More than Two Categories (1 of 3)
A skeptical
view of psychotherapy can be summed up as follows: one third better, one
third worse, one third the same. In other words, one third of the patients
improve as a result of psychotherapy, one third get worse, and one third
stay the same.
Consider a hypothetical study of 100 patients who underwent
psychotherapy. Assume that 45 were classified as having improved, 30 as
having gotten worse, and 25 as having stayed the same. Is this outcome
significantly different from that expected under the one third better, one
third worse, one third the same? The test is conducted by constructing a
table similar to the one used to test the difference between
p
and π .

Frequency 
Better 
45
(33.333) 
Worse 
25
(33.333) 
Same 
30
(33.333) 
Total 
100 
The first
column contains the three categories into which patients could be classified.
The top line in each cell in the second column contains the number of patients
in the category; the second line contains the number expected under the
null hypothesis. These "expected" frequencies are computed by
multiplying the total number of subjects (100) by the proportion expected
to be in the category (0.33333).