A repeated measures ANOVA is used to test whether a single group of people change over time by comparing distributions from the same group at different time periods, rather than comparing distributions from different groups. The overall F-ratio reveals if there are differences among time periods, and post hoc tests identify exactly where the differences occurred. In contrast, a one-way ANOVA compares distributions between two or more different groups to determine if there are statistical differences between them.

1. Repeated Measures (ANOVA)
Conceptual Explanation

2. How did you get here?

3. How did you get here?
So, you have decided to use a Repeated
Measures ANOVA.

4. How did you get here?
So, you have decided to use a Repeated
Measures ANOVA.
Let’s consider the decisions you made to get
here.

5. First of all, you must have noticed the problem
to be solved deals with generalizing from a
smaller sample to a larger population.

6. First of all, you must have noticed the problem
to be solved deals with generalizing from a
smaller sample to a larger population.

7. First of all, you must have noticed the problem
to be solved deals with generalizing from a
smaller sample to a larger population.
Sample of 30

8. First of all, you must have noticed the problem
to be solved deals with generalizing from a
smaller sample to a larger population.
Sample of 30

9. First of all, you must have noticed the problem
to be solved deals with generalizing from a
smaller sample to a larger population.
Large Population of 30,000
Sample of 30

10. First of all, you must have noticed the problem
to be solved deals with generalizing from a
smaller sample to a larger population.
Therefore, you would determine that the
problem deals with inferential not descriptive
statistics.
Large Population of 30,000
Sample of 30

11. Therefore, you would determine that the
problem deals with inferential not descriptive
statistics.

12. Therefore, you would determine that the
problem deals with inferential not descriptive
statistics.
Double check your
problem to see if
that is the case

13. Therefore, you would determine that the
problem deals with inferential not descriptive
statistics.
Inferential Descriptive
Double check your
problem to see if
that is the case

14. You would have also noticed that the problem
dealt with questions of difference not
Relationships, Independence nor Goodness of
Fit. Inferential Descriptive

15. You would have also noticed that the problem
dealt with questions of difference not
Relationships, Independence nor Goodness of
Fit.
Double check your
problem to see if
that is the case
Inferential Descriptive
Difference

16. You would have also noticed that the problem
dealt with questions of difference not
Relationships, Independence nor Goodness of
Fit.
Double check your
problem to see if
that is the case
Inferential Descriptive
Difference Relationship

17. You would have also noticed that the problem
dealt with questions of difference not
Relationships, Independence nor Goodness of
Fit.
Double check your
problem to see if
that is the case
Inferential Descriptive
DifferenceDifference Relationship

18. You would have also noticed that the problem
dealt with questions of difference not
Relationships, Independence nor Goodness of
Fit.
Double check your
problem to see if
that is the case
Inferential Descriptive
Difference Goodness of FitDifference Relationship

19. After checking the data, you noticed that the
data was ratio/interval rather than extreme
ordinal (1st, 2nd, 3rd place) or nominal (male,
female)
Double check your
problem to see if
that is the case
Inferential Descriptive
Difference Goodness of FitDifference Relationship

20. After checking the data, you noticed that the
data was ratio/interval rather than extreme
ordinal (1st, 2nd, 3rd place) or nominal (male,
female)
Double check your
problem to see if
that is the case
Inferential Descriptive
Difference Goodness of Fit
Ratio/Interval
Difference Relationship

21. After checking the data, you noticed that the
data was ratio/interval rather than extreme
ordinal (1st, 2nd, 3rd place) or nominal (male,
female)
Double check your
problem to see if
that is the case
Inferential Descriptive
Difference Goodness of Fit
OrdinalRatio/Interval
Difference Relationship

22. After checking the data, you noticed that the
data was ratio/interval rather than extreme
ordinal (1st, 2nd, 3rd place) or nominal (male,
female)
Double check your
problem to see if
that is the case
Inferential Descriptive
Difference Goodness of Fit
NominalOrdinalRatio/Interval
Difference Relationship

23. The distribution was more or less normal rather
than skewed or kurtotic.

24. The distribution was more or less normal rather
than skewed or kurtotic.

25. The distribution was more or less normal rather
than skewed or kurtotic.

26. The distribution was more or less normal rather
than skewed or kurtotic.

27. The distribution was more or less normal rather
than skewed or kurtotic.
Double check your
problem to see if
that is the case
Inferential Descriptive
Difference Goodness of Fit
Skewed
NominalOrdinalRatio/Interval
Difference Relationship

28. The distribution was more or less normal rather
than skewed or kurtotic.
Double check your
problem to see if
that is the case
Inferential Descriptive
Difference Goodness of Fit
Skewed Kurtotic
NominalOrdinalRatio/Interval
Difference Relationship

29. The distribution was more or less normal rather
than skewed or kurtotic.
Double check your
problem to see if
that is the case
Inferential Descriptive
Difference Goodness of Fit
Skewed Kurtotic Normal
NominalOrdinalRatio/Interval
Difference Relationship

30. Only one Dependent Variable (DV) rather than
two or more exist.

31. Only one Dependent Variable (DV) rather than
two or more exist.
DV #1
Chemistry
Test Scores

32. Only one Dependent Variable (DV) rather than
two or more exist.
DV #1 DV #2
Chemistry
Test Scores
Class
Attendance

33. Only one Dependent Variable (DV) rather than
two or more exist.
DV #1 DV #2 DV #3
Chemistry
Test Scores
Class
Attendance
Homework
Completed

34. Only one Dependent Variable (DV) rather than
two or more exist.
Inferential Descriptive
Difference Goodness of Fit
Skewed Kurtotic Normal
Double check your
problem to see if
that is the case
NominalOrdinalRatio/Interval
Difference Relationship

35. Only one Dependent Variable (DV) rather than
two or more exist.
Descriptive
Difference Goodness of Fit
Skewed Kurtotic Normal
1 DV
Double check your
problem to see if
that is the case
Inferential
NominalOrdinalRatio/Interval
Difference Relationship

36. Only one Dependent Variable (DV) rather than
two or more exist.
Inferential Descriptive
Difference Relationship Difference Goodness of Fit
Ratio/Interval Ordinal Nominal
Skewed Kurtotic Normal
1 DV 2+ DV
Double check your
problem to see if
that is the case

37. Only one Independent Variable (DV) rather than
two or more exist.

38. Only one Independent Variable (DV) rather than
two or more exist.
IV #1
Use of Innovative
eBook

39. Only one Independent Variable (DV) rather than
two or more exist.
IV #1 IV #2
Use of Innovative
eBook
Doing Homework
to Classical Music

40. Only one Independent Variable (DV) rather than
two or more exist.
IV #1 IV #2 IV #3
Use of Innovative
eBook
Doing Homework
to Classical Music
Gender

41. Only one Independent Variable (DV) rather than
two or more exist.
IV #1 IV #2 IV #3
Use of Innovative
eBook
Doing Homework
to Classical Music
Gender

42. Only one Independent Variable (DV) rather than
two or more exist.

43. Only one Independent Variable (DV) rather than
two or more exist. Descriptive
Difference Goodness of Fit
Skewed Kurtotic Normal
1 DV 2+ DV
Inferential
NominalOrdinalRatio/Interval
Difference Relationship

44. Only one Independent Variable (DV) rather than
two or more exist. Inferential Descriptive
Difference Goodness of Fit
Skewed Kurtotic Normal
1 DV 2+ DV
1 IV
Inferential
NominalOrdinalRatio/Interval
Difference Relationship

45. Only one Independent Variable (DV) rather than
two or more exist. Descriptive
Difference Goodness of Fit
Nominal
Skewed Kurtotic Normal
1 DV 2+ DV
1 IV 2+ IV
Inferential
NominalOrdinalRatio/Interval
Difference Relationship Difference

46. Only one Independent Variable (DV) rather than
two or more exist. Descriptive
Difference Goodness of Fit
Skewed Kurtotic Normal
1 DV 2+ DV
1 IV 2+ IV
Double check your
problem to see if
that is the case
Inferential
NominalOrdinalRatio/Interval
Difference Relationship Difference

47. There are three levels of the Independent
Variable (IV) rather than just two levels. Note –
even though repeated measures ANOVA can
analyze just two levels, this is generally analyzed
using a paired sample t-test.

48. There are three levels of the Independent
Variable (DV) rather than just two levels. Note –
even though repeated measures ANOVA can
analyze just two levels, this is generally analyzed
using a paired sample t-test.
Level 1
Before using the
innovative ebook

49. There are three levels of the Independent
Variable (DV) rather than just two levels. Note –
even though repeated measures ANOVA can
analyze just two levels, this is generally analyzed
using a paired sample t-test.
Level 1 Level 2
Before using the
innovative ebook
Using the
innovative ebook
for 2 months

50. There are three levels of the Independent
Variable (DV) rather than just two levels. Note –
even though repeated measures ANOVA can
analyze just two levels, this is generally analyzed
using a paired sample t-test.
Level 1 Level 2 Level 3
Before using the
innovative ebook
Using the
innovative ebook
for 2 months
Using the
innovative ebook
for 4 months

51. Descriptive
Difference Goodness of Fit
Skewed Kurtotic Normal
1 DV 2+ DVs
2+ IVs
Inferential
NominalOrdinalRatio/Interval
Difference Relationship
2 levels 3+ levels
1 IV
Difference

52. The samples are repeated rather than
independent. Notice that the same class (Chem
100 section 003) is repeatedly tested.

53. The samples are repeated rather than
independent. Notice that the same class (Chem
100 section 003) is repeatedly tested.
Chem 100
Section 003
January
Chem 100
Section 003
March
Chem 100
Section 003
May
Before using
the innovative
ebook
Using the
innovative ebook
for 2 months
Using the
innovative ebook
for 4 months

54. Descriptive
Difference Goodness of Fit
Skewed Kurtotic Normal
1 DV 2+ DVs
2+ IVs
Inferential
NominalOrdinalRatio/Interval
Difference Relationship
2 levels 3+ levels
1 IV
Difference
RepeatedIndependent

55. If this was the appropriate path for your
problem then you have correctly selected
Repeated-measures ANOVA to solve the
problem you have been presented.

56. Repeated Measures ANOVA –

57. Repeated Measures ANOVA –
Another use of analysis of variance is to test
whether a single group of people change over
time.

58. Repeated Measures ANOVA –
Another use of analysis of variance is to test
whether a single group of people change over
time.

59. In this case, the distributions that are compared
to each other are not from different groups

60. In this case, the distributions that are compared
to each other are not from different groups
versus
Group 1 Group 2

61. In this case, the distributions that are compared
to each other are not from different groups
versus
Group 1 Group 2

62. In this case, the distributions that are compared
to each other are not from different groups
But from different times.
versus
Group 1 Group 2

63. In this case, the distributions that are compared
to each other are not from different groups
But from different times.
versus
Group 1 Group 2
Group 1 Group 1:
Two Months Later
versus

64. For example, an instructor might administer the
same test three times throughout the semester
to ascertain whether students are improving in
their skills.

65. For example, an instructor might administer the
same test three times throughout the semester
to ascertain whether students are improving in
their skills.
January February
April
Exam 1
Exam 2
Exam 3

66. For example, an instructor might administer the
same test three times throughout the semester
to ascertain whether students are improving in
their skills.
The overall F-ratio will reveal whether there are
differences somewhere among three time
periods.
January February
April
Exam 1
Exam 2
Exam 3

67. For example, an instructor might administer the
same test three times throughout the semester
to ascertain whether students are improving in
their skills.
The overall F-ratio will reveal whether there are
differences somewhere among three time
periods.
January February
April
Exam 1
Exam 2
Exam 3

68. For example, an instructor might administer the
same test three times throughout the semester
to ascertain whether students are improving in
their skills.
The overall F-ratio will reveal whether there are
differences somewhere among three time
periods.
January February
April
Exam 1
Exam 2
Exam 3
Average
Score
Average
Score
Average
Score

69. For example, an instructor might administer the
same test three times throughout the semester
to ascertain whether students are improving in
their skills.
The overall F-ratio will reveal whether there are
differences somewhere among three time
periods.
January February
April
Exam 1
Exam 2
Exam 3
Average
Score
Average
Score
Average
Score

70. For example, an instructor might administer the
same test three times throughout the semester
to ascertain whether students are improving in
their skills.
The overall F-ratio will reveal whether there are
differences somewhere among three time
periods.
January February
April
Exam 1
Exam 2
Exam 3
Average
Score
Average
Score
Average
Score
There is a
difference but
we don’t
know where

71. Post hoc tests will reveal exactly where the
differences occurred.

72. Post hoc tests will reveal exactly where the
differences occurred.
January February
April
Exam 1
Exam 2
Exam 3
Average
Score 35
Average
Score 38
Average
Score 40

73. Post hoc tests will reveal exactly where the
differences occurred.
January February
April
Exam 1
Exam 2
Exam 3
Average
Score 35
Average
Score 38
Average
Score 40
There is a
statistically
significant
difference only
between Exam 1
and Exam 3

74. In contrast, with the One-way analysis of
Variance (ANOVA) we were attempting to
determine if there was a statistical difference
between 2 or more (generally 3 or more)
groups.

75. In contrast, with the One-way analysis of
Variance (ANOVA) we were attempting to
determine if there was a statistical difference
between 2 or more (generally 3 or more)
groups.
In our One-way ANOVA example in another
presentation we attempted to determine if
there was any statistically significant difference
in the amount of Pizza Slices consumed by three
different player types (football, basketball, and
soccer).

76. The data would be set up thus:

77. The data would be set up thus:
Football
Players
Pizza
Slices
Consumed
Basketball
Players
Pizza Slices
Consumed
Soccer
Players
Pizza Slices
Consumed
Ben 5 Cam 6 Dan 5
Bob 7 Colby 4 Denzel 8
Bud 8 Conner 8 Dilbert 8
Bubba 9 Custer 4 Don 1
Burt 10 Cyan 2 Dylan 2

78. The data would be set up thus:
Notice how the individuals in these groups are
different (hence different names)
Football
Players
Pizza
Slices
Consumed
Basketball
Players
Pizza Slices
Consumed
Soccer
Players
Pizza Slices
Consumed
Ben 5 Cam 6 Dan 5
Bob 7 Colby 4 Denzel 8
Bud 8 Conner 8 Dilbert 8
Bubba 9 Custer 4 Don 1
Burt 10 Cyan 2 Dylan 2

79. The data would be set up thus:
Notice how the individuals in these groups are
different (hence different names)
Football
Players
Pizza
Slices
Consumed
Basketball
Players
Pizza Slices
Consumed
Soccer
Players
Pizza Slices
Consumed
Ben 5 Cam 6 Dan 5
Bob 7 Colby 4 Denzel 8
Bud 8 Conner 8 Dilbert 8
Bubba 9 Custer 4 Don 1
Burt 10 Cyan 2 Dylan 2

80. The data would be set up thus:
Notice how the individuals in these groups are
different (hence different names)
A Repeated Measures ANOVA is different than a
One-Way ANOVA in one simply way: Only one
group of person or observations is being
measured, but they are measured more than
one time.
Football
Players
Pizza
Slices
Consumed
Basketball
Players
Pizza Slices
Consumed
Soccer
Players
Pizza Slices
Consumed
Ben 5 Ben 6 Ben 5
Bob 7 Bob 4 Bob 8
Bud 8 Bud 8 Bud 8
Bubba 9 Bubba 4 Bubba 1
Burt 10 Burt 2 Burt 2

81. The data would be set up thus:
Notice how the individuals in these groups are
different (hence different names)
A Repeated Measures ANOVA is different than a
One-Way ANOVA in one simply way: Only one
group of persons or observations is being
measured, but they are measured more than
one time.
Football
Players
Pizza
Slices
Consumed
Basketball
Players
Pizza Slices
Consumed
Soccer
Players
Pizza Slices
Consumed
Ben 5 Ben 6 Ben 5
Bob 7 Bob 4 Bob 8
Bud 8 Bud 8 Bud 8
Bubba 9 Bubba 4 Bubba 1
Burt 10 Burt 2 Burt 2

82. Notice the different times football player pizza
consumption is being measured.
Football
Players
Pizza
Slices
Consumed
Pizza Slices
Consumed
Pizza Slices
Consumed
Ben 5 Ben 6 Ben 5
Bob 7 Bob 4 Bob 8
Bud 8 Bud 8 Bud 8
Bubba 9 Bubba 4 Bubba 1
Burt 10 Burt 2 Burt 2

83. Notice the different times football player pizza
consumption is being measured.
Football
Players
Pizza
Slices
Consumed
Before the
Season
Pizza Slices
Consumed
During the
Season
Pizza Slices
Consumed
After the
Season
Ben 5 Ben 6 Ben 5
Bob 7 Bob 4 Bob 8
Bud 8 Bud 8 Bud 8
Bubba 9 Bubba 4 Bubba 1
Burt 10 Burt 2 Burt 2

84. Since only one group is being measured 3 times,
each time is dependent on the previous time.
By dependent we mean there is a relationship.

85. Since only one group is being measured 3 times,
each time is dependent on the previous time.
By dependent we mean there is a relationship.
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

86. Since only one group is being measured 3 times,
each time is dependent on the previous time.
By dependent we mean there is a relationship.
The relationship between the scores is that we
are comparing the same person across multiple
observations.
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

87. So, Ben’s before-season and during-season and
after-season scores have one important thing in
common:

88. So, Ben’s before-season and during-season and
after-season scores have one important thing in
common:
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

89. So, Ben’s before-season and during-season and
after-season scores have one important thing in
common: THESE SCORES ALL BELONG TO BEN.
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

90. So, Ben’s before-season and during-season and
after-season scores have one important thing in
common: THESE SCORES ALL BELONG TO BEN.
They are subject to all the factors that are
special to Ben when consuming pizza, including
how much he likes or dislikes, the toppings that
are available, the eating atmosphere, etc.
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

91. What we want to find out is – how much the
BEFORE, DURING, and AFTER season pizza
consuming sessions differ.

92. What we want to find out is – how much the
BEFORE, DURING, and AFTER season pizza
consuming sessions differ.
But we have to find a way to eliminate the
variability that is caused by individual
differences that linger across all three eating
sessions. Once again we are not interested in
the things that make Ben, Ben while eating pizza
(like he’s a picky eater). We are interested in the
effect of where we are in the season (BEFORE,
DURING, and AFTER on Pizza consumption.)

93. What we want to find out is – how much the
BEFORE, DURING, and AFTER season pizza
consuming sessions differ.
But we have to find a way to eliminate the
variability that is caused by individual
differences that linger across all three eating
sessions. Once again we are not interested in
the things that make Ben, Ben while eating pizza
(like he’s a picky eater). We are interested in the
effect of where we are in the season (BEFORE,
DURING, and AFTER on Pizza consumption.)

94. What we want to find out is – how much the
BEFORE, DURING, and AFTER season pizza
consuming sessions differ.
But we have to find a way to eliminate the
variability that is caused by individual
differences that linger across all three eating
sessions. Once again we are not interested in
the things that make Ben, Ben while eating pizza
(like he’s a picky eater). We are interested in the
effect of where we are in the season (BEFORE,
DURING, and AFTER on Pizza consumption.)

95. That way we can focus just on the differences
that are related to WHEN the pizza eating
occurred.

96. That way we can focus just on the differences
that are related to WHEN the pizza eating
occurred.
After running a repeated-measures ANOVA, this
is the output that we will get:

97. That way we can focus just on the differences
that are related to WHEN the pizza eating
occurred.
After running a repeated-measures ANOVA, this
is the output that we will get:
Tests of Within-Subjects Effects
Measure: Pizza slices
Source
Type III
Sum of
Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

98. This output will help us determine if we reject
the null hypothesis:

99. This output will help us determine if we reject
the null hypothesis:
There is no significant difference in the amount
of pizza consumed by football players before,
during, and/or after the season.

100. This output will help us determine if we reject
the null hypothesis:
There is no significant difference in the amount
of pizza consumed by football players before,
during, and/or after the season.
Or accept the alternative hypothesis:

101. This output will help us determine if we reject
the null hypothesis:
There is no significant difference in the amount
of pizza consumed by football players before,
during, and/or after the season.
Or accept the alternative hypothesis:
There is a significant difference in the amount of
pizza consumed by football players before,
during, and/or after the season.

102. To do so, let’s focus on the value .008

103. To do so, let’s focus on the value .008
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

104. To do so, let’s focus on the value .008
Tests of Within-Subjects Effects
Measure:
Pizza slices
consumed
Source
Type III
Sum of
Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

105. To do so, let’s focus on the value .008
This means that if we were to reject the null
hypothesis, the probability that we would be
wrong is 8 times out of 1000. As you remember,
if that were to happen, it would be called a Type
1 error.
Tests of Within-Subjects Effects
Measure:
Pizza slices
consumed
Source
Type III
Sum of
Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

106. To do so, let’s focus on the value .008
This means that if we were to reject the null
hypothesis, the probability that we would be
wrong is 8 times out of 1000. As you remember,
if that were to happen, it would be called a Type
1 error.
Tests of Within-Subjects Effects
Measure:
Pizza slices
consumed
Source
Type III
Sum of
Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

107. But it is so unlikely, that we would be willing to
take that risk and hence reject the null
hypothesis.

108. But it is so unlikely, that we would be willing to
take that risk and hence we reject the null
hypothesis.
There IS NO statistically significant difference
between the number of slices of pizza
consumed by football players before, during, or
after the football season.

109. But it is so unlikely, that we would be willing to
take that risk and hence we reject the null
hypothesis.
There IS NO statistically significant difference
between the number of slices of pizza
consumed by football players before, during, or
after the football season.

110. And accept the alternative hypothesis:

111. And accept the alternative hypothesis:
There IS A statistically significant difference
between the number of slices of pizza
consumed by football players before, during, or
after the football season.

112. And accept the alternative hypothesis:
There IS A statistically significant difference
between the number of slices of pizza
consumed by football players before, during, or
after the football season.

113. Now we do not know which of the three are
significantly different from one another or if all
three are different. We just know that a
difference exists.

114. Now we do not know which of the three are
significantly different from one another or if all
three are different. We just know that a
difference exists.
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

115. Now we do not know which of the three are
significantly different from one another or if all
three are different. We just know that a
difference exists.
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

116. Now we do not know which of the three are
significantly different from one another or if all
three are different. We just know that a
difference exists.
Later, we can run what is called a “Post-hoc” test
to determine where the difference lies.
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

117. From this point on – we will delve into the
actual calculations and formulas that produce a
Repeated-measures ANOVA. If such detail is of
interest or a necessity to know, please continue.

118. How was a significance value of .008 calculated?

119. How was a significance value of .008 calculated?
Let’s begin with the calculation of the various
sources of Sums of Squares

120. How was a significance value of .008 calculated?
Let’s begin with the calculation of the various
sources of Sums of Squares
Tests of Within-Subjects Effects
Measure:
Pizza slices
consumed
Source
Type III
Sum of
Squares df
Mean
Square F Sig.
Between
Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

121. We do this so that we can explain what is
causing the scores to vary or deviate.

122. We do this so that we can explain what is
causing the scores to vary or deviate.
• Is it error?

123. We do this so that we can explain what is
causing the scores to vary or deviate.
• Is it error?
• Is it differences between times (before,
during, and after)?

124. We do this so that we can explain what is
causing the scores to vary or deviate.
• Is it error?
• Is it differences between times (before,
during, and after)?
Remember, the full name for sum of squares is
the sum of squared deviations about the mean.
This will help us determine the amount of
variation from each of the possible sources.

125. Let’s begin by calculating the total sums of
squares.

126. Let’s begin by calculating the total sums of
squares.
𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑋)2

127. Let’s begin by calculating the total sums of
squares.
𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑋)2

128. Let’s begin by calculating the total sums of
squares.
𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑋)2
This means one pizza
eating observation for
person “I” (e.g., Ben) on
time “j” (e.g., before)

129. For example:

130. For example:
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

131. For example:
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

132. For example:
OR
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

133. For example:
OR
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

134. For example:
OR
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

135. For example:
OR
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

136. For example:
OR
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

137. For example:
ETC
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

138. 𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑿)2

139. 𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑿)2
This means the
average of all of the
observations

140. 𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑿)2
This means the
average of all of the
observations
This means one pizza
eating observation for
person “I” (e.g., Ben) on
time “j” (e.g., before)

141. 𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑿)2
This means the
average of all of the
observations
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
This means one pizza
eating observation for
person “I” (e.g., Ben) on
time “j” (e.g., before)

142. 𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑿)2
This means the
average of all of the
observations
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Average of All
Observations
This means one pizza
eating observation for
person “I” (e.g., Ben) on
time “j” (e.g., before)

143. 𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑿)2
This means
sum or add
everything up

144. 𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑿)2
This means
sum or add
everything up
This means
the average of
all of the
observations
𝑿𝑿

145. 𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑿)2
This means
sum or add
everything up
This means
the average of
all of the
observations
This means one pizza
eating observation for
person “I” (e.g., Ben) on
time “j” (e.g., before)

146. Let’s calculate total sums of squares with this
data set:

147. Let’s calculate total sums of squares with this
data set:
Pizza Slices Consumed
Football Players Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

148. To do so we will rearrange the data like so:

149. To do so we will rearrange the data like so:
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt

150. To do so we will rearrange the data like so:
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After

151. To do so we will rearrange the data like so:
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6

152. To do so we will rearrange the data like so:
We will subtract each of these
values from the grand mean,
square the result and sum them
all up.
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6

153. To do so we will rearrange the data like so:
We will subtract each of these
values from the grand mean,
square the result and sum them
all up.
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6
𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑋)2

154. To do so we will rearrange the data like so:
We will subtract each of these
values from the grand mean,
square the result and sum them
all up.
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6
𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑋)2
Each
observation

155. To do so we will rearrange the data like so:
We will subtract each of these
values from the grand mean,
square the result and sum them
all up.
Here is how we compute the
Grand Mean =
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6

156. To do so we will rearrange the data like so:
We will subtract each of these
values from the grand mean,
square the result and sum them
all up.
Here is how we compute the
Grand Mean =
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6

157. To do so we will rearrange the data like so:
We will subtract each of these
values from the grand mean,
square the result and sum them
all up.
Here is how we compute the
Grand Mean =
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

158. To do so we will rearrange the data like so:
We will subtract each of these
values from the grand mean,
square the result and sum them
all up.
Here is how we compute the
Grand Mean =
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Average of All
Observations =
6.3

159. To do so we will rearrange the data like so:
We will subtract each of these
values from the grand mean,
square the result and sum them
all up.
𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑋)2
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6
Football
Players
Season Slices of
Pizza
Ben Before 5 -
Bob Before 7 -
Bud Before 8 -
Bubba Before 9 -
Burt Before 10 -
Ben During 4 -
Bob During 5 -
Bud During 7 -
Bubba During 8 -
Burt During 7 -
Ben After 4 -
Bob After 5 -
Bud After 6 -
Bubba After 4 -
Burt After 6 -

160. To do so we will rearrange the data like so:
We will subtract each of these
values from the grand mean,
square the result and sum them
all up.
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6
Football
Players
Season Slices of
Pizza
Ben Before 5 -
Bob Before 7 -
Bud Before 8 -
Bubba Before 9 -
Burt Before 10 -
Ben During 4 -
Bob During 5 -
Bud During 7 -
Bubba During 8 -
Burt During 7 -
Ben After 4 -
Bob After 5 -
Bud After 6 -
Bubba After 4 -
Burt After 6 -
𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑋)2

161. 𝑆𝑆𝑡𝑜𝑡𝑎𝑙 = Σ(𝑋𝑖𝑗 − 𝑋)2
To do so we will rearrange the data like so:
We will subtract each of
these values from the
grand mean, square the
result and sum them all
up.
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6
Football
Players
Season Slices of
Pizza
Ben Before 5 -
Bob Before 7 -
Bud Before 8 -
Bubba Before 9 -
Burt Before 10 -
Ben During 4 -
Bob During 5 -
Bud During 7 -
Bubba During 8 -
Burt During 7 -
Ben After 4 -
Bob After 5 -
Bud After 6 -
Bubba After 4 -
Burt After 6 -
Football
Players
Season Slices of
Pizza
Grand
Mean
Ben Before 5 - 6.3
Bob Before 7 - 6.3
Bud Before 8 - 6.3
Bubba Before 9 - 6.3
Burt Before 10 - 6.3
Ben During 4 - 6.3
Bob During 5 - 6.3
Bud During 7 - 6.3
Bubba During 8 - 6.3
Burt During 7 - 6.3
Ben After 4 - 6.3
Bob After 5 - 6.3
Bud After 6 - 6.3
Bubba After 4 - 6.3
Burt After 6 - 6.3

162. To do so we will rearrange the data like so:
We will subtract each
of these values from
the grand mean,
square the result and
sum them all up.
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6
Football
Players
Season Slices of
Pizza
Ben Before 5 -
Bob Before 7 -
Bud Before 8 -
Bubba Before 9 -
Burt Before 10 -
Ben During 4 -
Bob During 5 -
Bud During 7 -
Bubba During 8 -
Burt During 7 -
Ben After 4 -
Bob After 5 -
Bud After 6 -
Bubba After 4 -
Burt After 6 -
Football
Players
Season Slices of
Pizza
Grand
Mean
Ben Before 5 - 6.3
Bob Before 7 - 6.3
Bud Before 8 - 6.3
Bubba Before 9 - 6.3
Burt Before 10 - 6.3
Ben During 4 - 6.3
Bob During 5 - 6.3
Bud During 7 - 6.3
Bubba During 8 - 6.3
Burt During 7 - 6.3
Ben After 4 - 6.3
Bob After 5 - 6.3
Bud After 6 - 6.3
Bubba After 4 - 6.3
Burt After 6 - 6.3
Football
Players
Season Slices of
Pizza
Grand
Mean
Ben Before 5 - 6.3 =
Bob Before 7 - 6.3 =
Bud Before 8 - 6.3 =
Bubba Before 9 - 6.3 =
Burt Before 10 - 6.3 =
Ben During 4 - 6.3 =
Bob During 5 - 6.3 =
Bud During 7 - 6.3 =
Bubba During 8 - 6.3 =
Burt During 7 - 6.3 =
Ben After 4 - 6.3 =
Bob After 5 - 6.3 =
Bud After 6 - 6.3 =
Bubba After 4 - 6.3 =
Burt After 6 - 6.3 =

163. To do so we will rearrange the data like so:
We will subtract each
of these values from
the grand mean,
square the result and
sum them all up.
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Football
Players
Season
Ben Before
Bob Before
Bud Before
Bubba Before
Burt Before
Ben During
Bob During
Bud During
Bubba During
Burt During
Ben After
Bob After
Bud After
Bubba After
Burt After
Football
Players
Season Slices of
Pizza
Ben Before 5
Bob Before 7
Bud Before 8
Bubba Before 9
Burt Before 10
Ben During 4
Bob During 5
Bud During 7
Bubba During 8
Burt During 7
Ben After 4
Bob After 5
Bud After 6
Bubba After 4
Burt After 6
Football
Players
Season Slices of
Pizza
Ben Before 5 -
Bob Before 7 -
Bud Before 8 -
Bubba Before 9 -
Burt Before 10 -
Ben During 4 -
Bob During 5 -
Bud During 7 -
Bubba During 8 -
Burt During 7 -
Ben After 4 -
Bob After 5 -
Bud After 6 -
Bubba After 4 -
Burt After 6 -
Football
Players
Season Slices of
Pizza
Grand
Mean
Ben Before 5 - 6.3
Bob Before 7 - 6.3
Bud Before 8 - 6.3
Bubba Before 9 - 6.3
Burt Before 10 - 6.3
Ben During 4 - 6.3
Bob During 5 - 6.3
Bud During 7 - 6.3
Bubba During 8 - 6.3
Burt During 7 - 6.3
Ben After 4 - 6.3
Bob After 5 - 6.3
Bud After 6 - 6.3
Bubba After 4 - 6.3
Burt After 6 - 6.3
Football
Players
Season Slices of
Pizza
Grand
Mean
Ben Before 5 - 6.3 =
Bob Before 7 - 6.3 =
Bud Before 8 - 6.3 =
Bubba Before 9 - 6.3 =
Burt Before 10 - 6.3 =
Ben During 4 - 6.3 =
Bob During 5 - 6.3 =
Bud During 7 - 6.3 =
Bubba During 8 - 6.3 =
Burt During 7 - 6.3 =
Ben After 4 - 6.3 =
Bob After 5 - 6.3 =
Bud After 6 - 6.3 =
Bubba After 4 - 6.3 =
Burt After 6 - 6.3 =
Football
Players
Season Slices
of Pizza
Grand
Mean
Deviation
Ben Before 5 - 6.3 = -1.3
Bob Before 7 - 6.3 = 0.7
Bud Before 8 - 6.3 = 1.7
Bubba Before 9 - 6.3 = 2.7
Burt Before 10 - 6.3 = 3.7
Ben During 4 - 6.3 = -2.3
Bob During 5 - 6.3 = -1.3
Bud During 7 - 6.3 = 0.7
Bubba During 8 - 6.3 = 1.7
Burt During 7 - 6.3 = 0.7
Ben After 4 - 6.3 = -2.3
Bob After 5 - 6.3 = -1.3
Bud After 6 - 6.3 = -0.3
Bubba After 4 - 6.3 = -2.3
Burt After 6 - 6.3 = -0.3

164. To do so we will rearrange the data like so:
We will subtract each of these values from the
grand mean, square the result and sum them all
up.
Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1

165. Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1
To do so we will rearrange the data like so:
We will subtract each of these values from the
grand mean, square the result and sum them all
up.
Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1
= 49.3

166. Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1
To do so we will rearrange the data like so:
Then –
Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1
= 49.3

167. To do so we will rearrange the data like so:
Then – we place the total sums of squares result
in the ANOVA table.
Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1
Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1
= 49.3

168. Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1
To do so we will rearrange the data like so:
Then – we place the total sums of squares result
in the ANOVA table.
Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1
= 49.3

169. Then – we place the total sums of squares result
in the ANOVA table.
Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1
= 49.3

170. Then – we place the total sums of squares result
in the ANOVA table.
Football
Players
Season Slices of
Pizza
Grand
Mean
Deviation Squared
Ben Before 5 - 6.3 = -1.3 1.8
Bob Before 7 - 6.3 = 0.7 0.4
Bud Before 8 - 6.3 = 1.7 2.8
Bubba Before 9 - 6.3 = 2.7 7.1
Burt Before 10 - 6.3 = 3.7 13.4
Ben During 4 - 6.3 = -2.3 5.4
Bob During 5 - 6.3 = -1.3 1.8
Bud During 7 - 6.3 = 0.7 0.4
Bubba During 8 - 6.3 = 1.7 2.8
Burt During 7 - 6.3 = 0.7 0.4
Ben After 4 - 6.3 = -2.3 5.4
Bob After 5 - 6.3 = -1.3 1.8
Bud After 6 - 6.3 = -0.3 0.1
Bubba After 4 - 6.3 = -2.3 5.4
Burt After 6 - 6.3 = -0.3 0.1
= 49.3Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

171. We have now calculated the total sums of
squares. This is a good starting point. Because
now we want to know of that total sums of
squares how many sums of squares are
generated from the following sources:

172. We have now calculated the total sums of
squares. This is a good starting point. Because
now we want to know of that total sums of
squares how many sums of squares are
generated from the following sources:
• Between subjects (this is the variance we
want to eliminate)

173. We have now calculated the total sums of
squares. This is a good starting point. Because
now we want to know of that total sums of
squares how many sums of squares are
generated from the following sources:
• Between subjects (this is the variance we
want to eliminate)
• Between Groups (this would be between
BEFORE, DURING, AFTER)

174. We have now calculated the total sums of
squares. This is a good starting point. Because
now we want to know of that total sums of
squares how many sums of squares are
generated from the following sources:
• Between subjects (this is the variance we
want to eliminate)
• Between Groups (this would be between
BEFORE, DURING, AFTER)
• Error (the variance that we cannot explain
with our design)

175. With these sums of squares we will be able to
compute our F ratio value and then statistical
significance.

176. With these sums of squares we will be able to
compute our F ratio value and then statistical
significance.
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

177. With these sums of squares we will be able to
compute our F ratio value and then statistical
significance.
Let’s calculate the sums of squares between
subjects.
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

178. Remember if we were just computing a one way
ANOVA the table would go from this:

179. Remember if we were just computing a one way
ANOVA the table would go from this:
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

180. Remember if we were just computing a one way
ANOVA the table would go from this:
To this:
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

181. Remember if we were just computing a one way
ANOVA the table would go from this:
To this:
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 2.669 .078
Error 29.600 8 3.700
Total 49.333 14

182. Remember if we were just computing a one way
ANOVA the table would go from this:
To this:
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 2.669 .078
Error 29.600 8 3.700
Total 49.333 14

183. All of that variability goes into the error or
within groups sums of squares (29.600) which
makes the F statistic smaller (from 9.548 to
2.669), the significance value no longer
significant (.008 to .078).

184. All of that variability goes into the error or
within groups sums of squares (29.600) which
makes the F statistic smaller (from 9.548 to
2.669), the significance value no longer
significant (.008 to .078).
But the difference in within groups variability is
not a function of error, it is a function of Ben,
Bob, Bud, Bubba, and Burt’s being different in
terms of the amount of slices they eat regardless
of when they eat!

185. All of that variability goes into the error or
within groups sums of squares (29.600) which
makes the F statistic smaller (from 9.548 to
2.669), the significance value no longer
significant (.008 to .078).
But the difference in within groups variability is
not a function of error, it is a function of Ben,
Bob, Bud, Bubba, and Burt’s being different in
terms of the amount of slices they eat regardless
of when they eat!
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 5 4 4 4.3
Bob 7 5 5 5.7
Bud 8 7 6 7.0
Bubba 9 8 4 7.0
Burt 10 7 6 7.7

186. Here is a data set where there are not between
group differences, but there is a lot of difference
based on when the group eats their pizza:

187. Here is a data set where there are not between
group differences, but there is a lot of difference
based on when the group eats their pizza:
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 1 5 9 5.0
Bob 2 5 8 5.0
Bud 3 5 7 5.0
Bubba 1 5 9 5.0
Burt 2 5 8 5.0

188. Here is a data set where there are not between
group differences, but there is a lot of difference
based on when the group eats their pizza:
There is no variability between subjects (they
are all 5.0).
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 1 5 9 5.0
Bob 2 5 8 5.0
Bud 3 5 7 5.0
Bubba 1 5 9 5.0
Burt 2 5 8 5.0

189. Look at the variability between groups:

190. Look at the variability between groups:
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 1 5 9 5.0
Bob 2 5 8 5.0
Bud 3 5 7 5.0
Bubba 1 5 9 5.0
Burt 2 5 8 5.0
1.8 5.0 8.2

191. Look at the variability between groups:
They are very different from one another.
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 1 5 9 5.0
Bob 2 5 8 5.0
Bud 3 5 7 5.0
Bubba 1 5 9 5.0
Burt 2 5 8 5.0
1.8 5.0 8.2

192. Here is what the ANOVA table would look like:

193. Here is what the ANOVA table would look like:
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 0.000 4
Between Groups 102.400 2 51.200 73.143 .000
Error 5.600 8 0.700
Total 49.333 14

194. Here is what the ANOVA table would look like:
Notice how there are no sum of squares values
for the between subjects source of variability!
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 0.000 4
Between Groups 102.400 2 51.200 73.143 .000
Error 5.600 8 0.700
Total 49.333 14

195. Here is what the ANOVA table would look like:
Notice how there are no sum of squares values
for the between subjects source of variability!
But there is a lot of sum of squares values for
the between groups.
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 0.000 4
Between Groups 102.400 2 51.200 73.143 .000
Error 5.600 8 0.700
Total 49.333 14

196. Here is what the ANOVA table would look like:
Notice how there are no sum of squares values
for the between subjects source of variability!
But there is a lot of sum of squares values for
the between groups.
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 0.000 4
Between Groups 102.400 2 51.200 73.143 .000
Error 5.600 8 0.700
Total 49.333 14
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 0.000 4
Between Groups 102.400 2 51.200 73.143 .000
Error 5.600 8 0.700
Total 49.333 14

197. What would the data set look like if there was
very little between groups (by season) variability
and a great deal of between subjects variability:

198. What would the data set look like if there was
very little between groups (by season) variability
and a great deal of between subjects variability:
Here it is:

199. What would the data set look like if there was
very little between groups (by season) variability
and a great deal of between subjects variability:
Here it is:
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 3 3 3 3.0
Bob 5 5 5 5.0
Bud 7 7 7 7.0
Bubba 8 8 8 8.0
Burt 12 12 13 12.3
Between
Subjects

200. In this case the between subjects (Ben, Bob, Bud
. . .), are very different.

201. In this case the between subjects (Ben, Bob, Bud
. . .), are very different.
When you see between SUBJECTS averages that
far away, you know that the sums of squares for
between groups will be very large.

202. In this case the between subjects (Ben, Bob, Bud
. . .), are very different.
When you see between SUBJECTS averages that
far away, you know that the sums of squares for
between groups will be very large.
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 148.267 4
Between Groups 0.133 2 0.067 1.000 .689
Error 0.533 8 0.067
Total 148.933 14

203. Notice, in contrast, as we compute the between
group (seasons) average how close they are.

204. Notice, in contrast, as we compute the between
group (seasons) average how close they are.
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 3 3 3 3.0
Bob 5 5 5 5.0
Bud 7 7 7 7.0
Bubba 8 8 8 8.0
Burt 12 12 13 12.3
7.0 7.0 7.2

205. Notice, in contrast, as we compute the between
group (seasons) average how close they are.
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 3 3 3 3.0
Bob 5 5 5 5.0
Bud 7 7 7 7.0
Bubba 8 8 8 8.0
Burt 12 12 13 12.3
7.0 7.0 7.2
Between
Groups

206. Notice, in contrast, as we compute the between
group (seasons) average how close they are.
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 3 3 3 3.0
Bob 5 5 5 5.0
Bud 7 7 7 7.0
Bubba 8 8 8 8.0
Burt 12 12 13 12.3
7.0 7.0 7.2
Between
Groups

207. When you see between group averages this
close you know that the sums of squares for
between groups will be very small.

208. When you see between group averages this
close you know that the sums of squares for
between groups will be very small.
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 148.267 4
Between Groups 0.133 2 0.067 1.000 .689
Error 0.533 8 0.067
Total 148.933 14

209. When you see between group averages this
close you know that the sums of squares for
between groups will be very small.
Now that we have conceptually considered the
sources of variability as described by the sum of
squares, let’s begin calculating between
subjects, between groups, and the error
sources.
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 148.267 4
Between Groups 0.133 2 0.067 1.000 .689
Error 0.533 8 0.067
Total 148.933 14

210. We will begin with calculating Between Subjects
sum of squares.

211. We will begin with calculating Between Subjects
sum of squares.
To do so, let’s return to our original data set:

212. We will begin with calculating Between Subjects
sum of squares.
To do so, let’s return to our original data set:
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

213. We will begin with calculating Between Subjects
sum of squares.
To do so, let’s return to our original data set:
Here is the formula for calculating SS between
subjects.
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

214. We will begin with calculating Between Subjects
sum of squares.
To do so, let’s return to our original data set:
Here is the formula for calculating SS between
subjects.
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑋 𝑏𝑠 − 𝑋)2

215. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2

216. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 5 4 4 4.3
Bob 7 5 5 5.7
Bud 8 7 6 7.0
Bubba 9 8 4 7.0
Burt 10 7 6 7.7

217. Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 5 4 4 4.3
Bob 7 5 5 5.7
Bud 8 7 6 7.0
Bubba 9 8 4 7.0
Burt 10 7 6 7.7
𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
This means the
average of between
subjects

218. Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average minus
Ben 5 4 4 4.3 -
Bob 7 5 5 5.7 -
Bud 8 7 6 7.0 -
Bubba 9 8 4 7.0 -
Burt 10 7 6 7.7 -
𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2

219. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
This means the
average of all of
the observations

220. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:

221. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Average of All
Observations =
6.3

222. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.

223. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Ben 5 4 4 4.3 - 6.3
Bob 7 5 5 5.7 - 6.3
Bud 8 7 6 7.0 - 6.3
Bubba 9 8 4 7.0 - 6.3
Burt 10 7 6 7.7 - 6.3
This means the
average of all of
the observations

224. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.
This gives us the person’s average score
deviation from the total or grand mean.

225. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.
This gives us the person’s average score
deviation from the total or grand mean.Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation
Ben 5 4 4 4.3 - 6.3 -2.0
Bob 7 5 5 5.7 - 6.3 -0.6
Bud 8 7 6 7.0 - 6.3 0.7
Bubba 9 8 4 7.0 - 6.3 0.7
Burt 10 7 6 7.7 - 6.3 1.4

226. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.
This gives us the person’s average score
deviation from the total or grand mean.
Now we will square the deviations.

227. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.
This gives us the person’s average score
deviation from the total or grand mean.
Now we will square the deviations.

228. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.
This gives us the person’s average score
deviation from the total or grand mean.
Now we will square the deviations
Pizza Slices Consumed
Football
Players
Before
the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation Squared
Ben 5 4 4 4.3 - 6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Bud 8 7 6 7.0 - 6.3 0.7 0.5
Bubba 9 8 4 7.0 - 6.3 0.7 0.5
Burt 10 7 6 7.7 - 6.3 1.4 1.9

229. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.
This gives us the person’s average score
deviation from the total or grand mean.
Now we will square the deviations.
Then we sum all of these squared deviations.

230. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.
This gives us the person’s average score
deviation from the total or grand mean.
Now we will square the deviations.
Then we sum all of these squared deviations.

231. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.
This gives us the person’s average score
deviation from the total or grand mean.
Now we will square the deviations.
Then we sum all of these squared deviations.
Pizza Slices Consumed
Football
Players
Before
the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation Squared
Ben 5 4 4 4.3 - 6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Bud 8 7 6 7.0 - 6.3 0.7 0.5
Bubba 9 8 4 7.0 - 6.3 0.7 0.5
Burt 10 7 6 7.7 - 6.3 1.4 1.9
7.1
Sum
up

232. Here is how we calculate the grand mean again:
Now we subtract each subject or person average
from the Grand Mean.
This gives us the person’s average score
deviation from the total or grand mean.
Now we will square the deviations.
Then we sum all of these squared deviations.
Finally, we multiply the sum all of these squared
deviations by the number of groups:

233. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Pizza Slices Consumed
Football
Players
Before
the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation Squared
Ben 5 4 4 4.3 - 6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Bud 8 7 6 7.0 - 6.3 0.7 0.5
Bubba 9 8 4 7.0 - 6.3 0.7 0.5
Burt 10 7 6 7.7 - 6.3 1.4 1.9
7.1
Times 3 groups
Sum of Squares Between Subjects 21.3

234. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Pizza Slices Consumed
Football
Players
Before
the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation Squared
Ben 5 4 4 4.3 - 6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Bud 8 7 6 7.0 - 6.3 0.7 0.5
Bubba 9 8 4 7.0 - 6.3 0.7 0.5
Burt 10 7 6 7.7 - 6.3 1.4 1.9
7.1
Times 3 groups
Sum of Squares Between Subjects 21.3
Number of
conditions

235. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Pizza Slices Consumed
Football
Players
Before
the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation Squared
Ben 5 4 4 4.3 - 6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Bud 8 7 6 7.0 - 6.3 0.7 0.5
Bubba 9 8 4 7.0 - 6.3 0.7 0.5
Burt 10 7 6 7.7 - 6.3 1.4 1.9
7.1
Times 3 groups
Sum of Squares Between Subjects 21.3

236. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Pizza Slices Consumed
Football
Players
Before
the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation Squared
Ben 5 4 4 4.3 - 6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Bud 8 7 6 7.0 - 6.3 0.7 0.5
Bubba 9 8 4 7.0 - 6.3 0.7 0.5
Burt 10 7 6 7.7 - 6.3 1.4 1.9
7.1
Times 3 groups
Sum of Squares Between Subjects 21.3

237. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Pizza Slices Consumed
Football
Players
Before
the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation Squared
Ben 5 4 4 4.3 - 6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Bud 8 7 6 7.0 - 6.3 0.7 0.5
Bubba 9 8 4 7.0 - 6.3 0.7 0.5
Burt 10 7 6 7.7 - 6.3 1.4 1.9
7.1
Times 3 groups
Sum of Squares Between Subjects 21.3
1 2 3

238. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Pizza Slices Consumed
Football
Players
Before
the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation Squared
Ben 5 4 4 4.3 - 6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Bud 8 7 6 7.0 - 6.3 0.7 0.5
Bubba 9 8 4 7.0 - 6.3 0.7 0.5
Burt 10 7 6 7.7 - 6.3 1.4 1.9
7.1
Times 3 groups
Sum of Squares Between Subjects 21.3
1 2 3

239. Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Pizza Slices Consumed
Football
Players
Before
the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation Squared
Ben 5 4 4 4.3 - 6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Bud 8 7 6 7.0 - 6.3 0.7 0.5
Bubba 9 8 4 7.0 - 6.3 0.7 0.5
Burt 10 7 6 7.7 - 6.3 1.4 1.9
7.1
Times 3 groups
Sum of Squares Between Subjects 21.3
1 2 3

240. 𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑠 = 𝑘 ∗ Σ(𝑿 𝒃𝒔 − 𝑋)2
Pizza Slices Consumed
Football
Players
Before
the
Season
During the
Season
After the
Season
Average minus Grand
Mean
Deviation Squared
Ben 5 4 4 4.3 - 6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Bud 8 7 6 7.0 - 6.3 0.7 0.5
Bubba 9 8 4 7.0 - 6.3 0.7 0.5
Burt 10 7 6 7.7 - 6.3 1.4 1.9
7.1
Times 3 groups
Sum of Squares Between Subjects 21.3
1 2 3
Tests of Within-Subjects Effects
Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

241. Now it is time to compute the between groups
(seasons) sum of squares.

242. Now it is time to compute the between groups’
(seasons) sum of squares.
Here is the equation we will use to compute it:

243. Now it is time to compute the between groups’
(seasons) sum of squares.
Here is the equation we will use to compute it:
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)

244. Let’s break this down with our data set:

245. Let’s break this down with our data set:
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)

246. Let’s break this down with our data set:
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

247. We begin by computing the mean of each
condition (k)
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

248. Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
We begin by computing the mean of each
condition (k)
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)

249. Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8
We begin by computing the mean of each
condition (k)
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)

250. Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8
We begin by computing the mean of each
condition (k)
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2

251. Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8
We begin by computing the mean of each
condition (k)
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2 5.0

252. Then subtract each condition mean from the
grand mean.
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2 5.0

253. Then subtract each condition mean from the
grand mean.
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2 5.0
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2 5.0
minus - - -

254. Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2 5.0
minus - - -
Grand
Mean
6.3 6.3 6.3
Then subtract each condition mean from the
grand mean.
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)

255. Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2 5.0
minus - - -
Grand
Mean
6.3 6.3 6.3
equals
Deviation 1.5 -0.1 -1.3
Then subtract each condition mean from the
grand mean.
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋)

256. Square the deviation.
𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋) 𝟐
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2 5.0
minus - - -
Grand
Mean
6.3 6.3 6.3
equals
Deviation 1.5 -0.1 -1.3
Squared
Deviation
2.2 0.0 1.8

257. Sum the Squared Deviations:

258. Sum the Squared Deviations: 𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋) 𝟐

259. Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2 5.0
minus - - -
Grand
Mean
6.3 6.3 6.3
equals
Deviation 1.5 -0.1 -1.3
Squared
Deviation
2.2 0.0 1.8
Sum
Sum the Squared Deviations: 𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋) 𝟐

260. Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
Condition
Mean
7.8 6.2 5.0
minus - - -
Grand
Mean
6.3 6.3 6.3
equals
Deviation 1.5 -0.1 -1.3
Squared
Deviation
2.2 0.0 1.8
Sum
Sum the Squared Deviations: 𝑛 ∗ Σ( 𝑋 𝑘 − 𝑋) 𝟐
3.95
Sum of Squared
Deviations

261. Multiply by the number of observations per
condition (number of pizza eating slices across
before, during, and after).

262. Multiply by the number of observations per
condition (number of pizza eating slices across
before, during, and after).
3.95
Sum of Squared
Deviations

263. Multiply by the number of observations per
condition (number of pizza eating slices across
before, during, and after).
3.95
Sum of Squared
Deviations

264. Multiply by the number of observations per
condition (number of pizza eating slices across
before, during, and after).
3.95
Sum of Squared
Deviations
5
Number of
observations

265. Multiply by the number of observations per
condition (number of pizza eating slices across
before, during, and after).
3.95
Sum of Squared
Deviations
5
Number of
observations

266. Multiply by the number of observations per
condition (number of pizza eating slices across
before, during, and after).
3.95
Sum of Squared
Deviations
5
Number of
observations
19.7
Weighted Sum of
Squared Deviations

267. Let’s return to the ANOVA table and put the
weighted sum of squared deviations.

268. Let’s return to the ANOVA table and put the
weighted sum of squared deviations.
Tests of Within-Subjects Effects
Measure: Pizza_slices
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

269. Let’s return to the ANOVA table and put the
weighted sum of squared deviations.
Tests of Within-Subjects Effects
Measure: Pizza_slices
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
3.95
Sum of Squared
Deviations
5
Number of
observations
19.7
Weighted Sum of
Squared Deviations

270. Let’s return to the ANOVA table and put the
weighted sum of squared deviations.
Tests of Within-Subjects Effects
Measure: Pizza_slices
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
3.95
Sum of Squared
Deviations
5
Number of
observations
19.7
Weighted Sum of
Squared Deviations

271. So far we have calculated Total Sum of Squares
along with Sum of Squares for Between
Subjects, and Between Groups.

272. So far we have calculated Total Sum of Squares
along with Sum of Squares along with Sum of
Squares for Between Subjects, Between Groups.
Tests of Within-Subjects Effects
Measure: Pizza_slices
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

273. Now we will calculate the sum of squares
associated with Error.

274. Now we will calculate the sum of squares
associated with Error.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

275. To do this we simply add the between subjects
and between groups sums of squares.

276. To do this we simply add the between subjects
and between groups sums of squares.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

277. To do this we simply add the between subjects
and between groups sums of squares.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
21.333
Between Subjects
Sum of Squares
19.733
Between Groups
Sum of Squares
41.600
Between Subjects &
Groups Sum of
Squares Combined

278. Then we subtract the Between Subjects & Group
Sum of Squares Combined (41.600) from the
Total Sum of Squares (49.333)

279. Then we subtract the Between Subjects & Group
Sum of Squares Combined (41.600) from the
Total Sum of Squares (49.333)
49.333
Total Sum of Squares
41.600
Between Subjects &
Groups Sum of Squares
Combined
8.267
Sum of Squares
Attributed to Error
or Unexplained

280. Then we subtract the Between Subjects & Group
Sum of Squares Combined (41.600) from the
Total Sum of Squares (49.333)
49.333
Total Sum of Squares
41.600
Between Subjects &
Groups Sum of Squares
Combined
8.267
Sum of Squares
Attributed to Error
or Unexplained
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

281. Now we have all of the information necessary to
determine if there is a statistically significant
difference between pizza slices consumed by
football players between three different eating
occasions (before, during or after the season).

282. Now we have all of the information necessary to
determine if there is a statistically significant
difference between pizza slices consumed by
football players between three different eating
occasions (before, during or after the season).
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

283. To calculate the significance level

284. To calculate the significance level
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

285. We must calculate the F ratio

286. We must calculate the F ratio
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

287. Which is calculated by dividing the Between
Groups Mean Square value (9.867) by the Error
Mean Square value (1.033).

288. Which is calculated by dividing the Between
Groups Mean Square value (9.867) by the Error
Mean Square value (1.033).
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
=

289. Which is calculated by dividing the sum of
squares between groups by its degrees of
freedom, as shown below:

290. Which is calculated by dividing the sum of
squares between groups by its degrees of
freedom, as shown below:
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
=

291. Which is calculated by dividing the sum of
squares between groups by its degrees of
freedom, as shown below:
And
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
=

292. Which is calculated by dividing the sum of
squares between groups by its degrees of
freedom, as shown below:
And
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
=
=

293. Which is calculated by dividing the sum of
squares between groups by its degrees of
freedom, as shown below:
And
Now we need to figure out how we calculate
degrees of freedom for each source of sums of
squares.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
=
=

294. Let’s begin with determining the degrees of
freedom Between Subjects.

295. Let’s begin with determining the degrees of
freedom Between Subjects.

296. Let’s begin with determining the degrees of
freedom Between Subjects.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

297. Let’s begin with determining the degrees of
freedom Between Subjects.
We take the number of subjects which, in this
case, is 5 – 1 = 4
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

298. Let’s begin with determining the degrees of
freedom Between Subjects.
We take the number of subjects which, in this
case, is 5 – 1 = 4
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

299. Let’s begin with determining the degrees of
freedom Between Subjects.
We take the number of subjects which, in this
case, is 5 – 1 = 4
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Average
Ben 3 3 3 3.0
Bob 5 5 5 5.0
Bud 7 7 7 7.0
Bubba 8 8 8 8.0
Burt 12 12 13 12.3
Between
Subjects
1
2
3
4
5

300. Now – onto Between Groups Degrees of
Freedom (df)

301. Now – onto Between Groups Degrees of
Freedom (df)
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

302. Now – onto Between Groups Degrees of
Freedom (df)
We take the number of groups which in this case
is 3 – 1 = 2
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

303. Now – onto Between Groups Degrees of
Freedom (df)
We take the number of groups which in this case
is 3 – 1 = 2
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

304. Now – onto Between Groups Degrees of
Freedom (df)
We take the number of groups which in this case
is 3 – 1 = 2
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
1 2 3

305. Now – onto Between Groups Degrees of
Freedom (df)
We take the number of groups which in this case
is 3 – 1 = 2
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
Pizza Slices Consumed
Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6
1 2 3

306. The error degrees of freedom are calculated by
multiplying the between subjects by the
between groups degrees of freedom.

307. The error degrees of freedom are calculated by
multiplying the between subjects by the
between groups degrees of freedom.
4
Between Subjects
Degrees of Freedom
2
Between Groups
Degrees of Freedom
8
Error Degrees of
Freedom

308. The error degrees of freedom are calculated by
multiplying the between subjects by the
between groups degrees of freedom.
4
Between Subjects
Degrees of Freedom
2
Between Groups
Degrees of Freedom
8
Error Degrees of
Freedom
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

309. The error degrees of freedom are calculated by
multiplying the between subjects by the
between groups degrees of freedom.
4
Between Subjects
Degrees of Freedom
2
Between Groups
Degrees of Freedom
8
Error Degrees of
Freedom
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

310. The degrees of freedom for total sum of squares
is calculated by adding all of the degrees of
freedom from the other three sources.

311. The degrees of freedom for total sum of squares
is calculated by adding all of the degrees of
freedom from the other three sources.
4 2 8 14

312. The degrees of freedom for total sum of squares
is calculated by adding all of the degrees of
freedom from the other three sources.
4 2 8 14
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Subjects 21.333 4
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14