```                      BIVARIATE TABLES AND CHI-SQUARE

I.  Bivariate tables.
A.  Bivariate tables (also called cross-classification or cross-
tabulation tables) present the joint frequency distributions of
two variables.
B.  The table has rows representing the frequency of one variable
and columns representing the frequency of another variable.
The rows and columns intersect creating cells which contain the
joint frequency distribution of the two variables.
1.  Bivariate tables are described by the notation r x c
where r equals the number of rows and c equals the number
of colums.  For example, a table with two rows and to
columns is a 2 x 2 table, a table with 2 rows and 3
columns is a 2 x 3 table, and so on.
2.  The number of cells in a table is equal to the number of
rows multiplied by the number of columns.
C.  The row totals contain the total frequencies for the row
variable and column totals contain the total frequencies for
the column variable.  These totals are called the marginal
frequencies or simply the marginals.
D.  Bivariate tables often contain percentages as well as
frequencies.  There are three kinds of percentages that may be
reported.
1.  Row percentages are calculated by dividing the number of
observations in a cell by the total number of
observations in the row of that cell and multiplying by
100.
2.  Column percentages are calculated by dividing the number
of observations in a cell by the total number of
observations in the column of that cell and multiplying
by 100.
3.  Total or overall percentages are calculated by dividing
the number of observations in a cell by the total sample
size and multiplying by 100.
E.  The distribution of one variable at a single level of the other
variable is called a conditional distribution.  There
conditional distributions of the row variable at each level of
the column variable and conditional distributions of the column
variable at each level of the row variable.

II.  Chi-square test of independence.
A.  If the conditional distributions of one variable are the same
proportionally at every level of the other variable, the two
variables are considered independent.  That is, the disribution
of one variable does not affect the distribution of the other
variable.  Simply put, there is no relationship between the two
variables.
B.  If the conditional distributions of one variable differ
proportionally at different levels of the other variable, the
two variables are considered dependent.  That is, the
distribution of one variable affects the distribution of the
other variable. Simply put, there is a relationship between the
two variables.
C.  The Chi-square test of independence is designed to test whether
two variables are independent.  The steps are:
1.  State the null hypothesis--the two variables are
independent.
2.  Choose a statistical test--the chi-square test of
independence.
3.  Check assumptions.
a.  Sample is a random probability sample.
b.  Both variables are categorical variables and can be
arranged in a bivariate table.
c.  Expected frequencies of each cell will be at least
5 in 2x2 tables.  In 2x3 or larger tables, the
expected frequencies of each cell should be at
least 5 in at least 75% of the cells.
4.  Select an alpha level.
5.  Calculate the test statistic, determine the probability
associated with the statistic, and make a decision about
the null.
a.  Calculating chi-square.
b.  Finding the probability.
i.  To determine the probability, you must use
the chi-square table.  To use the table you
must know the the degrees of freedom.
ii.  Calculating degrees of freedom.