Chi-Squared Tests: Inference for Two-Way Tables

Chi-squared ("Chi" rhymes with "pie") tests have many uses. This page discusses the basis for all chi-squared tests. There are links at the bottom of this page for specific chi-squared tests.

**Purpose:** To test whether there is an association
between two or more categorical variables.

**Requirements:** Set up a two-way table of observed
counts. Use it to compute the expected counts for each cell in the table, thus:.
The required conditions for the tests are

(1) The data are from one or more (independent) simple random samples. (The exact randomness requirements depend on the actual design of the experiment.)

(2) All expected counts are greater than 1.

(3) No more than 20% of the expected counts are less than 5.

**Hypotheses:**

*There is no association between
the two variables, *versus
*There is an association between
the two variables*. These hypotheses can be specialized to a stunning
variety of more specific hypotheses. See the links at the bottom of this page
for more information.

**Test Statistic:** We recommend using software such as
SPSS to calculate the
test
statistic, the degrees of freedom, and the p-value. However, you __can__ use
the formula
,
with degrees
of freedom, and get the p-value from a
table
such as Table F in Moore & McCabe’s __The Practice of Business
Statistics__.

**WARNING: The
test
statistic is very sensitive to round-off errors made during calculation. **
We
recommend using the above formula instead of the one usually found in textbooks,
as it suffers less from this problem. Also, we recommend entering the entire
calculation in your calculator all at once, rather than computing each term
separately, to help minimize round-off error.

**Instructions on using SPSS for chi-squared tests:**
Click here.

** **

**Related topics:**

Tests of homogeneity (several proportions)

Goodness-of-fit (comparing populations when one population is “known”)

**Statistics Reference Pages page:**
Click here.

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