What is a one way anova?

We will now venture into the world of One-way
Analysis of Variance.

How did we get here?
We will begin with a real-world problem to consider
how we got to a One-Way ANOVA.

We want to determine if there is a statistically
significant difference between English
comprehension test scores

comprehension test scores of students who
have an English speaking instructor

have an English speaking instructor who is a
native German,

native German, French,

native German, French, or Dutch Speaker.

First, we have to determine if the problem we are
solving poses an inferential or descriptive question.

Problem

Problem
Inferential Descriptive

Problem
We want to determine if
there is a statistically
significant difference
between English
comprehension test scores
of students who have an
English speaking instructor
who is a native German,
French or Dutch Speaker.

Second, we determine if the inferential problem poses
a difference, relationship, independence or goodness
of fit question.

of fit question.
Problem
Difference Relationship Goodness
of Fit
Independence

of fit question.
Problem
Difference Relationship Goodness
of Fit
Independence
We want to determine if there is a
statistically significant difference
between English comprehension
test scores of students who have an
English speaking instructor who is a
native German, French or Dutch
Speaker.

Third, we determine if we will use parametric or
nonparametric analytical methods by determining if
the distributions are normal, skewed or kurtotic.

Problem
Difference Relationship
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence

Let’s say we find that the
distribution is NORMAL
Problem
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence

Problem
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence
distribution is NORMAL

Fourth, we determine if we will use parametric or
the data are ratio/interval, ordinal, or nominal.

Problem
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence
Data: Ratio/Interval Data: Ordinal Data: Nominal

Problem
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence
data are Ratio/Interval

Fifth, we determine the number of dependent
variables.

variables.
Problem
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence
1 Dependent Variable 2 or more Dependent Variables

variables.
Problem
Speaker.
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence

Sixth, we determine the number of independent
variables.

variables.
Problem
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence

variables.
Problem
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence
1 Independent Variable 2 or more Independent Variables

variables.
Problem
Speaker.
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence

Seventh, we determine the number of levels of the
independent variable.

Problem
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence
2 levels 3 or more levels

Problem
Speaker.
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence

Problem
Speaker.
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence
3

Problem
Distributions
Normal
Distributions
Skewed or Kurtotic
Goodness
of Fit
Independence
These elements point to a
one-way ANOVA as an
appropriate method to
answer this question.
Speaker.

Now let’s consider the concepts that support the use of
a One-Way Analysis of Variance

Once we have made the leap from simple means-differences
embedded in t-tests to the logic of sums of
squares, a wide range of procedures opens up to us.

mean 1 mean 2
Simple Mean Difference

mean 1 mean 2 mean 3
Sums of Squares Logic
mean 1 mean 2

mean 1 mean 2
The possibilities are endless!

mean 1 mean 2
The possibilities are endless!
(well, almost … )

As reminder, t-tests can compare means only between
two levels of one independent variable.

First example:
One Independent variable: Athlete
Two levels: Soccer player and Basketball player

First example:
Second example:
One Independent variable: Religious Affiliation
Two levels: Catholic and Protestant

First example:
Second example:
Third example:
One Independent variable: Age
Two levels: 19-45 yrs old and 31-64 yrs old

Fourth example:
One Independent variable: Test takers in a stats class
Two levels: Pre-test and posttest

Fourth example:
Fifth example:
One Independent variable: People from Texas
Two levels: Population of Texans and a sample of Texans who
ate raw fish

… across one dependent variable.

First example:

First example:
One Dependent variable: degree of flexibility

First example:
Second example:

First example:
Second example:
One Dependent variable: level of education

First example:
Second example:
Third example:

First example:
Second example:
Third example:
One Dependent variable: level of comfort with technology

Fourth example:
One Dependent variable: scores on a basic statistics test

Fourth example:
Fifth example:
ate raw fish

Fourth example:
Fifth example:
ate raw fish
One Dependent variable: IQ scores

With a one way analysis of variance, on the other hand,
we are generally comparing more than two levels of an

First example:
Two levels: Soccer players and Basketball players

First example:
Three levels: Soccer players and Basketball players and Football players

First example:
Three levels: Soccer players and Basketball players and Football Players
Second example:

First example:
Second example:
Four levels: Catholic and Protestant and Mormon and Muslim

First example:
Second example:
Third example:

First example:
Second example:
Third example:
Three levels: 19-30 yrs old and 31-64 yrs old and 65-79 yrs old

Fourth example:
Three levels: Pre-test and midterm test and posttest

Fourth example:
Fifth example:
Two levels: Population of Texans and a sample of Texans who ate raw fish

Fourth example:
Fifth example:
Three levels: Population of Texans and a sample of Texans who ate raw
fish and a sample of Texans who ate cooked fish

While there is a whole family of procedures under the
umbrella of Analysis of Variance we will begin with the
most basic: One-Way Analysis of Variance.

First, let’s remind ourselves what variance is before we
do an “analysis” of it.

First, let’s remind ourselves what variance is before we
do an “analysis” of it.
Technically, the variance is the average squared
distance of the scores around the mean.

Variance
Conceptually, the variance represents the average
squared deviations of scores from the mean.

Variance
So, what does that mean?

Variance
Here is a visual depiction of the variance:

Variance
1 2 3 4 5 6 7 8 9
5
4
3
2
1

Variance
1 2 3 4 5 6 7 8 9
5
4
3
2
1
So, the mean of this
distribution is 5

Variance
And this observation
is 0 units from the
is 1 unit from the
is -1 units from the
is 0 units from the
And this ombseearvna. tion
is 2 units from the
is 0 units from the
is 1 unit from the
And this obmseeravna.tion
is 0 units from the
And this ombseearnva. mean.
tion
is 2 units from the
is 1 unit from the
mean.
mean.
mean.
1 2 3 4 5 6 7 8 9
5
4
3
2
1
So we subtract this
observation “9”
from the mean “5”
and we get +4
This means that this
is 3 units And from this
the
mean.
mean.
observation is
mean.
9
is 4
units from the
mean.
mean.
mean.
mean.
mean.
mean.
mean.
mean.

Variance
So to avoid this from happening, we square each deviation
(-42 = 16, -32 = 9, -22 = 4, -12 = 1, 12 = 1, 22 = 4, 32 = 9, 42 = 16).
You might think that the variance is
just the average of all of the
deviations (the average distance between
each score from the mean), right?
Then we add up all of the squared deviations and divide that
number by the number of observations
1 2
2 units above the
mean squared = 4
1 2 3 4 5 6 7 8 9
5
4
3
2
1
But, because half of the
deviations are positive and
the other half are negative,
if you take the average it
will come to zero
1 2 3 4
1
4 3 2 1
4 units above the mean
squared = 16
1 unit below the
mean squared = 1
4 units above the mean
squared = 16
Now we sum all of the squared deviations
16 + 9 + 4 + 4 + 1 + 1 + 1 + 0 + 0 + 0 + 0 + 1+ 1 + 1 + 4 + 4 + 9 + 16 =
116
Remember – these
values represent the
distance or deviation
from the mean,
squared.

The “one way” in one way analysis of variance indicates
that we have only one dependent variable

that we have only one dependent variable
Dependent Variable: amount of pizza eaten

that we have only one dependent variable and only one
independent variable

independent variable
Independent Variable: athletes

independent variable with at least two levels.

Level one: football player

Level two: basketball player

Note that an independent sample t-test could be run as
well. The results will be the same, but one-way ANOVA
can also analyze at least three levels.

Level one: football player Level two: basketball player

Level one: football player Level two: basketball player Level three: soccer player

We begin with a question: Who eats more slices of
pizza in one sitting, football players, basketball players
or soccer players?

We begin with a question: Who eats more slices of
pizza in one sitting, football players, basketball players
or soccer players? How would you convert this question
into a null hypothesis?

Null hypothesis:
There is no statistically significant difference in the
quantity of pizza consumed in one sitting among
football, basketball, and soccer players.

Null hypothesis:
To test whether the null hypothesis will be retained or
rejected we compare a value called the F statistic or F
ratio with what we call the F critical value.

We will compute the F ratio from the following data
set:

set:
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices

set:
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
And then check to see if it is larger than the F-critical. If
it is, then we will reject our null hypothesis:

Otherwise, we will retain the null hypothesis:

So, let’s say after doing our Analysis of Variance
calculation we have an F ratio or F value of 22.17 (nice
round number, right? ).

We would then compare it with the F critical value.

We would then compare it with the F critical value. If
the F ratio is larger than the F critical value then we
would consider it to be a rare occurrence and reject the
null hypothesis.

null hypothesis.
So, let’s see if it is

null hypothesis.
So, let’s see if it is by looking at what is called the Table
of Probabilities for the F Distribution.

null hypothesis.
So, let’s see if it is by looking at what is called the Table
of Probabilities for the F Distribution. This table will
help us locate the F critical for our data set.

Let’s say that the F critical value turns out to be 5.14
(we’ll show you how we got that value later).

So, with an F critical of 5.14 and an F ratio of 22.17, (we
will show you how to calculate this F ratio from the
data set below),

data set below),
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices

data set below),
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
we would have to say that that is a rare occurrence at
the .05 alpha level and we would reject the null
hypothesis.

Obviously, if the critical F value had been, say, 9.55,

Obviously, if the critical F value had been, say, 9.55, we
would have to say that that is a not a rare occurrence at
the .05 alpha level and we would retain the null
hypothesis.

Obviously, if the critical F value had been, say, 9.55, we
would have to say that that is a not a rare occurrence at
the .05 alpha level and we would retain the null
hypothesis.
So how did we calculate the 22.17 F ratio in the first
place? And how was the F critical determined so we
could reject or retain our null-hypothesis?

Here are the steps we follow to determine to reject or
retain the null-hypothesis: (Note – there will be new
terminology. Don’t be too concerned about it. At a
certain point each concept below will be explained to
you)

you)
Step 1 – calculate the sums of squares between groups

you)
Step 2 – calculate the sums of squares within groups

you)
Step 3 – place the sums of squares values in an ANOVA table

you)
Step 4 - determine the degrees of freedom

you)
Step 4 - determine the degrees of freedom
Step 5 – divide the between and the within sums of squares
by their corresponding degrees of freedom to get the means
square values for both the between and within groups.

Step 6 – Divide the between groups means square by the
within groups mean square to get the F-ratio

Step 7 – locate the F critical on the F Distribution Table

Step 8 – determine which is bigger

Step 9 – retain or reject the null hypothesis

Step 10 – if the null is rejected conduct a posthoc test to see
where the differences lie (this will be shown in another
presentation)

Step 1 - calculate the sums of squares between groups

Let’s illustrate this statement with a data set:

Let’s illustrate this statement with a data set:
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices

First we calculate the mean for each group.
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices

football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
average
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
average
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
average

football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
average 18
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
average 9
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
average 5

Then we create a new column of group means

football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
average 18
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
average 9
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
average 5

football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
average 18
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
average 9
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
average 5
player group
means
football 18
b-ball 9
soccer 5
average

football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
average 18
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
average 9
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
average 5
player group
means
football 18
b-ball 9
soccer 5
average
Another way to state Average of Averages is Mean of
Means.

football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
average 18
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
average 9
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
average 5
player group
means
football 18
b-ball 9
soccer 5
mean of
means
10.7
Means.

football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
average 18
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
average 9
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
average 5
player group
means
football 18
b-ball 9
soccer 5
mean of
means
10.7
Means. Average of Averages or Mean of Means is also
called the Grand Mean.

football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
average 18
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
average 9
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
average 5
player group
means
football 18
b-ball 9
soccer 5
Grand
Mean
10.7
Means. Average of Averages or Mean of Means is also
called the Grand Mean.

Now we will compute the sum of squared deviations
like we’ve done before with variance but this time
between group means and the grand mean.

player group
means
football 18
b-ball 9
soccer 5
Grand
Mean
10.7

group
means
18
9
5
football
b-ball
soccer

group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer

So, here is the deviation between each of the group
means and grand mean.
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer

So, here is the deviation between each of the group
means and grand mean.
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=

Now we square the deviations between groups and
grand mean. If we didn’t, when we try to sum the
deviations they will come to zero.
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=

Now we square the deviations between groups and
grand mean. If we didn’t, when we try to sum the
deviations they will come to zero.
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =

Finally, we multiply these squared deviations by the
number of students in each group.
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =

Finally, we multiply these squared deviations by the
number of students in each group.
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =
# in each
group
3
3
3
x
x
x

Why do we do this?

Why do we do this? We do this so as to provide greater
weight to those groups with more students.

weight to those groups with more students. As an
example, if there were 6 soccer players, then we would
do the following:

do the following:
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =
# in each
group
3
3
6
x
x
x

do the following:
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =
x
x
x
# in each
group
3
3
6

Let’s go back to our original data set:

group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =
x
x
x
# in each
group
3
3
3

group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =
x
x
x
# in each
group
3
3
3
weighted
sq. dev.
161.3
8.3
96.3
=
=
=

So, the Sum of the Weighted Squared Deviations
between groups and grand mean is:
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =
x
x
x
# in each
group
3
3
3
weighted
sq. dev.
161.3
8.3
96.3
=
=
=

So, the Sum of the Weighted Squared Deviations
between groups and grand mean is:
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =
x
x
x
# in each
group
3
3
3
weighted
sq. dev.
161.3
8.3
96.3
266.0
=
=
=
sum

Watch what happens to the sum of squared deviations
between group and grand mean when the group
means are closer.

means are closer.
So let’s say:
• Football players ate 4 slices;
• B-ball players ate 5 slices;
• Soccer players ate 7 slices

means are closer.
So let’s say:
group
means
4
5
7
football
b-ball
soccer

means are closer.
So let’s say:
group
means
4
5
7
5.3
football
b-ball
soccer
grand
mean

means are closer.
So let’s say:
group
means
4
5
7
grand
mean
5.3
5.3
5.3
–
–
–
football
b-ball
soccer

means are closer.
So let’s say:
group
means
4
5
7
grand
mean
5.3
5.3
5.3
–
–
–
football
b-ball
soccer
deviation
- 1.3
- 0.3
1.7
=
=
=

means are closer.
So let’s say:
group
means
4
5
7
grand
mean
5.3
5.3
5.3
–
–
–
football
b-ball
soccer
deviation
- 1.3
- 0.3
1.7
=
=
=
squared
deviation
1.8
0.1
2.8
2 =
2 =
2 =

means are closer.
So let’s say:
group
means
4
5
7
grand
mean
5.3
5.3
5.3
–
–
–
football
b-ball
soccer
deviation
- 1.3
- 0.3
1.7
=
=
=
squared
deviation
1.8
0.1
2.8
2 =
2 =
2 =
x
x
x
# in each
group
3
3
3

means are closer.
So let’s say:
group
means
4
5
7
grand
mean
5.3
5.3
5.3
–
–
–
football
b-ball
soccer
deviation
- 1.3
- 0.3
1.7
=
=
=
squared
deviation
1.8
0.1
2.8
2 =
2 =
2 =
x
x
x
# in each
group
3
3
3
weighted
sq. dev.
5.3
0.3
8.3
=
=
=

means are closer.
So let’s say:
group
means
4
5
7
grand
mean
5.3
5.3
5.3
–
–
–
football
b-ball
soccer
deviation
- 1.3
- 0.3
1.7
=
=
=
squared
deviation
1.8
0.1
2.8
2 =
2 =
2 =
x
x
x
# in each
group
3
3
3
weighted
sq. dev.
5.3
0.3
8.3
14.0
=
=
=
sum

Notice how when the group means are closer, their
sum of squares between groups is much smaller than
when they are further apart like our original analysis.
(See below)

Notice how when the group means are closer, their
sum of squares between groups is much smaller than
when they are further apart like our original analysis.
(See below)
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =
x
x
x
# in each
group
3
3
3
weighted
sq. dev.
161.3
8.3
96.3
266.0
=
=
=
sum

The weighted sums of squares between the groups
represents how spread apart the levels

represents how spread apart the levels (football,
basketball, and soccer players)

basketball, and soccer players) of the independent
variable

variable (athletes)

variable (athletes) are from one another.

We believe we know the source of the differences
among the groups.

We believe we know the source of the differences
among the groups.
In this case we believe that the variance in the amount
of pizza eaten is a function of whether an athlete plays
football, basketball or soccer. This is our alternative
hypothesis.

Important note: To simplify the phrase “sum of
weighted squares deviations between the group means
and grand mean” we simply state: Between Groups
Sum of Squares.

Important note: To simplify the phrase “sum of
weighted squares deviations between the group means
and grand mean” we simply state: Between Groups
Sum of Squares.
Now on to Step 2 – calculate the sums of squares
within groups

On the other hand, the sum of squares within the
groups represents variance for which we have not
accounted. We don’t know the source of the variance
within the groups.

within the groups.
So, we compute the within groups sum of squares.

within the groups.
So, we compute the within groups sum of squares. Let’s
think about why do we do this.

Based on the group means alone (2, 5, 9) we still do not
know how much variability there is within each group.

For all we know, the scores are clustered around their
means like this:

means like this:
mean = 5 mean = 9 mean = 18

means like this:
grand mean = 10.67

means like this:
grand mean = 10.67
(Hint: if this were the case then there most likely would be a significant difference
between the means.)

means like this: Or, they could be clustered around
their means like this (very spread out and overlapping):

grand mean = 10.67

grand mean = 10.67
(Hint: if this were the case then there would most likely NOT be a significant
difference between the means, because there is too much overlap between the
distributions.)

By computing the within groups sums of squares we
will be able to consider how narrow or wide these
sample distributions are.

As you recall, our between groups sums of squares
value is 266:

value is 266:
group
means
18
9
5
grand
mean
10.7
10.7
10.7
–
–
–
football
b-ball
soccer
deviation
7.3
- 1.7
- 5.7
=
=
=
squared
deviation
53.8
2.8
32.1
2 =
2 =
2 =
x
x
x
# in each
group
3
3
3
weighted
sq. dev.
161.3
8.3
96.3
266.0
=
=
=
sum

value is 266:
There are two ways to compute the within groups sums
of squares:

value is 266:
of squares:
1. The short way

value is 266:
of squares:
1. The short way
2. The long way

We will begin with the long way.

We will begin with the long way.
The long way is calculated by computing the within
sums of squares for football players plus the sums of
squares for basketball players plus the sums of squares
with soccer players.

Now, we compute the sums of squares within football
players:

players:
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices

players: then within basketball players:
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices

players: then within basketball players:
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices

players: then within basketball players: finally, within
soccer players:
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices

players: then within basketball players: finally, within
soccer players:
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices

We begin with our football players.

We begin with our football players.
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices

We begin with our football players. Compute the mean.
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices

football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
mean 18 slices

sample
mean
18
18
18
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
mean 18 slices

Subtract the mean from each player’s slices.
sample
mean
18
18
18
–
–
–
=
=
=
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
mean 18 slices

Subtract the mean from each player’s slices.
sample
mean
18
18
18
–
–
–
deviation
- 1
0
1
=
=
=
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
mean 18 slices

Subtract the mean from each player’s slices. Square
each deviation score.
sample
mean
18
18
18
–
–
–
deviation
- 1
0
1
=
=
=
2 =
2 =
2 =
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
mean 18 slices

each deviation score.
sample
mean
18
18
18
–
–
–
deviation
- 1
0
1
=
=
=
squared
deviation
1
0
1
2 =
2 =
2 =
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
mean 18 slices

each deviation score. The sum of the squared
deviations for the football player group …
sample
mean
18
18
18
–
–
–
deviation
- 1
0
1
=
=
=
squared
deviation
1
0
1
2 =
2 =
2 =
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
mean 18 slices

deviations for the football player group …
sample
mean
18
18
18
–
–
–
deviation
- 1
0
1
=
=
=
squared
deviation
1
0
1
2 =
2 =
2 =
=
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
mean 18 slices sum

deviations for the football player group is 2.
sample
mean
18
18
18
–
–
–
deviation
- 1
0
1
=
=
=
squared
deviation
1
0
1
2
2 =
2 =
2 =
=
football
player
pizza
eaten
1 17 slices
2 18 slices
3 19 slices
mean 18 slices sum

Now we will do the same for the basketball player
group.

group.
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices

group.
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
mean 9 slices

group.
sample
mean
9
9
9
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
mean 9 slices

group.
sample
mean
9
9
9
–
–
–
=
=
=
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
mean 9 slices

group.
sample
mean
9
9
9
–
–
–
deviation
- 1
0
1
=
=
=
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
mean 9 slices

group.
sample
mean
9
9
9
–
–
–
deviation
- 1
0
1
=
=
=
2 =
2 =
2 =
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
mean 9 slices

group.
sample
mean
9
9
9
–
–
–
deviation
- 1
0
1
=
=
=
squared
deviation
1
0
1
2 =
2 =
2 =
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
mean 9 slices

group.
sample
mean
9
9
9
–
–
–
deviation
- 1
0
1
=
=
=
squared
deviation
1
0
1
2 =
2 =
2 =
=
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
mean 9 slices sum

group.
sample
mean
9
9
9
–
–
–
deviation
- 1
0
1
=
=
=
squared
deviation
1
0
1
2
2 =
2 =
2 =
=
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
mean 9 slices sum

group.
sample
mean
9
9
9
–
–
–
deviation
- 1
0
1
=
=
=
squared
deviation
1
0
1
2
2 =
2 =
2 =
=
basketball
player
pizza
eaten
1 8 slices
2 9 slices
3 10 slices
mean 9 slices sum
So the sum of squares within the basketball player
group is 2.

Now we will do the same for the soccer player group.

soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices

soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
mean 5 slices

sample
mean
5
5
5
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
mean 5 slices

sample
mean
5
5
5
–
–
–
=
=
=
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
mean 5 slices

sample
mean
5
5
5
–
–
–
deviation
- 4
0
4
=
=
=
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
mean 5 slices

sample
mean
5
5
5
–
–
–
deviation
- 4
0
4
=
=
=
2 =
2 =
2 =
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
mean 5 slices

sample
mean
5
5
5
–
–
–
deviation
- 4
0
4
=
=
=
squared
deviation
16
0
16
2 =
2 =
2 =
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
mean 5 slices

sample
mean
5
5
5
–
–
–
deviation
- 4
0
4
=
=
=
squared
deviation
16
0
16
2 =
2 =
2 =
=
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
mean 5 slices sum

sample
mean
5
5
5
–
–
–
deviation
- 4
0
4
=
=
=
squared
deviation
16
0
16
32
2 =
2 =
2 =
=
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
mean 5 slices sum

sample
mean
5
5
5
–
–
–
deviation
- 4
0
4
=
=
=
squared
deviation
16
0
16
32
2 =
2 =
2 =
=
soccer
player
pizza
eaten
1 1 slices
2 5 slices
3 9 slices
mean 5 slices sum
So the sum of squares within the soccer player group is
32.

Let’s summarize the sum of squares within groups:

Sum of squares within the football player group = 2 .

Sum of squares within the basketball player group = 2 .

Sum of squares within the soccer player group = 32 .

Now lets’ sum up these within groups sum of squares

36 .

Another way to calculate the total sums of squares is to
put all of the scores in a column:

Another way to calculate the total sums of squares is to
put all of the scores in a column:
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9

Calculate the grand mean:
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9

Calculate the grand mean:
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9
grand
mean
10.7

And subtract
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9
grand
mean
10.7

And subtract the grand mean from them:
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9
grand
mean
10.7
grand mean
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
–
–
–
–
–
–
–
–
–

Which equals
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9
grand
mean
10.7
grand mean
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
–
–
–
–
–
–
–
–
–

Which equals the deviation
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9
grand
mean
10.7
grand mean
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
–
–
–
–
–
–
–
–
–
deviation
6.3
7.3
8.3
- 2.7
- 1.7
- 0.7
- 9.7
- 5.7
- 1.7
=
=
=
=
=
=
=
=
=

Then square those deviations
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9
grand
mean
10.7
grand mean
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
–
–
–
–
–
–
–
–
–
deviation
6.3
7.3
8.3
- 2.7
- 1.7
- 0.7
- 9.7
- 5.7
- 1.7
=
=
=
=
=
=
=
=
=

Then square those deviations
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9
grand
mean
10.7
grand mean
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
–
–
–
–
–
–
–
–
–
deviation
6.3
7.3
8.3
- 2.7
- 1.7
- 0.7
- 9.7
- 5.7
- 1.7
=
=
=
=
=
=
=
=
=
deviation
39.7
53.3
68.9
7.3
2.9
0.5
94.1
32.5
2.9
2 =
2 =
2 =
2 =
2 =
2 =
2 =
2 =
2 =

Then sum up the squared deviations from the mean
and you get the total sums of squares
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9
grand
mean
10.7
grand mean
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
–
–
–
–
–
–
–
–
–
deviation
6.3
7.3
8.3
- 2.7
- 1.7
- 0.7
- 9.7
- 5.7
- 1.7
=
=
=
=
=
=
=
=
=
deviation
39.7
53.3
68.9
7.3
2.9
0.5
94.1
32.5
2.9
2 =
2 =
2 =
2 =
2 =
2 =
2 =
2 =
2 =

Then sum up the squared deviations from the mean
and you get the total sums of squares
players slices
F1 17
F2 18
F3 19
B1 8
B2 9
B3 10
S1 1
S2 5
S3 9
grand
mean
10.7
grand mean
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
10.7
–
–
–
–
–
–
–
–
–
deviation
6.3
7.3
8.3
- 2.7
- 1.7
- 0.7
- 9.7
- 5.7
- 1.7
=
=
=
=
=
=
=
=
=
deviation
39.7
53.3
68.9
7.3
2.9
0.5
94.1
32.5
2.9
302.0
2 =
2 =
2 =
2 =
2 =
2 =
2 =
2 =
2 =
sum2 =

You have probably figured out the short way to
compute the within groups sums of squares:

1. Compute the between group sum of squares
(266.0)

1. Compute the between group sum of squares
(266.0)
2. Subtract it from the total sums of squares (302.0 –
266.0), which equals the within group sums of
squares (36.0)

Step 3 – place the sums of squares values in an ANOVA
table
As you recall

table
To calculate total sums of squares we simply add the
between groups and within groups sums of squares.

table
To calculate total sums of squares we simply add the
between groups and within groups sums of squares.
Group Sum of Squares
Between groups 266.0
Within groups 36.0
total 302.0

Step 4 – determine the degrees of freedom
To calcul

We then divide the sum of squares by their degrees of
freedom to get the mean square value for both groups.

Here is how we calculate the number of degrees of
freedom:

freedom:
Sums of Squares BETWEEN Groups Degrees of Freedom
# of groups minus one equals

freedom:
Sums of Squares WITHIN Groups Degrees of Freedom
# of persons minus # of groups equals

freedom:
3
3

freedom:
3 –
3

freedom:
3 – 1
3

freedom:
3 – 1 =
3

freedom:
3 – 1 = 2
3

freedom:
3 – 1 = 2
9 3

freedom:
3 – 1 = 2
9 – 3

freedom:
3 – 1 = 2
9 – 3 =

freedom:
3 – 1 = 2
9 – 3 = 6

Back to the table:

Back to the table:
Group Sum of Squares Deg. of F.
Between groups 266.0
Within groups 36.0

Back to the table:
Group Sum of Squares Deg. of F.
Between groups 266.0 2
Within groups 36.0 6

Step 5 – divide the between and the within groups
sums of squares by their corresponding degrees of
freedom to get the means square values for both the
between and within groups.
Back to the table:

The mean square is like an average. So dividing the
sum of squares by the degrees freedom is like dividing
a data set total (1, 2, 3, 4, 5 = 15) by the number of
data points (5). (15/5 = 3). But in this case, the data
points are the squared deviations.
Back to the table:

For example, you may remember that the weighted
squared deviations between groups were:
– 161.3
– 8.3
– 96.3
Back to the table:

For example, you may remember that the weighted
squared deviations between groups were:
– 161.3
– 8.3
– 96.3
To take the average of this you would divide it by three;
however, in this case we divide it by the degrees of
freedom (why we do this is explained in another
presentation).

So 161.3 + 8.3 + 96.3 divided by 2 degrees of freedom =

132.95

132.95
As you may also remember, the variance is the average
of the sums of squared deviations. Well, guess what?
The mean square is essentially the variance of the
between group.

Group Sum of Squares Deg. of F. Mean Square
Between groups 266.0 2

Between groups 266.0 2 133.0

Within groups 36.0 6 6.0

Also
known as
the
variance

Step 6 – divide the between groups means square by
the within groups mean square to get the F-ratio

Finally, we divide the Between Groups Mean Square
(133.0) by the Within Groups Mean Square (6.0) to get
the F-ratio.

the F-ratio.
Group Sum of Squares Deg. of F. Mean Square F-ratio

the F-ratio.
22.17

What is the F-ratio?

As just explained, the ratio of the mean sum of squares
between groups to the mean sum of squares within
groups generates an F-statistic.

groups generates an F-statistic. It is this F-statistic that
we will use to test our null hypothesis:

There is no statistically significant difference in the quantity
of pizza consumed in one sitting among football, basketball,
and soccer players.
If

There is no statistically significant difference in the quantity
of pizza consumed in one sitting among football, basketball,
and soccer players.
If the F critical value is greater than our F-statistic of
22.17

value

This is done by using the number of degrees of
freedom in the denominator (within groups = 6) and
the degrees of freedom in the numerator (between
groups =2) and determining where these two intersect.

This

This
And we find the F
critical value, which is
5.14

This

It just so happens that our F-ratio is 22.17, which
means it is bigger than the F critical value.

It

We will therefore reject the null hypothesis at the .05
alpha level, or in other words, with a probability of
Type-I error less than .05.

We will therefore reject the null hypothesis at the .05
alpha level, or in other words, with a probability of
Type-I error less than .05.
If the F-ratio had been 4.38, then we would fail to
reject or in other words, retain the null hypothesis.

Do you see what makes a small or large F-ratio value?

Here is what makes the difference:

Here is what makes the difference: When the groups or
their means are closer together,

their means are closer together,

their means are closer together, then the between
groups mean square will be smaller

groups mean square will be smaller
Within groups ?

groups mean square will be smaller. And to compound
the situation, if the within groups are large (meaning
there is a lot of difference or variability within groups),

then the within groups mean square will be smaller

then the within groups mean square will be smaller,

then the F ratio will be extremely smaller than what we
found with the pizza eating athlete groups:

0.3

With an F critical still at 5.14 and an F ratio of 0.3, we
would retain the null hypothesis.

Note: if there are really no differences among three
groups in terms of some dependent variable, the mean
sum of squares between groups will be very similar to
the mean sum of squares within groups.

The ratio of such mean squares will be close to 1. As
the differences among the groups increases, the ratio
of mean sums of squares will increase above 1.

of mean sums of squares will increase above 1. At
some point, the ratio of mean sums of squares (F) will
be large enough that we will conclude that there are
probably systematic differences among the groups.

of mean sums of squares will increase above 1. At
some point, the ratio of mean sums of squares (F) will
be large enough that we will conclude that there are
probably systematic differences among the groups. We
will reject the null hypothesis of no differences and
investigate where the differences occur.

Step 10 – if the null is rejected, conduct a post hoc test
to see where the differences lie (this will be shown in
another presentation).
We will

Is the difference between football and basketball
players or is it between football and soccer players or
basketball and soccer players or all three? The post hoc
will provide this information. This will be shown in
another presentation.

In order to discover what pattern of differences
generated the significant F-statistic, we conduct post
hoc tests, which use different logics and calculations,
some of which are more conservative than others in
how they protect against cumulative Type-I error across
multiple tests. In essence, each post hoc test is
comparing each group to every other group in a series
of two-group tests. The results of the series of two-group
tests identifies which of the many possible
patterns generated the significant F-statistic and where
the differences lie.

In summary, One way Analysis of Variance is a method
that will help you determine if there is a statistically
significant difference between more than two group
means. It does not tell you which groups differ, just
that they do. Further testing will determine which
groups differ.

What is a one way anova?

More Related Content

Viewers also liked (20)

Similar to What is a one way anova? (20)

More from Ken Plummer (20)

Recently uploaded (20)

What is a one way anova?