The document discusses factorial analysis of variance (ANOVA). It explains how total sums of squares can be partitioned into explained and unexplained components. An example shows an F ratio of 5.0 for one data set, indicating variation between groups is rare. This allows rejecting the null hypothesis with a low probability of Type I error. Finally, it describes how factorial ANOVA can analyze the effects of multiple independent variables on a single dependent variable.
3. Having made the jump to sums of squares logic, …
(here’s an example of sums of squares calculation:)
4. Having made the jump to sums of squares logic, …
(here’s an example of sums of squares calculation:)
Scenario 1:
Deviatio
Person Scores Mean
n
Squared
Bob 1 – 4 = - 3 2 = 9
Sally 4 – 4 = 0 2 = 0
Val 7 – 4 = + 4 2 = 16
Average 4 sum of
squares 25
5. Having made the jump to sums of squares logic, …
(here’s an example of sums of squares calculation:)
Scenario 1:
Person Scores Mean
Scenario 2:
Deviatio
n
Squared
Bob 1 – 4 = - 3 2 = 9
Sally 4 – 4 = 0 2 = 0
Val 7 – 4 = + 4 2 = 16
Average 4 sum of
squares 25
Person Scores Mean
Deviatio
n
Squared
Bob 3 – 4 = - 1 2 = 1
Sally 4 – 4 = 0 2 = 0
Val 5 – 4 = + 1 2 = 1
Average 4 sum of
squares 2
6. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components …
7. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components …
For example:
8. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components …
For example:
• Explained Sums of Squares component (variation
explained by differences between groups) = 30
9. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components …
For example:
• Explained Sums of Squares component (variation
explained by differences between groups) = 30
• Unexplained Sums of Squares component (variation
explained by differences within groups) = 6
10. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components … and having discovered that the ratio of
explained to unexplained variance can render a
coefficient that can be evaluated for its rarity …
11. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components … and having discovered that the ratio of
explained to unexplained variance can render a
coefficient that can be evaluated for its rarity …
Explained Variance (30)
Unexplained Variance (6)
= 5.0
12. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components … and having discovered that the ratio of
explained to unexplained variance can render a
coefficient that can be evaluated for its rarity …
Explained Variance (30)
Unexplained Variance (6)
= 5.0
Wow, for this data
set an F ratio of 5.0
is pretty rare!
13. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components … and having discovered that the ratio of
explained to unexplained variance can render a
coefficient that can be evaluated for its rarity …
Explained Variance (30)
Unexplained Variance (6)
= 5.0
– OR –
14. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components … and having discovered that the ratio of
explained to unexplained variance can render a
coefficient that can be evaluated for its rarity …
Explained Variance (30)
Unexplained Variance (6)
= 5.0
Explained Variance (2)
Unexplained Variance (2)
= 1.0
– OR –
15. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components … and having discovered that the ratio of
explained to unexplained variance can render a
coefficient that can be evaluated for its rarity …
Explained Variance (30)
Unexplained Variance (6)
= 5.0
Explained Variance (2)
Unexplained Variance (2)
= 1.0
– OR –
Wow, for this data
set an F ratio of 1.0
is not rare at all but
pretty common!
16. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components … and having discovered that the ratio of
explained to unexplained variance can render a
coefficient that can be evaluated for its rarity … to
make decisions about the probability of Type I error
when rejecting a null hypothesis, …
17. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components … and having discovered that the ratio of
explained to unexplained variance can render a
coefficient that can be evaluated for its rarity … to
make decisions about the probability of Type I error
when rejecting a null hypothesis, …
Hmm, an F ratio of 5.0 for this
data set is so rare that there is
a .02 chance that I’m wrong to
reject the null hypothesis (this
would be a Type I error).
I can live with those odds. So
I’ll reject the Null hypothesis!
18. Having made the jump to sums of squares logic, … and
having observed that the total sums of squares can be
partitioned into “explained” and “unexplained”
components … and having discovered that the ratio of
explained to unexplained variance can render a
coefficient that can be evaluated for its rarity … to
make decisions about the probability of Type I error
when rejecting a null hypothesis, …
Hmm, an F ratio of 5.0 for this
data set is so rare that there is
a .02 chance that I’m wrong to
reject the null hypothesis (this
would be a Type I error).
I can live with those odds. So
I’ll reject the Null hypothesis!
19. We can then extend those principles to a wide range of
applications.
20. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
21. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
22. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
23. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
means square
24. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
means square
F ratio & F critical
25. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
means square
F ratio & F critical
hypothesis testing
26. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
means square
F ratio & F critical
hypothesis testing
one-way
ANOVA
27. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
means square
F ratio & F critical
hypothesis testing
factorial
ANOVA
one-way
ANOVA
28. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
means square
F ratio & F critical
hypothesis testing
factorial
ANOVA
split plot
ANOVA
one-way
ANOVA
29. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
means square
F ratio & F critical
hypothesis testing
factorial
ANOVA
split plot
ANOVA
repeated measures
ANOVA
one-way
ANOVA
30. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
means square
F ratio & F critical
hypothesis testing
factorial
ANOVA
split plot
ANOVA
repeated measures
ANOVA
ANCOVA
one-way
ANOVA
31. We can then extend those principles to a wide range of
applications.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
means square
F ratio & F critical
hypothesis testing
one-way
ANOVA
factorial
ANOVA
split plot
ANOVA
repeated measures
ANOVA
ANCOVA
MANOVA
32. Thus far we have only considered one dependent
variable and one independent variable that was
categorized into several levels
33. Thus far we have only considered one dependent
variable and one independent variable that was
categorized into several levels
One dependent variable
34. Thus far we have only considered one dependent
variable and one independent variable that was
categorized into several levels
One dependent variable
Dependent Variable: Amount of pizza eaten
35. Thus far we have only considered one dependent
variable and one independent variable that was
categorized into several levels
One dependent variable
Dependent Variable: Amount of pizza eaten
One independent variable
36. Thus far we have only considered one dependent
variable and one independent variable that was
categorized into several levels
One dependent variable
Dependent Variable: Amount of pizza eaten
One independent variable
Independent Variable: Athletes
37. Thus far we have only considered one dependent
variable and one independent variable that was
categorized into several levels
One dependent variable
Dependent Variable: Amount of pizza eaten
One independent variable
Independent Variable: Athletes
Categorized into several levels
38. Thus far we have only considered one dependent
variable and one independent variable that was
categorized into several levels
One dependent variable
Dependent Variable: Amount of pizza eaten
One independent variable
Independent Variable: Athletes
Categorized into several levels
Level 1:
Football Player
39. Thus far we have only considered one dependent
variable and one independent variable that was
categorized into several levels
One dependent variable
Dependent Variable: Amount of pizza eaten
One independent variable
Independent Variable: Athletes
Categorized into several levels
Level 1:
Football Player
Level 2:
Basketball Player
40. Thus far we have only considered one dependent
variable and one independent variable that was
categorized into several levels
One dependent variable
Dependent Variable: Amount of pizza eaten
One independent variable
Independent Variable: Athletes
Categorized into several levels
Level 1:
Football Player
Level 2:
Basketball Player
Level 3:
Soccer Player
41. We can consider the effect of multiple independent
variables on a single dependent variable.
42. We can consider the effect of multiple independent
variables on a single dependent variable.
For example:
43. We can consider the effect of multiple independent
variables on a single dependent variable.
For example:
First Independent Variable: Athletes
Level 1:
Football Player
Level 2:
Basketball Player
Level 3:
Soccer Player
44. We can consider the effect of multiple independent
variables on a single dependent variable.
For example:
First Independent Variable: Athletes
Level 1:
Football Player
Level 2:
Basketball Player
Level 3:
Soccer Player
Second Independent Variable: Age
45. We can consider the effect of multiple independent
variables on a single dependent variable.
For example:
First Independent Variable: Athletes
Level 1:
Football Player
Level 2:
Basketball Player
Level 3:
Soccer Player
Second Independent Variable: Age
Level 1:
Adults
Level 2:
Teenagers
46. We can consider the effect of multiple independent
variables on a single dependent variable.
For example: the differences in number of slices of
pizza consumed (this is the single independent variable)
among 3 different athlete groups (Football, Basketball,
& Soccer) at two different age levels (Adults &
Teenagers).
47. We can consider the effect of multiple independent
variables on a single dependent variable.
For example: the differences in number of slices of
pizza consumed (this is the single independent variable)
among 3 different athlete groups (Football, Basketball,
& Soccer) at two different age levels (Adults &
Teenagers). Now, rather than comparing only 3 groups,
we will be comparing 6 groups (3 levels of athlete x 2
levels of age groups).
48. We can consider the effect of multiple independent
variables on a single dependent variable.
For example: the differences in number of slices of
pizza consumed (this is the single independent variable)
among 3 different athlete groups (Football, Basketball,
& Soccer) at two different age levels (Adults &
Teenagers). Now, rather than comparing only 3 groups,
we will be comparing 6 groups (3 levels of athlete x 2
levels of age groups).
Let’s see what this data set might look like.
50. First we list our three levels of athletes
Athletes
Football Player 1
Football Player 2
Football Player 3
Football Player 4
Football Player 5
Football Player 6
Basketball Player 1
Basketball Player 2
Basketball Player 3
Basketball Player 4
Basketball Player 5
Basketball Player 6
Soccer Player 1
Soccer Player 2
Soccer Player 3
Soccer Player 4
Soccer Player 5
Soccer Player 6
51. Then our two age groups
Athletes
Football Player 1
Football Player 2
Football Player 3
Football Player 4
Football Player 5
Football Player 6
Basketball Player 1
Basketball Player 2
Basketball Player 3
Basketball Player 4
Basketball Player 5
Basketball Player 6
Soccer Player 1
Soccer Player 2
Soccer Player 3
Soccer Player 4
Soccer Player 5
Soccer Player 6
52. Then our two age groups
Athletes Adults Teenagers
Football Player 1
Football Player 2
Football Player 3
Football Player 4
Football Player 5
Football Player 6
Basketball Player 1
Basketball Player 2
Basketball Player 3
Basketball Player 4
Basketball Player 5
Basketball Player 6
Soccer Player 1
Soccer Player 2
Soccer Player 3
Soccer Player 4
Soccer Player 5
Soccer Player 6
53. Now we add our dependent variable - pizza consumed
Athletes Adults Teenagers
Football Player 1
Football Player 2
Football Player 3
Football Player 4
Football Player 5
Football Player 6
Basketball Player 1
Basketball Player 2
Basketball Player 3
Basketball Player 4
Basketball Player 5
Basketball Player 6
Soccer Player 1
Soccer Player 2
Soccer Player 3
Soccer Player 4
Soccer Player 5
Soccer Player 6
54. Now we add our dependent variable - pizza consumed
Athletes Adults Teenagers
Football Player 1 9
Football Player 2 10
Football Player 3 12
Football Player 4 12
Football Player 5 15
Football Player 6 17
Basketball Player 1 1
Basketball Player 2 5
Basketball Player 3 9
Basketball Player 4 3
Basketball Player 5 6
Basketball Player 6 8
Soccer Player 1 1
Soccer Player 2 2
Soccer Player 3 3
Soccer Player 4 2
Soccer Player 5 3
Soccer Player 6 5
55. The procedure by which we analyze the sums of
squares among the 6 groups based on 2 independent
variables (Age Group and Athlete Category) is called
Factorial ANOVA.
56. The procedure by which we analyze the sums of
squares among the 6 groups based on 2 independent
variables (Age Group and Athlete Category) is called
Factorial ANOVA.
sums of squares
between groups
sums of squares
within groups
degrees of freedom
means square
F ratio & F critical
hypothesis testing
one-way
ANOVA
factorial
ANOVA
57. Factorial ANOVA partitions the total sums of squares
into the unexplained variance and the variance
explained by the main effects of each of the
independent variables and the interaction of the
independent variables.
58. Factorial ANOVA partitions the total sums of squares
into the unexplained variance and the variance
explained by the main effects of each of the
independent variables and the interaction of the
independent variables.
Main Effect Interaction Effect Error
Explained Variance Type of Athlete
Age group
Type of Athlete by
Age Group
Unexplained Variance Within Groups
60. Continuing our example:
• The type of athlete may have an effect on the
number of slices of pizza eaten.
61. Continuing our example:
• The type of athlete may have an effect on the
number of slices of pizza eaten.
• But also the age group might as well have an effect
on the number of slices eaten.
62. Continuing our example:
• The type of athlete may have an effect on the
number of slices of pizza eaten.
• But also the age group might as well have an effect
on the number of slices eaten.
• And the interaction of type of athlete and age group
may have an effect on slices eaten as well
63. Continuing our example:
• The type of athlete may have an effect on the
number of slices of pizza eaten.
• But also the age group might as well have an effect
on the number of slices eaten.
• And the interaction of type of athlete and age group
may have an effect on slices eaten as well
In other words, some age groups within different athlete
categories may consume different amounts of pizza. For
example, maybe football and basketball adults eat much
more than football and basketball teenagers, while adult
soccer players eat much less than teenage soccer players.
64. Continuing our example:
• The type of athlete may have an effect on the
number of slices of pizza eaten.
• But also the age group might as well have an effect
on the number of slices eaten.
• And the interaction of type of athlete and age group
may have an effect on slices eaten as well
In other words, some age groups within different athlete
categories may consume different amounts of pizza. For
example, maybe football and basketball adults eat much
more than football and basketball teenagers, while adult
soccer players eat much less than teenage soccer players.
65. Continuing our example:
• The type of athlete may have an effect on the
number of slices of pizza eaten.
• But also the age group might as well have an effect
on the number of slices eaten.
• And the interaction of type of athlete and age group
may have an effect on slices eaten as well
In other words, some age groups within different athlete
categories may consume different amounts of pizza. For
example, maybe football and basketball adults eat much
more than football and basketball teenagers, while adult
soccer players eat much less than teenage soccer players.
66. In that case, the soccer players did not follow the trend
of the football and basketball players. This would be
considered an interaction effect between age group
and type of athlete.
67. In that case, the soccer players did not follow the trend
of the football and basketball players. This would be
considered an interaction effect between age group
and type of athlete.
Of course, there are 6 (3 x 2) possible combinations of
age groups and types of athletes any one of which may
not follow the direct main effect trend of age group or
type of athlete.
68. In that case, the soccer players did not follow the trend
of the football and basketball players. This would be
considered an interaction effect between age group
and type of athlete.
Of course, there are 6 (3 x 2) possible combinations of
age groups and types of athletes any one of which may
not follow the direct main effect trend of age group or
type of athlete.
• Adult Football Player
• Teenage Football Player
• Adult Basketball Player
• Teenage Basketball Player
• Adult Soccer Player
• Teenage Soccer Player
70. You could also order them this way:
• Adult Football Player
• Adult Basketball Player
• Adult Soccer Player
• Teenage Football Player
• Teenage Basketball Player
• Teenage Soccer Player
71. You could also order them this way:
• Adult Football Player
• Adult Basketball Player
• Adult Soccer Player
• Teenage Football Player
• Teenage Basketball Player
• Teenage Soccer Player
The order doesn’t really matter.
72. When subgroups respond differently under different
conditions, we say that an interaction has occurred.
73. When subgroups respond differently under different
conditions, we say that an interaction has occurred.
Adult Football Players
eat 19 slices on average Teenage Football Players
eat 12 slices on average
74. When subgroups respond differently under different
conditions, we say that an interaction has occurred.
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
75. When subgroups respond differently under different
conditions, we say that an interaction has occurred.
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Do you see the trend here?
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
76. When subgroups respond differently under different
conditions, we say that an interaction has occurred.
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Do you see the trend here?
• Football players consume more pizza slices in one sitting
than do basketball players
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
77. When subgroups respond differently under different
conditions, we say that an interaction has occurred.
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Do you see the trend here?
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
• Football players consume more pizza slices in one sitting
than do basketball players
• And adults consume more pizza slices than do teenagers
78. When subgroups respond differently under different
conditions, we say that an interaction has occurred.
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Do you see the trend here?
• Football players consume more pizza slices in one sitting
than do basketball players
• And adults consume more pizza slices than do teenagers
Now let’s add the soccer players
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
79. When subgroups respond differently under different
conditions, we say that an interaction has occurred.
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Do you see the trend here?
• Football players consume more pizza slices in one sitting
than do basketball players
• And adults consume more pizza slices than do teenagers
Now let’s add the soccer players
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 6 slices on average
Teenage Soccer Players eat
8 slices on average
80. Because the soccer players do not follow the trend of
the other two groups, this is called an interaction effect
between type of athlete and age group.
81. So in the case below there would be no interaction
effect because all of the trends are the same:
82. So in the case below there would be no interaction
effect because all of the trends are the same:
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 8 slices on average Teenage Soccer Players eat
6 slices on average
83. So in the case below there would be no interaction
effect because all of the trends are the same:
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
• As you get older you eat more slices of pizza
• If you play football you eat more than basketball and
soccer players
• etc.
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 8 slices on average Teenage Soccer Players eat
6 slices on average
84. But in our first case there is an interaction effect
because one of the subgroups is not following the
trend:
85. But in our first case there is an interaction effect
because one of the subgroups is not following the
trend:
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 6 slices on average
Teenage Soccer Players eat
8 slices on average
86. But in our first case there is an interaction effect
because one of the subgroups is not following the
trend:
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
• Soccer players do not follow the trend of the older you
are the more pizza you eat.
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 6 slices on average
Teenage Soccer Players eat
8 slices on average
87. A factorial ANOVA will have at the very least three null
hypotheses. In the simplest case of two independent
variables, there will be three.
88. A factorial ANOVA will have at the very least three null
hypotheses. In the simplest case of two independent
variables, there will be three.
Here they are:
89. A factorial ANOVA will have at the very least three null
hypotheses. In the simplest case of two independent
variables, there will be three.
Here they are:
• Main Effect for Age Group: There is no significant
difference between the amount of pizza slices eaten by
adults and teenagers in one sitting.
90. A factorial ANOVA will have at the very least three null
hypotheses. In the simplest case of two independent
variables, there will be three.
Here they are:
• Main Effect for Age Group: There is no significant
difference between the amount of pizza slices eaten by
adults and teenagers in one sitting.
• Main Effect for Type of Athlete: There is no significant
difference between the amount of pizza slices eaten by
football, basketball, and soccer players in one sitting.
91. A factorial ANOVA will have at the very least three null
hypotheses. In the simplest case of two independent
variables, there will be three.
Here they are:
• Main Effect for Age Group: There is no significant
difference between the amount of pizza slices eaten by
adults and teenagers in one sitting.
• Main Effect for Type of Athlete: There is no significant
difference between the amount of pizza slices eaten by
football, basketball, and soccer players in one sitting.
• Interaction Effect Between Age Group and Type of
Athlete: There is no significant interaction between the
amount of pizza eaten by football, basketball and soccer
players in one sitting.
93. Let’s begin with the main effect for Age Group
Adults
eat 13 slices on average Teenagers
eat 11 slices on average
94. Let’s begin with the main effect for Age Group
Adults
eat 13 slices on average Teenagers
eat 11 slices on average
So adults eat 2 slices on average more than teenagers.
Is this a statistically significant difference? That’s what
we will find out using sums of squares logic.
96. Now let’s look at main effect for Type of Athlete
Football Players
eat 15.5 slices on average
Basketball Players
eat 10 slices on average
Soccer Players
eat 7slices on average
97. Now let’s look at main effect for Type of Athlete
Football Players
eat 15.5 slices on average
Basketball Players
eat 10 slices on average
Soccer Players
eat 7slices on average
So Football Players eat on average 5.5 slices more than
Basketball Players; Basketball Players eat 3 more slices
on average than Soccer Players; and Football Players
eat 8.5 slices on average more than Soccer Players.
98. Now let’s look at main effect for Type of Athlete
Football Players
eat 15.5 slices on average
Basketball Players
eat 10 slices on average
Soccer Players
eat 7slices on average
So Football Players eat on average 5.5 slices more than
Basketball Players; Basketball Players eat 3 more slices
on average than Soccer Players; and Football Players
eat 8.5 slices on average more than Soccer Players. Is
this a statistically significant difference? That’s what we
will find out using sums of squares logic.
100. Finally let’s consider the interaction effect
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 6 slices on average
Teenage Soccer Players eat
8 slices on average
101. Finally let’s consider the interaction effect
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 6 slices on average
Teenage Soccer Players eat
8 slices on average
As noted in this example earlier, it appears that there
will be an interaction effect between Age Group and
Types of Athletes.
102. So how do we test these possibilities statistically?
103. So how do we test these possibilities statistically?
Factorial ANOVA will produce an F-ratio for each main
effect and for each interaction.
104. So how do we test these possibilities statistically?
Factorial ANOVA will produce an F-ratio for each main
effect and for each interaction.
• Main effect: Age Group
105. So how do we test these possibilities statistically?
Factorial ANOVA will produce an F-ratio for each main
effect and for each interaction.
• Main effect: Age Group – F ratio.
106. So how do we test these possibilities statistically?
Factorial ANOVA will produce an F-ratio for each main
effect and for each interaction.
• Main effect: Age Group – F ratio.
• Main effect: Type of Athlete
107. So how do we test these possibilities statistically?
Factorial ANOVA will produce an F-ratio for each main
effect and for each interaction.
• Main effect: Age Group – F ratio.
• Main effect: Type of Athlete – F ratio.
108. So how do we test these possibilities statistically?
Factorial ANOVA will produce an F-ratio for each main
effect and for each interaction.
• Main effect: Age Group – F ratio.
• Main effect: Type of Athlete – F ratio.
• Interaction effect: Age Group by Type of Athlete
109. So how do we test these possibilities statistically?
Factorial ANOVA will produce an F-ratio for each main
effect and for each interaction.
• Main effect: Age Group – F ratio.
• Main effect: Type of Athlete – F ratio.
• Interaction effect: Age Group by Type of Athlete – F ratio
110. So how do we test these possibilities statistically?
Factorial ANOVA will produce an F-ratio for each main
effect and for each interaction.
• Main effect: Age Group – F ratio.
• Main effect: Type of Athlete – F ratio.
• Interaction effect: Age Group by Type of Athlete – F ratio
Each of these F ratios will be compared with their
individual F-critical values on the F distribution table to
determine if the null hypothesis will be retained or
rejected.
111. Always interpret the F-ratio for the interactions effect
first, before considering the F-ratio for the main effects.
112. Always interpret the F-ratio for the interactions effect
first, before considering the F-ratio for the main effects.
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 6 slices on average
Teenage Soccer Players eat
8 slices on average
113. Always interpret the F-ratio for the interactions effect
first, before considering the F-ratio for the main effects.
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 6 slices on average
Teenage Soccer Players eat
8 slices on average
If the F-ratio for the interaction is significant, the results
for the main effects may be moot.
114. If the interaction is significant, it is extremely helpful to
plot the interaction to determine where the effect is
occurring.
115. If the interaction is significant, it is extremely helpful to
plot the interaction to determine where the effect is
occurring.
116. If the interaction is significant, it is extremely helpful to
plot the interaction to determine where the effect is
occurring.
Notice how you can tell
visually that soccer players
are not following the age
trend as is the case with
football and basketball
players.
118. This looks a lot like our earlier image:
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 6 slices on average
Teenage Soccer Players eat
8 slices on average
119. There are many possible combinations of effects that
can render a significant F-ratio for the interaction. In
our example, one of the 6 groups might respond very
differently than the others …
120. There are many possible combinations of effects that
can render a significant F-ratio for the interaction. In
our example, one of the 6 groups might respond very
differently than the others … or 2, or 3, or … it can be
very complex.
121. If the interaction is significant, it is the primary focus of
interpretation.
122. If the interaction is significant, it is the primary focus of
interpretation.
However, sometimes the main effects may be
significant and meaningful; even the presence of the
significant interaction. The plot will help you decide if it
is meaningful.
123. If the interaction is significant, it is the primary focus of
interpretation.
However, sometimes the main effects may be
significant and meaningful; even the presence of the
significant interaction. The plot will help you decide if it
is meaningful.
For example, if all players increase in pizza consumption
as they age but some increase much faster in than
others, both the interaction and the main effect for age
may be important.
124. If the interaction is not significant, it can be ignored and
the interpretation of the main effects is
straightforward,
125. If the interaction is not significant, it can be ignored and
the interpretation of the main effects is
straightforward, as would be the case in this example:
126. If the interaction is not significant, it can be ignored and
the interpretation of the main effects is
straightforward, as would be the case in this example:
Adult Football Players
eat 19 slices on average
Adult Basketball Players
eat 14 slices on average
Teenage Football Players
eat 12 slices on average
Teenage Basketball Players
eat 10 slices on average
Adult Soccer Players
eat 8 slices on average Teenage Soccer Players eat
6 slices on average
127. You will now see how to calculate a Factorial ANOVA by
hand. Normally you will use a statistical software
package to do this calculation. That being said, it is
important to see what is going on behind the scenes.
128. Here is the data set we will be working with:
129. Here is the data set we will be working with:
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
130. First we will compute the between group sums of squares for Age Group
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
131. First we will compute the between group sums of squares for Age Group
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
132. Then we will compute the between group sums of squares for Type of Player
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
133. Then we will compute the between group sums of squares for Type of Player
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
134. And then the sums of squares for the interaction effect
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
135. And then the sums of squares for the interaction effect
Age Group Slices of Pizza Eaten Type of Player
Adult 17 Football Player
Adult 19 Football Player
Adult 21 Football Player
Adult 13 Basketball Player
Adult 14 Basketball Player
Adult 15 Basketball Player
Adult 2 Soccer Player
Adult 6 Soccer Player
Adult 8 Soccer Player
Teenage 11 Football Player
Teenage 12 Football Player
Teenage 13 Football Player
Teenage 8 Basketball Player
Teenage 10 Basketball Player
Teenage 12 Basketball Player
Teenage 7 Soccer Player
Teenage 8 Soccer Player
Teenage 9 Soccer Player
137. Then, we’ll round it off with the total sums of squares.
Once we have all of the sums of squares we can
produce an ANOVA table …
138. Then, we’ll round it off with the total sums of squares.
Once we have all of the sums of squares we can
produce an ANOVA table …
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
139. Then, we’ll round it off with the total sums of squares.
Once we have all of the sums of squares we can
produce an ANOVA table …
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
140. Then, we’ll round it off with the total sums of squares.
Once we have all of the sums of squares we can
produce an ANOVA table …
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
… that will make it possible to find the F-ratios we’ll
need to determine if we will reject or retain the null
hypothesis.
141. Then, we’ll round it off with the total sums of squares.
Once we have all of the sums of squares we can
produce an ANOVA table …
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
… that will make it possible to find the F-ratios we’ll
need to determine if we will reject or retain the null
hypothesis.
142. Then, we’ll round it off with the total sums of squares.
Once we have all of the sums of squares we can
produce an ANOVA table …
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
… that will make it possible to find the F-ratios we’ll
need to determine if we will reject or retain the null
hypothesis.
143. We begin with calculating Age Group Sums of Squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
144. We begin with calculating Age Group Sums of Squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
145. We begin with calculating Age Group Sums of Squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
Here’s how we do it:
147. We organize the data set with Age Groups in the
headers,
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
148. We organize the data set with Age Groups in the
headers, then calculate the mean for each age group
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
149. We organize the data set with Age Groups in the
headers, then calculate the mean for each age group
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean
150. We organize the data set with Age Groups in the
headers, then calculate the mean for each age group
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78
151. We organize the data set with Age Groups in the
headers, then calculate the mean for each age group
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
152. Then calculate the grand mean (which is the average of
all of the data)
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
153. Then calculate the grand mean (which is the average of
all of the data)
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean
154. Then calculate the grand mean (which is the average of
all of the data)
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39
155. Then calculate the grand mean (which is the average of
all of the data)
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
156. We subtract the grand mean from each age group
mean to get the deviation score
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
157. We subtract the grand mean from each age group
mean to get the deviation score
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score
158. We subtract the grand mean from each age group
mean to get the deviation score
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39
159. We subtract the grand mean from each age group
mean to get the deviation score
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
160. Then we square the deviations
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
161. Then we square the deviations
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev.
162. Then we square the deviations
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93
163. Then we square the deviations
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93 1.93
164. Then multiply each squared deviation by the number of persons (9). This is
called weighting the squared deviations. The more person, the heavier the
weighting, or larger the weighted squared deviation values.
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93 1.93
165. Then multiply each squared deviation by the number of persons (9). This is
called weighting the squared deviations. The more person, the heavier the
weighting, or larger the weighted squared deviation values.
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93 1.93
wt. sq. dev.
166. Then multiply each squared deviation by the number of persons (9). This is
called weighting the squared deviations. The more person, the heavier the
weighting, or larger the weighted squared deviation values.
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93 1.93
wt. sq. dev. 17.36
167. Then multiply each squared deviation by the number of persons (9). This is
called weighting the squared deviations. The more person, the heavier the
weighting, or larger the weighted squared deviation values.
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93 1.93
wt. sq. dev. 17.36 17.36
168. Finally, sum up the weighted squared deviations to get
the sums of squares for age group.
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93 1.93
wt. sq. dev. 17.36 17.36
169. Finally, sum up the weighted squared deviations to get
the sums of squares for age group.
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93 1.93
wt. sq. dev. 17.36 17.36
170. Finally, sum up the weighted squared deviations to get
the sums of squares for age group.
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
mean 12.78 10.00
grand mean 11.39 11.39
dev.score 1.39 - 1.39
sq.dev. 1.93 1.93
wt. sq. dev. 17.36 17.36 34.722
171. Note – this is the value from the ANOVA Table shown
previously:
172. Note – this is the value from the ANOVA Table shown
previously:
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
174. Next we calculate the Type of Player Sums of Squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
175. We reorder the data so that we can calculate sums of
squares for Type of Player
176. We reorder the data so that we can calculate sums of
squares for Type of Player
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
177. Calculate the mean for each Type of Player
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
178. Calculate the mean for each Type of Player
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
179. Calculate the grand mean (average of all of the scores)
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
180. Calculate the grand mean (average of all of the scores)
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
grand mean 11.4 11.4 11.4
181. Calculate the deviation between each group mean and
the grand mean(subtract grand mean from each
mean).
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
grand mean 11.4 11.4 11.4
182. Calculate the deviation between each group mean and
the grand mean(subtract grand mean from each
mean).
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
grand mean 11.4 11.4 11.4
dev.score 4.11 0.61 - 4.72
183. Square the deviations
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
grand mean 11.4 11.4 11.4
dev.score 4.11 0.61 - 4.72
184. Square the deviations
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
grand mean 11.4 11.4 11.4
dev.score 4.11 0.61 - 4.72
sq.dev. 16.9 0.4 22.3
185. Weight the squared deviations by multiplying the
squared deviations by 9
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
grand mean 11.4 11.4 11.4
dev.score 4.11 0.61 - 4.72
sq.dev. 16.9 0.4 22.3
186. Weight the squared deviations by multiplying the
squared deviations by 9
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
grand mean 11.4 11.4 11.4
dev.score 4.11 0.61 - 4.72
sq.dev. 16.9 0.4 22.3
wt. sq. dev. 101.4 2.2 133.8
187. Sum the weighted squared deviations
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
grand mean 11.4 11.4 11.4
dev.score 4.11 0.61 - 4.72
sq.dev. 16.9 0.4 22.3
wt. sq. dev. 101.4 2.2 133.8
188. Sum the weighted squared deviations
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
grand mean 11.4 11.4 11.4
dev.score 4.11 0.61 - 4.72
sq.dev. 16.9 0.4 22.3
wt. sq. dev. 101.4 2.2 133.8
189. Sum the weighted squared deviations
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
mean 15.50 12.00 6.67
grand mean 11.4 11.4 11.4
dev.score 4.11 0.61 - 4.72
sq.dev. 16.9 0.4 22.3
wt. sq. dev. 101.4 2.2 133.8 237.444
191. Here is the ANOVA table again:
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
192. Here is how we reorder the data to calculate the within
groups sums of squares
193. Here is how we reorder the data to calculate the within groups sums of squares
Type of Player Age Group Slices of Pizza Eaten
Football Player Adult 17
Football Player Adult 19
Football Player Adult 21
Football Player Teenage 11
Football Player Teenage 12
Football Player Teenage 13
Basketball Player Adult 13
Basketball Player Adult 14
Basketball Player Adult 15
Basketball Player Teenage 8
Basketball Player Teenage 10
Basketball Player Teenage 12
Soccer Player Adult 2
Soccer Player Adult 6
Soccer Player Adult 8
Soccer Player Teenage 7
Soccer Player Teenage 8
Soccer Player Teenage 9
194. Calculate the mean for each subgroup
Type of Player Age Group Slices of Pizza Eaten
Football Player Adult 17
Football Player Adult 19
Football Player Adult 21
Football Player Teenage 11
Football Player Teenage 12
Football Player Teenage 13
Basketball Player Adult 13
Basketball Player Adult 14
Basketball Player Adult 15
Basketball Player Teenage 8
Basketball Player Teenage 10
Basketball Player Teenage 12
Soccer Player Adult 2
Soccer Player Adult 6
Soccer Player Adult 8
Soccer Player Teenage 7
Soccer Player Teenage 8
Soccer Player Teenage 9
195. Calculate the mean for each subgroup
Type of Player Age Group Slices of Pizza Eaten Group Average
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 8
196. Calculate the mean for each subgroup
Type of Player Age Group Slices of Pizza Eaten Group Average
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 8
197. Calculate the mean for each subgroup
Type of Player Age Group Slices of Pizza Eaten Group Average
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 8
198. Calculate the mean for each subgroup
Type of Player Age Group Slices of Pizza Eaten Group Average
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 8
199. Calculate the mean for each subgroup
Type of Player Age Group Slices of Pizza Eaten Group Average
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 8
200. Calculate the mean for each subgroup
Type of Player Age Group Slices of Pizza Eaten Group Average
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 8
201. Calculate the mean for each subgroup
Type of Player Age Group Slices of Pizza Eaten Group Average
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 8
202. Calculate the deviations by subtracting the group average from each athlete’s pizza eaten:
Type of Player Age Group Slices of Pizza Eaten Group Average
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 8
203. Calculate the deviations by subtracting the group average from each athlete’s pizza eaten:
Type of Player Age Group Slices of Pizza Eaten Group Average
Football Player Adult 17 19
Football Player Adult 19 19
Football Player Adult 21 19
Football Player Teenage 11 12
Football Player Teenage 12 12
Football Player Teenage 13 12
Basketball Player Adult 13 14
Basketball Player Adult 14 14
Basketball Player Adult 15 14
Basketball Player Teenage 8 10
Basketball Player Teenage 10 10
Basketball Player Teenage 12 10
Soccer Player Adult 2 5
Soccer Player Adult 6 5
Soccer Player Adult 8 5
Soccer Player Teenage 7 8
Soccer Player Teenage 8 8
Soccer Player Teenage 9 8
204. Calculate the deviations by subtracting the group average from each athlete’s pizza eaten:
Type of Player Age Group Slices of Pizza Eaten Group Average Deviations
Football Player Adult 17 19 - 2.0
Football Player Adult 19 19 0
Football Player Adult 21 19 2.0
Football Player Teenage 11 12 - 1.0
Football Player Teenage 12 12 0
Football Player Teenage 13 12 1.0
Basketball Player Adult 13 14 - 1.0
Basketball Player Adult 14 14 0
Basketball Player Adult 15 14 1.0
Basketball Player Teenage 8 10 - 2.0
Basketball Player Teenage 10 10 0
Basketball Player Teenage 12 10 2.0
Soccer Player Adult 2 5 - 3.3
Soccer Player Adult 6 5 0.7
Soccer Player Adult 8 5 2.7
Soccer Player Teenage 7 8 - 1.0
Soccer Player Teenage 8 8 0
Soccer Player Teenage 9 8 1.0
205. Square the deviations
Type of Player Age Group Slices of Pizza Eaten Group Average Deviations
Football Player Adult 17 19 - 2.0
Football Player Adult 19 19 0
Football Player Adult 21 19 2.0
Football Player Teenage 11 12 - 1.0
Football Player Teenage 12 12 0
Football Player Teenage 13 12 1.0
Basketball Player Adult 13 14 - 1.0
Basketball Player Adult 14 14 0
Basketball Player Adult 15 14 1.0
Basketball Player Teenage 8 10 - 2.0
Basketball Player Teenage 10 10 0
Basketball Player Teenage 12 10 2.0
Soccer Player Adult 2 5 - 3.3
Soccer Player Adult 6 5 0.7
Soccer Player Adult 8 5 2.7
Soccer Player Teenage 7 8 - 1.0
Soccer Player Teenage 8 8 0
Soccer Player Teenage 9 8 1.0
206. Square the deviations
Type of Player Age Group Slices of Pizza Eaten Group Average Deviations Squared
Football Player Adult 17 19 - 2.0 4.0
Football Player Adult 19 19 0 0
Football Player Adult 21 19 2.0 4.0
Football Player Teenage 11 12 - 1.0 1.0
Football Player Teenage 12 12 0 0
Football Player Teenage 13 12 1.0 1.0
Basketball Player Adult 13 14 - 1.0 1.0
Basketball Player Adult 14 14 0 0
Basketball Player Adult 15 14 1.0 1.0
Basketball Player Teenage 8 10 - 2.0 4.0
Basketball Player Teenage 10 10 0 0
Basketball Player Teenage 12 10 2.0 4.0
Soccer Player Adult 2 5 - 3.3 11.1
Soccer Player Adult 6 5 0.7 0.4
Soccer Player Adult 8 5 2.7 7.1
Soccer Player Teenage 7 8 - 1.0 1.0
Soccer Player Teenage 8 8 0 0
Soccer Player Teenage 9 8 1.0 1.0
207. Sum the squared deviations
Type of Player Age Group Slices of Pizza Eaten Group Average Deviations Squared
Football Player Adult 17 19 - 2.0 4.0
Football Player Adult 19 19 0 0
Football Player Adult 21 19 2.0 4.0
Football Player Teenage 11 12 - 1.0 1.0
Football Player Teenage 12 12 0 0
Football Player Teenage 13 12 1.0 1.0
Basketball Player Adult 13 14 - 1.0 1.0
Basketball Player Adult 14 14 0 0
Basketball Player Adult 15 14 1.0 1.0
Basketball Player Teenage 8 10 - 2.0 4.0
Basketball Player Teenage 10 10 0 0
Basketball Player Teenage 12 10 2.0 4.0
Soccer Player Adult 2 5 - 3.3 11.1
Soccer Player Adult 6 5 0.7 0.4
Soccer Player Adult 8 5 2.7 7.1
Soccer Player Teenage 7 8 - 1.0 1.0
Soccer Player Teenage 8 8 0 0
Soccer Player Teenage 9 8 1.0 1.0
208. Sum the squared deviations
Type of Player Age Group Slices of Pizza Eaten Group Average Deviations Squared
Football Player Adult 17 19 - 2.0 4.0
Football Player Adult 19 19 0 0
Football Player Adult 21 19 2.0 4.0
Football Player Teenage 11 12 - 1.0 1.0
Football Player Teenage 12 12 0 0
Football Player Teenage 13 12 1.0 1.0
Basketball Player Adult 13 14 - 1.0 1.0
Basketball Player Adult 14 14 0 0
Basketball Player Adult 15 14 1.0 1.0
Basketball Player Teenage 8 10 - 2.0 4.0
Basketball Player Teenage 10 10 0 0
Basketball Player Teenage 12 10 2.0 4.0
Soccer Player Adult 2 5 - 3.3 11.1
Soccer Player Adult 6 5 0.7 0.4
Soccer Player Adult 8 5 2.7 7.1
Soccer Player Teenage 7 8 - 1.0 1.0
Soccer Player Teenage 8 8 0 0
Soccer Player Teenage 9 8 1.0 1.0
sum of squares
209. Sum the squared deviations
Type of Player Age Group Slices of Pizza Eaten Group Average Deviations Squared
Football Player Adult 17 19 - 2.0 4.0
Football Player Adult 19 19 0 0
Football Player Adult 21 19 2.0 4.0
Football Player Teenage 11 12 - 1.0 1.0
Football Player Teenage 12 12 0 0
Football Player Teenage 13 12 1.0 1.0
Basketball Player Adult 13 14 - 1.0 1.0
Basketball Player Adult 14 14 0 0
Basketball Player Adult 15 14 1.0 1.0
Basketball Player Teenage 8 10 - 2.0 4.0
Basketball Player Teenage 10 10 0 0
Basketball Player Teenage 12 10 2.0 4.0
Soccer Player Adult 2 5 - 3.3 11.1
Soccer Player Adult 6 5 0.7 0.4
Soccer Player Adult 8 5 2.7 7.1
Soccer Player Teenage 7 8 - 1.0 1.0
Soccer Player Teenage 8 8 0 0
Soccer Player Teenage 9 8 1.0 1.0
sum of squares 40.7
210. Sum the squared deviations
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
211. Here is a simple way we go about calculating sums of
squares for the interaction between type of athlete
and age group
212. Here is a simple way we go about calculating sums of
squares for the interaction between type of athlete
and age group
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
213. We simply sum up the total sums of squares and then
subtract it from the other sums of squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
214. We simply sum up the total sums of squares and then
subtract it from the other sums of squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
Total Age Type of Player Error Age * Player
– – – =
215. We simply sum up the total sums of squares and then
subtract it from the other sums of squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
Total Age Type of Player Error Age * Player
386.278 – – – =
216. We simply sum up the total sums of squares and then
subtract it from the other sums of squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
Total Age Type of Player Error Age * Player
386.278 – 34.722 – – =
217. We simply sum up the total sums of squares and then
subtract it from the other sums of squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – =
218. We simply sum up the total sums of squares and then
subtract it from the other sums of squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 =
219. We simply sum up the total sums of squares and then
subtract it from the other sums of squares
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 = 73.444
221. We line up our data in one column:
Slices of Pizza Eaten
17
19
21
13
14
15
2
6
8
11
12
13
8
10
12
7
8
9
222. Then we compute the grand mean (which the average of all of the scores) and
subtract the grand mean from each of
Slices of Pizza Eaten the scores.
17
19
21
13
14
15
2
6
8
11
12
13
8
10
12
7
8
9
223. Then we compute the grand mean (which the average of all of the scores) and
subtract the grand mean from each of
Slices of Pizza Eaten Grand Mean the scores.
17 – 11.4
19 – 11.4
21 – 11.4
13 – 11.4
14 – 11.4
15 – 11.4
2 – 11.4
6 – 11.4
8 – 11.4
11 – 11.4
12 – 11.4
13 – 11.4
8 – 11.4
10 – 11.4
12 – 11.4
7 – 11.4
8 – 11.4
9 – 11.4
224. This gives us the deviation scores between each score and the grand mean
Slices of Pizza Eaten Grand Mean
17 – 11.4
19 – 11.4
21 – 11.4
13 – 11.4
14 – 11.4
15 – 11.4
2 – 11.4
6 – 11.4
8 – 11.4
11 – 11.4
12 – 11.4
13 – 11.4
8 – 11.4
10 – 11.4
12 – 11.4
7 – 11.4
8 – 11.4
9 – 11.4
225. This gives us the deviation scores between each score and the grand mean
Slices of Pizza Eaten Grand Mean Deviations
17 – 11.4 = 5.6
19 – 11.4 = 7.6
21 – 11.4 = 9.6
13 – 11.4 = 1.6
14 – 11.4 = 2.6
15 – 11.4 = 3.6
2 – 11.4 = - 9.4
6 – 11.4 = - 5.4
8 – 11.4 = - 3.4
11 – 11.4 = - 0.4
12 – 11.4 = 0.6
13 – 11.4 = 1.6
8 – 11.4 = - 3.4
10 – 11.4 = - 1.4
12 – 11.4 = 0.6
7 – 11.4 = - 4.4
8 – 11.4 = - 3.4
9 – 11.4 = - 2.4
231. Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
232. And that’s how we calculate the total sums of squares
along with the interaction between Age Group and
Type of Player.
233. And that’s how we calculate the total sums of squares
along with the interaction between Age Group and
Type of Player.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 = 73.444
234. And that’s how we calculate the total sums of squares
along with the interaction between Age Group and
Type of Player.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 = 73.444
235. And that’s how we calculate the total sums of squares
along with the interaction between Age Group and
Type of Player.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
Total Age Type of Player Error Age * Player
386.278 – 34.722 – 237.444 – 40.667 = 73.444
236. We then determine the degrees of freedom for each
source of variance:
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
237. We then determine the degrees of freedom for each
source of variance:
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
238. We then determine the degrees of freedom for each
source of variance:
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
239. Why do we need to determine the degrees of
freedom?
240. Why do we need to determine the degrees of
freedom? Because this will make it possible to test our
three null hypotheses:
241. Why do we need to determine the degrees of
freedom? Because this will make it possible to test our
three null hypotheses:
• Main effect for Age Group: There is NO significant difference
between the amount of pizza slices eaten by adults and
teenagers in one sitting.
242. Why do we need to determine the degrees of
freedom? Because this will make it possible to test our
three null hypotheses:
• Main effect for Age Group: There is NO significant difference
between the amount of pizza slices eaten by adults and
teenagers in one sitting.
• Main effect for Type of Player: There is NO significant
difference between the amount of pizza slices eaten by
football, basketball, and soccer players in one sitting.
243. Why do we need to determine the degrees of
freedom? Because this will make it possible to test our
three null hypotheses:
• Main effect for Age Group: There is NO significant difference
between the amount of pizza slices eaten by adults and
teenagers in one sitting.
• Main effect for Type of Player: There is NO significant
difference between the amount of pizza slices eaten by
football, basketball, and soccer players in one sitting.
• Interaction effect between Age Group and Type of Athlete:
There is NO significant interaction between the amount of
pizza slices eaten by football, basketball, and soccer players
in one sitting.
244. By dividing the sums of squares by the degrees of
freedom we can compute a mean square from which
we can compute an F ratio which can be compared to
the F critical.
245. By dividing the sums of squares by the degrees of
freedom we can compute a mean square from which
we can compute an F ratio which can be compared to
the F critical.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
246. By dividing the sums of squares by the degrees of
freedom we can compute a mean square from which
we can compute an F ratio which can be compared to
the F critical.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
247. By dividing the sums of squares by the degrees of
freedom we can compute a mean square from which
we can compute an F ratio which can be compared to
the F critical.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
248. By dividing the sums of squares by the degrees of
freedom we can compute a mean square from which
we can compute an F ratio which can be compared to
the F critical.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
249. By dividing the sums of squares by the degrees of
freedom we can compute a mean square from which
we can compute an F ratio which can be compared to
the F critical.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
If the F ratio is greater than the F critical, we would reject the null hypothesis
and determine that the result is statistically significant.
250. By dividing the sums of squares by the degrees of
freedom we can compute a mean square from which
we can compute an F ratio which can be compared to
the F critical.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
If the F ratio is greater than the F critical, we would reject the null hypothesis
and determine that the result is statistically significant. If the F ratio is smaller
than the F critical then we would fail to reject the null hypothesis.
251. Most statistical packages report statistical significance.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
252. Most statistical packages report statistical significance.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
253. Most statistical packages report statistical significance.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
This means that if we
took 1000 samples we
would be wrong 1 time.
We just don’t know if
this is that time.
254. Most statistical packages report statistical significance.
But it is important to know where this value came
from.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
This means that if we
took 1000 samples we
would be wrong 1 time.
We just don’t know if
this is that time.
255. So let’s calculate the number of degrees of freedom
beginning with Age_Group.
256. So let’s calculate the number of degrees of freedom
beginning with Age_Group. When determining the
degrees of freedom for main effects, we take the
number of levels and subtract them by one.
257. So let’s calculate the number of degrees of freedom
beginning with Age_Group. When determining the
degrees of freedom for main effects, we take the
number of levels and subtract them by one. How many
levels of age are there?
258. So let’s calculate the number of degrees of freedom
beginning with Age_Group. When determining the
degrees of freedom for main effects, we take the
number of levels and subtract them by one. How many
levels of age are there?
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
259. So let’s calculate the number of degrees of freedom
beginning with Age_Group. When determining the
degrees of freedom for main effects, we take the
number of levels and subtract them by one. How many
levels of age are there?
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
260. So let’s calculate the number of degrees of freedom
beginning with Age_Group. When determining the
degrees of freedom for main effects, we take the
number of levels and subtract them by one. How many
levels of age are there?
Adults Teens
17 11
19 12
21 13
13 8
14 10
15 12
2 7
6 8
8 9
2 – 1 = 1 degree of freedom for age
261. So let’s calculate the number of degrees of freedom
beginning with Age_Group. When determining the
degrees of freedom for main effects, we take the
number of levels and subtract them by one. How many
levels of age are there?
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
263. Now we determine the degrees of freedom for Type of
Player. How many levels of Type of Player are there?
264. Now we determine the degrees of freedom for Type of
Player. How many levels of Type of Player are there?
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
265. Now we determine the degrees of freedom for Type of
Player. How many levels of Type of Player are there?
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
266. Now we determine the degrees of freedom for Type of
Player. How many levels of Type of Player are there?
Football Basketball Soccer
17 13 2
19 14 6
21 15 8
11 8 7
12 10 8
13 12 9
3 – 1 = 2 degrees of freedom for
type of player
267. Now we determine the degrees of freedom for Type of
Player. How many levels of Type of Player are there?
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
268. To determine the degrees of freedom for the
interaction effect between age and type of player you
multiply the degrees of freedom for age by the degrees
of freedom for type of player.
269. To determine the degrees of freedom for the
interaction effect between age and type of player you
multiply the degrees of freedom for age by the degrees
of freedom for type of player.
1 * 2 = 2 degrees of freedom for
interaction effect
270. To determine the degrees of freedom for the
interaction effect between age and type of player you
multiply the degrees of freedom for age by the degrees
of freedom for type of player.
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
272. We now determine the degrees of freedom for error.
Here we take the number of subjects (18) and subtract
that number by the number of subgroups (6):
273. We now determine the degrees of freedom for error.
Here we take the number of subjects (18) and subtract
that number by the number of subgroups (6):
• Adult Football Player
• Adult Basketball Player
• Adult Soccer Player
• Teenage Football Player
• Teenage Basketball Player
• Teenage Soccer Player
274. We now determine the degrees of freedom for error.
Here we take the number of subjects (18) and subtract
that number by the number of subgroups (6):
• Adult Football Player
• Adult Basketball Player
• Adult Soccer Player
• Teenage Football Player
• Teenage Basketball Player
• Teenage Soccer Player
18 – 6 = 12 degrees of freedom for error
275. We now determine the degrees of freedom for error.
Here we take the number of subjects (18) and subtract
that number by the number of subgroups (6):
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
276. To determine the total degrees of freedom we simply
add up all of the other degrees of freedom
277. To determine the total degrees of freedom we simply
add up all of the other degrees of freedom
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
278. To determine the total degrees of freedom we simply
add up all of the other degrees of freedom
Tests of Between-Subjects Effects
Dependent Variable: Pizza_Slices
Source Type III Sum of Squares df Mean Square F Sig.
Age_Group 34.722 1 34.722 10.25 0.01
Type of Player 237.444 2 118.722 35.03 0.00
Age_Group * Type of Player 73.444 2 36.722 10.84 0.00
Error 40.667 12 3.389
Total 386.278 17
280. We now calculate the mean square. The reason this
value is called mean square because it represents the
average squared deviation of scores from the mean.
281. We now calculate the mean square. The reason this
value is called mean square because it represents the
average squared deviation of scores from the mean.
You will notice that this is actually the definition for
variance.
283. So the mean square is a variance.
• The mean square for Age_Group is the variance between the two ages
(adult and teenager) and the grand mean. (This is explained variance or
variance explained by whether you are an adult or a teenager)
284. So the mean square is a variance.
• The mean square for Age_Group is the variance between the two ages
(adult and teenager) and the grand mean. (This is explained variance or
variance explained by whether you are an adult or a teenager)
• The mean square for Type of Player is the variance between the three
types of player (football, basketball, and soccer) and the grand mean.
(This is explained variance or variance explained by whether you are a
football, basketball, or soccer player)
285. So the mean square is a variance.
• The mean square for Age_Group is the variance between the two ages
(adult and teenager) and the grand mean. (This is explained variance or
variance explained by whether you are an adult or a teenager)
• The mean square for Type of Player is the variance between the three
types of player (football, basketball, and soccer) and the grand mean.
(This is explained variance or variance explained by whether you are a
football, basketball, or soccer player)
• The mean square for the interaction effect represents the variance
between each subgroup and the grand mean. (This is explained variance
or variance explained by the interaction between Age and Type of Player
effects)