Step 5: Decide if chi-square is statistically significant
The final step of the chi-square test of significance is to determine if the value
of the chi-square test statistic is large enough to reject the null hypothesis
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For the purpose of this analysis, only the Pearson Chi-Square statistic is
needed
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000, which is smaller than the
alpha level of
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Therefore, there is enough evidence to reject the null hypothesis
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A table in a statistics textbook could also be used to conduct the test by hand:
The chi-square value of 28
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84, so
the null hypothesis can be rejected
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If the test is significant, it is important to look at the data to
learn the nature of the relationship
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Forty undergraduate students preferred face-to-face instruction, while the
expected value was 27
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Researchers can conclude that graduate students prefer online instruction,
whereas undergraduate students prefer face-to-face instruction
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Therefore, it is important to
check that the counts are large enough to result in a trustworthy p-value
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Cell Counts Required for the Chi-Square Test
You can safely use the chi-square test with critical values from the chi-square
distribution when no more than 20% of the expected counts are less than 5 and all
individual expected counts are 1 or greater
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Uses of the Chi-Square Test
One of the most useful properties of the chi-square test is that it tests the null
hypothesis “the row and column variables are not related to each other”
whenever this hypothesis makes sense for a two-way variable
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A single random sample, with each individual classified according to both of
two categorical variables