Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

578 CHAPTER 17 | The Chi-Square Statistic: Tests for Goodness of Fit and Independence
A Simple Formula for Determining Expected Frequencies Although expected
frequencies are derived directly from the null hypothesis and the sample characteristics,
it is not necessary to go through extensive calculations to find f e
values. In fact, there is a
simple formula that determines f e
for any cell in the frequency distribution matrix:
f e
5 f c
f r
n
(17.4)
where f c
is the frequency total for the column (column total), f r
is the frequency total for
the row (row total), and n is the number of individuals in the entire sample. To demonstrate
this formula, we compute the expected frequency for introverts selecting yellow in
Table 17.6. First, note that this cell is located in the top row and second column in the table.
The column total is f c
= 20, the row total is f r
= 50, and the sample size is n = 200. Using
these values in formula 17.4, we obtain
f e
5 f c
f r
n 5 20s50d
200 5 5
This is identical to the expected frequency we obtained using percentages from the overall
distribution.
■ The Chi-Square Statistic and Degrees of Freedom
The chi-square test of independence uses exactly the same chi-square formula as the test
for goodness of fit:
2 5S s f o 2 f e d2
f e
As before, the formula measures the discrepancy between the data ( f o
values) and the
hypothesis ( f e
values). A large discrepancy produces a large value for chi-square and
indicates that H 0
should be rejected. To determine whether a particular chi-square statistic
is significantly large, you must first determine degrees of freedom (df) for the statistic
and then consult the chi-square distribution in the appendix. For the chi-square test of
independence, degrees of freedom are based on the number of cells for which you can
freely choose expected frequencies. Recall that the f e
values are partially determined by
the sample size (n) and by the row totals and column totals from the original data. These
various totals restrict your freedom in selecting expected frequencies. This point is illustrated
in Table 17.7. Once three of the f e
values have been selected, all the other f e
values in
the table are also determined. In general, the row totals and the column totals restrict the
final choices in each row and column. As a result, we may freely choose all but one f e
in
each row and all but one f e
in each column. If R is the number of rows and C is the number
of columns, and you remove the last column and the bottom row from the matrix, you
are left with a smaller matrix that has C – 1 columns and R – 1 rows. The number of cells
in the smaller matrix determines the df value. Thus, the total number of f e
values that you
can freely choose is (R – 1)(C – 1), and the degrees of freedom for the chi-square test of
independence are given by the formula
df = (R – 1)(C – 1) (17.5)
Also note that once you calculate the expected frequencies to fill the smaller matrix, the
rest of the f e
values can be found by subtraction.