equality. Chi square will determine whether the
observed pattern differs significantly from the
daily expected 40.

8
CHI SQUARE FORMULA
The formula for chi square is the summation
for each cell
:
(O - E)
E
2
Where:
O
= observed frequency
- the data observed in our
research/survey
E
= expected frequency, and
= the summation over all the cells in the table
=
Chi
2

9
FORMAT OF CELL
•
Each cell follows the pattern:
Observed
Expected
O – E
(O - E)
2

10
EXAMPLE OF GOODNESS OF
FIT
observed 200 sick leave absences - the expected frequency in each cell must
be 200/5 = 40
Monday
Tuesday
Wednesday
Thursday
Friday
64
40
29
40
15
40
20
40
72
40
24
576
11
121
25
625
20
400
32
1024
Chi square =
(O - E)
2
=
576
+ 121
+ 625
+ 400
+
1024
E
40
40
40
40
40
Chi square =
68.65
p < .01
i.e. a significant
association between absence and particular days of week

11
INTERPRETATION OF
GOODNESS OF FIT EXAMPLE
•
We can reject the null hypothesis with
confidence, and accept the alternate hypothesis
that sick leave is not randomly distributed
through the week.
•
To specify how it is distributed, you must return
to inspect the original data where you can
readily appreciate that absences are much
higher on Mondays and Fridays and much lower
on other days of the week. I leave the
interpretation and speculation of why to you!

12
INTERPRETATION OF
GOODNESS OF FIT EXAMPLE
•
A chi square of zero indicates that the
observed and expected frequencies match
exactly.
•
Chi square can never be negative since
differences between the observed and
expected are always squared.

13
SPSS EXAMPLE OF GOODNESS
OF FIT
Is there any specific preference for one of three
drinks? Null hypothesis claims any variation is
simply random
1.
Click on
Analyze
and select
Nonparametric
Tests
from the drop-down menu.
2.
Choose
Chi-square
...
which opens the
Chi-
Square Test
dialogue
box.
3.
Select the variable (in this example
‘drink’)
then
click on the arrow button which transfers this
variable to
the
Test Variable List
: box.
4.
Select
OK
.
The results of the analysis are
displayed in next slides.

14
SPSS Example

15
SPSS Output
Equality
of choice
Actual choices
Residuals are
difference
between
observed and
expected

16
SPSS Output
Significant as p<.05

17
How to Interpret Output
•
The observed choice frequencies are presented in the
second column.
•
The expected frequencies of cases are displayed in the
third column. The expected frequency for each of the
four drinks with 40 personal choices is 40/4, i.e. 10.
•
The residual column displays the differences between
the observed and expected frequencies.
•
The second box presents the value of chi square, its
degrees of freedom and its significance. Chi square is
8.4, its degrees of freedom are 3 (i.e. 4 choices - 1) and
its significance level is 0.038.
This indicates that there is
a statistically significant deviation from the expected
distribution of equality beyond p<.05. Coke is most
popular while Solo and Sprite are significantly less
preferred.

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- Summer '14
- Dr.EugeneKaciak
- Chi-Square Test, Statistical hypothesis testing, Chi-square distribution, Pearson's chi-square test, Fisher's exact test, CHI SQUARE