Page 266 - Statistics II for Dummies

250 Part IV: Building Strong Connections with Chi-Square Tests

Putting all these Chi-square test statistics together gives you what’s called a
Chi-square distribution. You find your particular test statistic on that distribu-
tion (step six of the Chi-square test), and see where it stands compared to
the rest.

If your test statistic is large enough that it appears way out on the right tail
of the Chi-square distribution (boldly going where no test statistic has gone
before), you reject Ho and conclude the two variables are not independent. If
the test statistic isn’t that far out, you can’t reject Ho.

In the next sections, you find out more about the Chi-square distribution and
how it behaves, so you can make a decision about the independence of your
two variables based on your Chi-square statistic.
Determining degrees of freedom

Each type of two-way table has its own Chi-square distribution, depending on
the number of rows and columns it has, and each Chi-square distribution is
identified by its degrees of freedom.

In general, a two-way table with r rows and c columns uses a Chi-square
distribution with (r – 1) * (c – 1) degrees of freedom. A two-way table with
two rows and two columns uses a Chi-square distribution with one degree of
freedom. Notice that 1 = (2 – 1) * (2 – 1). A two-way table with three rows and
two columns uses a Chi-square distribution with (3 – 1) * (2 – 1) = 2 degrees
of freedom.

Understanding why degrees of freedom are calculated this way is likely to be
beyond the scope of your statistics class. But if you really want to know, the
degrees of freedom represents the number of cells in the table that are flexi-
ble, or free, given all the marginal row and column totals.

For example, suppose that a two-way table has all row and column totals
equal to 100 and the upper-left cell is 70. Then the upper-right cell must be
100 (row total) – 30 = 70. Because the column one total is 100, and the upper-
left cell count is 70, the lower-left cell count must be 100 – 70 = 30. Similarly,
the lower-right cell count must be 70.

So you have only one free cell in a two-way table after you have the marginal
totals set up. That’s why the degree of freedom for a two-way table is 1. In
general, you always lose one row and one column because of knowing the
marginal totals. That’s because the last row and column values can be calcu-
lated through subtraction. That’s where the formula (r – 1) * (c – 1) comes
from. (That’s more than you wanted to know, isn’t it?)

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