What is a one-way repeated measures ANOVA?

Repeated Measures (ANOVA)
Conceptual Explanation

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

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.

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

Large Population of 30,000
Sample of 30

Large Population of 30,000
Sample of 30
Therefore, you would determine that the
problem deals with inferential not descriptive
statistics.

statistics.

statistics.
Double check your
problem to see if
that is the case

statistics.
Inferential Descriptive
Double check your
problem to see if
that is the case

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

Fit.
Double check your
problem to see if
that is the case
Difference

Fit.
Double check your
problem to see if
that is the case
Difference Relationship

Fit.
Double check your
problem to see if
that is the case
Difference Relationship Difference

Fit.
Double check your
problem to see if
that is the case
Difference Relationship Difference Goodness of Fit

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

female)
Double check your
problem to see if
that is the case
Difference Goodness of Fit
Ratio/Interval

female)
Double check your
problem to see if
that is the case
Ratio/Interval Ordinal

female)
Double check your
problem to see if
that is the case
Ratio/Interval Ordinal Nominal

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

Double check your
problem to see if
that is the case
Skewed

Double check your
problem to see if
that is the case
Skewed Kurtotic

Double check your
problem to see if
that is the case
Skewed Kurtotic Normal

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

two or more exist.
DV #1
Chemistry
Test Scores

two or more exist.
DV #1 DV #2
Chemistry
Test Scores
Class
Attendance

two or more exist.
DV #1 DV #2 DV #3
Chemistry
Test Scores
Class
Attendance
Homework
Completed

two or more exist.
Double check your
problem to see if
that is the case

two or more exist.
Descriptive
1 DV
Double check your
problem to see if
that is the case
Inferential

two or more exist.
1 DV 2+ DV
Double check your
problem to see if
that is the case

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

two or more exist.
IV #1
Use of Innovative
eBook

two or more exist.
IV #1 IV #2
Use of Innovative
eBook
Doing Homework
to Classical Music

two or more exist.
IV #1 IV #2 IV #3
Use of Innovative
eBook
Doing Homework
to Classical Music
Gender

two or more exist. Descriptive
Inferential
1 DV 2+ DV

two or more exist. Inferential Descriptive
1 DV 2+ DV
1 IV
Inferential

Nominal
1 DV 2+ DV
1 IV 2+ IV
Inferential

1 DV 2+ DV
1 IV 2+ IV
Double check your
problem to see if
that is the case
Inferential

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.

Variable (DV) rather than just two levels. Note –
Level 1
Before using the
innovative ebook

Level 1 Level 2
Before using the
innovative ebook
Using the
innovative ebook
for 2 months

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

Descriptive
Goodness of Fit
1 DV 2+ DVs
2+ IVs
Inferential
1 IV
2 levels 3+ levels
Difference

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

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

Descriptive
Goodness of Fit
1 DV 2+ DVs
2+ IVs
Inferential
1 IV
2 levels 3+ levels
Difference
Independent Repeated

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.

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

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

versus
Group 1 Group 2

versus
Group 1 Group 2
But from different times.

versus
Group 1 Group 2
But from different times.
Group 1 Group 1:
Two Months Later
versus

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

their skills.
January February
April
Exam 1
Exam 2
Exam 3

their skills.
Exam 1
Exam 2
January February
Exam 3
April
The overall F-ratio will reveal whether there are
differences somewhere among three time
periods.

their skills.
Exam 1
Exam 2
Average
Score
January February
Exam 3
April
Average
Score
Average
Score
periods.

their skills.
Exam 1
Exam 2
Average
Score
January February
Exam 3
April
Average
Score
Average
Score
There is a
difference but
we don’t
know where
periods.

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

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

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 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).

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

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
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 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
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
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.

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

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

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

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
The relationship between the scores is that we
are comparing the same person across multiple
observations.

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

common:
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

common: THESE SCORES ALL BELONG TO BEN.
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

common: THESE SCORES ALL BELONG TO BEN.
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
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.

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

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.)

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

occurred.
After running a repeated-measures ANOVA, this
is the output that we will get:

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

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:

Or accept the alternative hypothesis:
There is a significant difference in the amount of
pizza consumed by football players before,

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

Measure: Pizza slices consumed
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

Measure:
Pizza slices
consumed
Source
Type III
Sum of
Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

Measure:
Pizza slices
consumed
Source
Type III
Sum of
Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
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.

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

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.

And accept the alternative hypothesis:

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.

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.

difference exists.
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

difference exists.
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
Later, we can run what is called a “Post-hoc” test
to determine where the difference lies.

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.

How was a significance value of .008 calculated?

Let’s begin with the calculation of the various
sources of Sums of Squares

Let’s begin with the calculation of the various
sources of Sums of Squares
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

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

• Is it error?

• Is it error?
• Is it differences between times (before,
during, and after)?

• 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.

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

squares.
푆푆푡표푡푎푙 = Σ(푋푖푗 − 푋 )2

squares.
This means one pizza
eating observation for
person “I” (e.g., Ben) on
time “j” (e.g., before)

For example:
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

For example:
OR
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

For example:
OR
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
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

For example:
Season
During the
Season
ETC
After the
Season
Ben 5 4 4
Bob 7 5 5
Bud 8 7 6
Bubba 9 8 4
Burt 10 7 6

푆푆푡표푡푎푙 = Σ(푋푖푗 − 푿)2

This means the
average of all of the
observations

This means the
observations

This means the
observations
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 the
observations
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
sum or add
everything up

푆푆푡표푡푎푙 = Σ(푋푖푗 − 푿
)2
This means
sum or add
everything up
This means
the average of
all of the
observations

This means
sum or add
everything up
This means
the average of
all of the
observations

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

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

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
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Season
Before
Before
Before
Before
Before
During
During
During
During
During
After
After
After
After
After

Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Season
Before
Before
Before
Before
Before
During
During
During
During
During
After
After
After
After
After
Slices of
Pizza
5
7
8
9
10
4
5
7
8
7
4
5
6
4
6

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
Season
Before
Before
Before
Before
Before
During
During
During
During
During
After
After
After
After
After
Slices of
Pizza
5
7
8
9
10
4
5
7
8
7
4
5
6
4
6

all up.
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Season
Before
Before
Before
Before
Before
During
During
During
During
During
After
After
After
After
After
Slices of
Pizza
5
7
8
9
10
4
5
7
8
7
4
5
6
4
6 푆푆푡표푡푎푙 = Σ(푋푖푗 − 푋 )2

all up.
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Season
Before
Before
Before
Before
Before
During
During
During
During
During
After
After
After
After
After
Slices of
Pizza
5
7
8
9
10
4
5
7
Each
8
7
observation
4
5
6
4
푆푆2
6 푡표푡푎푙 = Σ(푋푖푗 − 푋 )

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
Season
Before
Before
Before
Before
Before
During
During
During
During
During
After
After
After
After
After
Slices of
Pizza
5
7
8
9
10
4
5
7
8
7
4
5
6
4
6

all up.
Grand Mean =
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Season
Before
Before
Before
Before
Before
During
During
During
During
During
After
After
After
After
After
Slices of
Pizza
5
7
8
9
10
4
5
7
8
7
4
5
6
4
6
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

all up.
Grand Mean =
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Season
Before
Before
Before
Before
Before
During
During
During
During
During
After
After
After
After
After
Slices of
Pizza
5
7
8
9
10
4
5
7
8
7
4
5
6
4
6
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

all up.
Football
Players
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Ben
Bob
Bud
Bubba
Burt
Season
Before
Before
Before
Before
Before
During
During
During
During
During
After
After
After
After
After
Slices of
Pizza
5
7
8
9
10
4
5
7
8
7
4
5
6
4
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-

We will subtract each of
these values from the
grand mean, square the
result and sum them all
up.
Bob
Bob
Before
During
7
5
-
-
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

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
Season
Before
Before
Before
Before
Before
During
During
During
During
During
After
After
After
After
After
Slices of
Pizza
5
7
8
9
10
4
5
7
8
7
4
5
6
4
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
=

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

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
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
We will subtract each of these values from the
grand mean, square the result and sum them all
up.

Football
Players
Season Slices of
Then –
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

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
Then – we place the total sums of squares result
in the ANOVA table.

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

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 Tests of Within-Subjects Effects
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

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)

want to eliminate)
• Between Groups (this would be between
BEFORE, DURING, AFTER)

want to eliminate)
• Between Groups (this would be between
BEFORE, DURING, AFTER)
• Error (the variance that we cannot explain
with our design)

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

significance.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

significance.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
Let’s calculate the sums of squares between
subjects.

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

Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

To this:
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

To this:
Source
Type III Sum
of Squares df
Mean
Square F Sig.
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 Groups 19.733 2 9.867 2.669 .078
Error 29.600 8 3.700
Total 49.333 14

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!

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 Pizza slices Slices Consumed
they eat regardless
Football
Before the
During the
After the
Average
of when they eat!
Players
Season
Season
Season
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

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:

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

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
There is no variability between subjects (they
are all 5.0).

Look at the variability between groups:

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

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
They are very different from one another.

Here is what the ANOVA table would look like:

Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 102.400 2 51.200 73.143 .000
Error 5.600 8 0.700
Total 49.333 14

Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 102.400 2 51.200 73.143 .000
Error 5.600 8 0.700
Total 49.333 14
Notice how there are no sum of squares values
for the between subjects source of variability!

Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 102.400 2 51.200 73.143 .000
Error 5.600 8 0.700
Total 49.333 14
But there is a lot of sum of squares values for
the between groups.

Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 102.400 2 51.200 73.143 .000
Error 5.600 8 0.700
Total 49.333 14
But there is a lot of sum of squares values for
the between groups.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 102.400 2 51.200 73.143 .000
Error 5.600 8 0.700
Total 49.333 14

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:

Here it is:
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

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.

When you see between SUBJECTS averages that
far away, you know that the sums of squares for
between groups will be very large.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 0.133 2 0.067 1.000 .689
Error 0.533 8 0.067
Total 148.933 14

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

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

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

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

Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 0.133 2 0.067 1.000 .689
Error 0.533 8 0.067
Total 148.933 14

Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 0.133 2 0.067 1.000 .689
Error 0.533 8 0.067
Total 148.933 14
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.

We will begin with calculating Between Subjects
sum of squares.

sum of squares.
To do so, let’s return to our original data set:

sum of squares.
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

sum of squares.
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
Here is the formula for calculating SS between
subjects.

sum of squares.
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
Here is the formula for calculating SS between
subjects.
푆푆푏푒푡푤푒푒푛 푠푢푏푗푒푐푡푠 = 푘 ∗ Σ(푋푏푠 − 푋 )2

푆푆푏푒푡푤푒푒푛 푠푢푏푗푒푐푡푠 = 푘 ∗ Σ(푿풃풔 − 푋 )2

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

This means the
average of between
Football
Players
Before the
Season
During the
Season
After the
Season
Average
subjects
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

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 -

This means the
average of all of
the observations

Here is how we calculate the grand mean again:

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

Now we subtract each subject or person average
from the Grand Mean.

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

This gives us the person’s average score
deviation from the total or grand mean.

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

Now we will square the deviations.

deviation Pizza Slices from Consumed
the total or grand mean.
Football
Before
During the
After the
Average minus Grand
Deviation Squared
Players
the
Season
Season
Mean
Now Season
we will square the deviations
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

Then we sum all of these squared deviations.

This gives Pizza Slices us Consumed
the person’s average score
deviation Football
Before
During from the
After the the
Average minus Grand
Deviation Squared
Players
the
Season
Season
total or Mean
grand mean.
Season
Now Ben we 5 will 4 square 4 4.3 the - deviations.
6.3 -2.0 3.9
Bob 7 5 5 5.7 - 6.3 -0.6 0.4
Then Bud we 8 sum 7 all 6 of these 7.0 - 6.3 squared 0.7 deviations.
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

Then we sum all of these squared deviations.
Finally, we multiply the sum all of these squared
deviations by the number of groups:

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

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
Number of
conditions
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

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
1 2 3

Football
Players
Before
the
Season
During the
Season
Source
Type III Sum
of Squares df
After the
Season
Average minus Grand
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
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
1 2 3

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
1 2 3
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

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

Now it is time to compute the between groups’
Here is the equation we will use to compute it:

Now it is time to compute the between groups’
Here is the equation we will use to compute it:
푛 ∗ Σ(푋 푘 − 푋 )

Let’s break this down with our data set:

푛 ∗ Σ(푋 푘 − 푋 )

푛 ∗ Σ(푋 푘 − 푋 )
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

We begin by computing the mean of each
condition (k)
푛 ∗ Σ(푋 푘 − 푋 )
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 (k)
푛 ∗ Σ(푋 푘 − 푋 )
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

condition (k)
푛 ∗ Σ(푋 푘 − 푋 )
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

condition (k)
푛 ∗ Σ(푋 푘 − 푋 )
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

condition (k)
푛 ∗ Σ(푋 푘 − 푋 )
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

Then subtract each condition mean from the
grand mean.
푛 ∗ Σ(푋 푘 − 푋 )
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

grand mean.
푛 ∗ Σ(푋 푘 − 푋 )
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.
푛 ∗ Σ(푋 푘 − 푋 )
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

grand mean.
푛 ∗ Σ(푋 푘 − 푋 )
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

Square the deviation.
푛 ∗ Σ(푋 푘 − 푋 )ퟐ
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 the Squared Deviations: 푛 ∗ Σ(푋 푘 − 푋 )ퟐ

Sum the Squared Deviations: 푛 ∗ Σ(푋 푘 − 푋 )ퟐ
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

Football
Players
Before the
Season
During the
Season
After the
Season
Ben 5 4 4
Sum the Squared Deviations: Bob 7 5 푛 ∗ 5
Σ(푋 푘 − 푋 )ퟐ
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
3.95
Sum of Squared
Deviations

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

3.95
Sum of Squared
Deviations

3.95
Sum of Squared
Deviations
5
Number of
observations

3.95
Sum of Squared
Deviations
5
Number of
observations
19.7
Weighted Sum of
Squared Deviations

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

Measure: Pizza_slices
Source
Type III Sum
of Squares df
Mean
Square F Sig.
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 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

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

So far we have calculated Total Sum of Squares
along with Sum of Squares along with Sum of
Squares for Between Subjects, Between Groups.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

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

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

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 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 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

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

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 Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

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).

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 Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

To calculate the significance level

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

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

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

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 Groups 19.733 2 9.867 =
9.548 .008
Error 8.267 8 1.033
Total 49.333 14

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 Groups 19.733 2 =
9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

And
Source
Type III Sum
of Squares df
Mean
Square F Sig.
9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

=
And
Source
Type III Sum
of Squares df
Mean
Square F Sig.
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 Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 =
1.033
Total 49.333 14

Type III Sum
of Squares df
9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
And
Source
Mean
Square F Sig.
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 =
1.033
Total 49.333 14
Now we need to figure out how we calculate
degrees of freedom for each source of sums of
squares.

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

Source
Type III Sum
of Squares df
Mean
Square F Sig.
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 Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
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 Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14
We take the number of subjects which, in this
case, is 5 – 1 = 4
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

Now – onto Between Groups Degrees of
Freedom (df)

Freedom (df)
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

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

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

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

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 Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

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

4 2 8 14
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Between Groups 19.733 2 9.867 9.548 .008
Error 8.267 8 1.033
Total 49.333 14

We will compute the Mean Square values for
just the Between Groups and Error. We are not
interested in what is happening with Between
Subjects. We calculated the Between Subjects
sum of squares only take out any differences
that are a function of differences that would
exist regardless of what group we were looking
at.

Once again, if we had not pulled out Between
Subjects sums of squares, then the Between
Subjects would be absorbed in the error value:

Source
Type III Sum
of Squares df
Mean
Square F Sig.
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 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 Groups 19.733 2 9.867 4.000 .047
Within Groups 29.600 8 1.033
Total 49.333 14

Source
Type III Sum
of Squares df
Mean
Square F Sig.
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 Groups 19.733 2 9.867 4.000 .047
Within Groups 29.600 8 1.033
Total 49.333 14
Within Groups is
another way of
saying Error

And that would have created a larger error
mean square value:

And that would have created a larger error
mean square value:
Source
Type III Sum
of Squares df
Mean
Square F Sig.
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 Groups 19.733 2 9.867 4.000 .047
Error 29.600 12 2.467
Total 49.333 14

Which in turn would have created a smaller F
value:

value:
Source
Type III Sum
of Squares df
Mean
Square F Sig.
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 Groups 19.733 2 9.867 4.000 .047
Error 29.600 12 2.467
Total 49.333 14

value:
Source
Type III Sum
of Squares df
Mean
Square F Sig.
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 Groups 19.733 2 9.867 =
4.000 .047
Error 29.600 12 2.467
Total 49.333 14

Which in turn would have created a larger
significance value:

Which in turn would have created a larger
significance value:
Source
Type III Sum
of Squares df
Mean
Square F Sig.
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 Groups 19.733 2 9.867 4.000 .047
Error 29.600 12 2.467
Total 49.333 14

What is a one-way repeated measures ANOVA?

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