what is an ancova?
Post on 16-Dec-2014
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One-way Analysis of Covariance(ANCOVA)
Conceptual Tutorial
First
How did we get here?
Consider a similar problem
A pizza café owner wants to know which type of high school athlete she should market to.
A pizza café owner wants to know which type of high school athlete she should market to. Should she market to football, basketball or soccer players?
A pizza café owner wants to know which type of high school athlete she should market to. Should she market to football, basketball or soccer players? So she measures the ounces of pizza eaten by 12 football, 12 basketball, and 12 soccer players in one sitting.
Here are the raw data:Football Players Basketball Players Soccer Players
29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
The owner wondered how much these athletes like pizza to begin with and how that might affect the results.
The owner wondered how much these athletes like pizza to begin with and how that might affect the results. She surveyed them prior to their eating the pizza.
The Survey
On a scale of 1 to 10 how much do you like pizza?
1 2 3 4 5 6 7 8 9 10
Here were the results:Football Basketball Soccer
7.0 3.0 7.55.0 8.0 4.53.5 4.5 3.59.0 9.5 6.07.0 6.5 6.08.0 7.0 4.56.5 7.5 6.07.5 9.0 1.52.5 8.5 6.59.0 4.0 5.08.0 7.5 5.55.0 8.0 4.0
Based on this information, let’s determine how we got here.
Here is the problem again:
A pizza café owner wants to know which type of high school athlete she should market to, by
comparing how many ounces of pizza are consumed across all three athlete groups.
She will control for pizza preference.
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is This:
Inferential Descriptiveor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is This:
Inferential Descriptiveor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is This:
Inferential Descriptiveor
Based on the data set of 36 athletes, this is a sample from which the owner would like to make generalizations about potential athlete customers.
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is This:
Inferential Descriptiveor
Based on the data set of 36 athletes, this is a sample from which the owner would like to make generalizations about potential athlete customers.
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is This Question of:
Relationship Differenceor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is This Question of:
Relationship Differenceor
Because the owner wants to compare groups differences, we are dealing with DIFFERENCE.
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is This Question of:
Relationship Differenceor
Because the owner wants to compare groups differences, we are dealing with DIFFERENCE.
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Inferential Descriptive
InferentialDescriptive
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is The Distribution:
Normal Not Normalor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is The Distribution:
Normal Not Normalor
After graphing each column we find that the distributions are mostly normal.
Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is The Distribution:
Normal Not Normalor
After graphing each column we find that the distributions are mostly normal.
Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Inferential Descriptive
InferentialDescriptive
Normal Not Normal
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Are the Data:
Scaled? (ratio/interval/ordinal)
Categorical?(ordinal)
or
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Are the Data:
Scaled? (ratio/interval/ordinal)
Categorical?(ordinal)
or
The data is interval (ounces of Pizza)
Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Are the Data:
Scaled? (ratio/interval/ordinal)
Categorical?(ordinal)
or
The data is interval (ounces of Pizza)
Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Inferential Descriptive
InferentialDescriptive
Normal Not Normal
Scaled Categorical
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is there:
1 DV 2 or more DVor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is there:
1 DV 2 or more DVor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is there:
1 DV 2 or more DVor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Inferential Descriptive
InferentialDescriptive
Normal Not Normal
Scaled Categorical
1 DV 2 or more DV
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is there:
1 IV 2 or more IVsor
[Type of Athlete is the only Independent Variable (IV)]
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is there:
1 IV 2 or more IVsor
[Type of Athlete is the only Independent Variable (IV)]
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is there:
1 IV 2 or more IVsor
[Type of Athlete is the only Independent Variable (IV)]
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Inferential Descriptive
InferentialDescriptive
Normal Not Normal
Scaled Categorical
1 DV 2 or more DV
1 IV 2 or more IV
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is there:
1 IV Level 2 or more IV Levelsor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is there:
1 IV Level 2 or more IV Levelsor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is there:
1 IV Level 2 or more IV Levelsor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Inferential Descriptive
InferentialDescriptive
Normal Not Normal
Scaled Categorical
1 DV 2 or more DV
1 IV 2 or more IV
2 or more IV Levels1 IV Level
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Are the Samples:
Repeated Independentor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Are the Samples:
Repeated Independentor
No individual is in more than one group
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Are the Samples:
Repeated Independentor
No individual is in more than one group
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Inferential Descriptive
InferentialDescriptive
Normal Not Normal
Scaled Categorical
1 DV 2 or more DV
1 IV 2 or more IV
2 or more IV Levels1 IV Level
IndependentRepeated
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is There:
A Covariate Not a Covariateor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is There:
A Covariate Not a Covariateor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Is There:
A Covariate Not a Covariateor
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Inferential Descriptive
InferentialDescriptive
Normal Not Normal
Scaled Categorical
1 DV 2 or more DV
1 IV 2 or more IV
2 or more IV Levels1 IV Level
IndependentRepeated
A Covariate Not a Covariate
Now that we know how we got here, let’s consider what Analysis of Covariance is.
Now that we know how we got here, let’s consider what Analysis of Covariance is.First, . . . what is covariance?
Now that we know how we got here, let’s consider what Analysis of Covariance is.First, . . . what is covariance?As you know, variance is a statistic that helps you determine how much the data in a distribution varies.
Now that we know how we got here, let’s consider what Analysis of Covariance is.First, . . . what is covariance?As you know, variance is a statistic that helps you determine how much the data in a distribution varies.
6 75
Number of Pizza Slices
eaten by Basketball
Players
Not much variance
Now that we know how we got here, let’s consider what Analysis of Covariance is.First, . . . what is covariance?As you know, variance is a statistic that helps you determine how much the data in a distribution varies.
6 75
Number of Pizza Slices
eaten by Basketball
Players
Not much variance
6 754 8 932 10
Number of Pizza Slices
eaten by Soccer Players
A lot of variance
Covariance is a statistic that helps us determine how much two distributions that have some relationship covary.
Let’s imagine that students take a math test and their ordered scores look like this.
Let’s imagine that students take a math test and their ordered scores look like this.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Let’s imagine that students take a math test and their ordered scores look like this. Then let’s imagine they take a math anxiety survey.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Let’s imagine that students take a math test and their ordered scores look like this. Then let’s imagine they take a math anxiety survey.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
How much do they covary?
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
Bambi has the highest Math Test Score and the highest Math Anxiety Score
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
Belle has the 2nd highest Math Test Score and the 2nd highest Math Anxiety Score
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
Billy has the 3rd highest Math Test Score and the 3rd highest Math Anxiety Score
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
Boston has the 4th highest Math Test Score and the 4th highest Math Anxiety Score
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
Bryne has the 5th highest Math Test Score and the 5th highest Math Anxiety Score
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
Bubba has the 6th highest Math Test Score and the 6th highest Math Anxiety Score
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
These two data sets perfectly covary.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
This means as one changes the other changes.
These two data sets perfectly covary.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
This means as one changes the other changes.
Either in the same direction
These two data sets perfectly covary.
Student Test Scores
Bambi 98
Belle 92
Billy 84
Boston 77
Bryne 73
Bubba 68
Etc
Math Anxiety Scores
6
5
4
4
3
2
Etc.
This means as one changes the other changes.
Or opposite directions
Math Anxiety Scores
2
3
4
4
5
6
Etc.
These two data sets perfectly covary.
Covariance is a statistic that describes that relationship.
Covariance is a statistic that describes that relationship. The larger the covariance statistic (either positive or negative), the more the two samples covary.
Covariance is a statistic that describes that relationship. The larger the covariance statistic (either positive or negative), the more the two samples covary. Let’s demonstrate how to calculate covariance by hand.
Covariance is a statistic that describes that relationship. The larger the covariance statistic (either positive or negative), the more the two samples covary. Let’s demonstrate how to calculate covariance by hand.
(Although most statistical software can do it for you automatically.)
Here is the data set
StudentTest
ScoresMath Anxiety
Scores Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
iX iY
Here is the sum of each column:
StudentTest
ScoresMath Anxiety
Scores Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24
iX iY
StudentTest
ScoresMath Anxiety
Scores Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
And the mean:
StudentTest
ScoresMath Anxiety
Scores Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
Remember – Covariance can only be computed between two or more variables (e.g., test questions, test scores, ect.) with scores across each variable for each
person.
Here is the formula for covariance:
i iX Y
XY
X Y
N
With an explanation for each value:
i iX Y
XY
X Y
N
i iX Y
XY
X Y
N
Anxiety Scores
i iX Y
XY
X Y
N
Test Scores
i iX Y
XY
X Y
N
Each Test Scores
i iX Y
XY
X Y
N
i iX Y
XY
X Y
N
Or in this case is the mean for Test
Scores (82)
i iX Y
XY
X Y
N
i iX Y
XY
X Y
N
Each Anxiety Score
i iX Y
XY
X Y
N
i iX Y
XY
X Y
N
Or in this case is the mean for
Anxiety Scores (24)
i iX Y
XY
X Y
N
Let the Covariance Calculations Begin!
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
98
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
98 - 82
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
16
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
1692 - 82
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
161084 - 82
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
1610277 - 82
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
16102-573 - 82
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
16102-5-968 - 82
Deviation between each math score and the mean.
i XX Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
16102-5-9-14
Deviation between each math score and the mean.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX Student
StudentTest
ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX Student
StudentTest
ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
Deviation between each Anxiety score and its mean.
i YY
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX Student
StudentTest
ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i YY
5
Deviation between each Anxiety score and its mean.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX Student
StudentTest
ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i YY
5 - 4
Deviation between each Anxiety score and its mean.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX Student
StudentTest
ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i YY
1
Deviation between each Anxiety score and its mean.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX Student
StudentTest
ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i YY
16 - 4
Deviation between each Anxiety score and its mean.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX Student
StudentTest
ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i YY
124 - 4
Deviation between each Anxiety score and its mean.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX Student
StudentTest
ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i YY
1204 - 4
Deviation between each Anxiety score and its mean.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX Student
StudentTest
ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i YY
12003 - 4
Deviation between each Anxiety score and its mean.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX Student
StudentTest
ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i YY
1200
-12 - 4
Deviation between each Anxiety score and its mean.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i iX YX Y
Multiply the two paired Deviations to get what is called “the cross
product”
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i iX YX Y
16
Multiply the two paired Deviations to get what is called “the cross
product”
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i iX YX Y
16 x 1
Multiply the two paired Deviations to get what is called “the cross
product”
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i iX YX Y
16
Multiply the two paired Deviations to get what is called “the cross
product”
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i iX YX Y
1610 x 2
Multiply the two paired Deviations to get what is called “the cross
product”
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i iX YX Y
16202 x 0
Multiply the two paired Deviations to get what is called “the cross
product”
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i iX YX Y
16200
-5 x 0
Multiply the two paired Deviations to get what is called “the cross
product”
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i iX YX Y
162000
-9 x -1
Multiply the two paired Deviations to get what is called “the cross
product”
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i iX YX Y
1620009
-14 x -2
Multiply the two paired Deviations to get what is called “the cross
product”
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX i iX YX Y
1620009
28
Multiply the two paired Deviations to get what is called “the cross
product”
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
Here’s the covariance equation again:
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
i iX Y
XY
X Y
N
i iX Y
XY
X Y
N
So far we have calculated a portion of the numerator of this equation.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
i iX Y
XY
X Y
N
Now we will sum or add up the cross products.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
i iX Y
XY
X Y
N
Now we will sum or add up the cross products.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
i iX Y
XY
X Y
N
Now we will sum or add up the cross products.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
Add up
i iX Y
XY
X Y
N
Now we will sum or add up the cross products.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
73
Add up
i iX Y
XY
X Y
N
Now we will sum or add up the cross products.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
73
i iX Y
XY
X Y
N
Then we divide the result by the number of subjects, which in this case is (6)
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
73
i iX Y
XY
X Y
N
Then we divide the result by the number of subjects, which in this case is (6)
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
73 / 6
i iX Y
XY
X Y
N
Then we divide the result by the number of subjects, which in this case is (6)
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
73 / 6 = 12.2
i iX Y
XY
X Y
N
Covariance = 12.2
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
73 / 6 = 12.2
12.2
Let’s see what the covariance looks like when the direction of the data goes in the opposite direction:
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
BEFORE
Student
StudentTest
ScoresMath
Anxiety Bambi 98 2Belle 92 3Billy 84 4Boston 77 4Bryne 73 6Bubba 68 5
Sum 492 24Mean 82 4
iX iYStudent
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
BEFORE AFTER
Student
StudentTest
ScoresMath
Anxiety Bambi 98 2Belle 92 3Billy 84 4Boston 77 4Bryne 73 6Bubba 68 5
Sum 492 24Mean 82 4
iX iYStudent
StudentTest
ScoresMath
Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2
Sum 492 24Mean 82 4
iX iY
BEFORE AFTER
First we calculate the deviations from the mean:
Student
StudentTest
ScoresMath
Anxiety Bambi 98 2Belle 92 3Billy 84 4Boston 77 4Bryne 73 6Bubba 68 5
Sum 492 24Mean 82 4
iX iY
First we calculate the deviations from the mean:
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Bambi 98 2 16 - 2Belle 92 3 10 - 1Billy 84 4 2 0Boston 77 4 - 5 0Bryne 73 6 - 9 2Bubba 68 5 - 14 1
Sum 492 24Mean 82 4
iX iY i XX i YY
We now compute the Cross Products
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Bambi 98 2 16 - 2Belle 92 3 10 - 1Billy 84 4 2 0Boston 77 4 - 5 0Bryne 73 6 - 9 2Bubba 68 5 - 14 1
Sum 492 24Mean 82 4
iX iY i XX i YY
x =x =x = x =x =x =
We now compute the Cross Products
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Cross ProductsBambi 98 2 16 - 2 -32Belle 92 3 10 - 1 -10Billy 84 4 2 0 0Boston 77 4 - 5 0 0Bryne 73 6 - 9 2 -18Bubba 68 5 - 14 1 -14
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
We sum the cross products and then divide it by the number of students
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Cross ProductsBambi 98 2 16 - 2 -32Belle 92 3 10 - 1 -10Billy 84 4 2 0 0Boston 77 4 - 5 0 0Bryne 73 6 - 9 2 -18Bubba 68 5 - 14 1 -14
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
We sum the cross products and then divide it by the number of students
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Cross ProductsBambi 98 2 16 - 2 -32Belle 92 3 10 - 1 -10Billy 84 4 2 0 0Boston 77 4 - 5 0 0Bryne 73 6 - 9 2 -18Bubba 68 5 - 14 1 -14
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
-73
We sum the cross products and then divide it by the number of students
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Cross ProductsBambi 98 2 16 - 2 -32Belle 92 3 10 - 1 -10Billy 84 4 2 0 0Boston 77 4 - 5 0 0Bryne 73 6 - 9 2 -18Bubba 68 5 - 14 1 -14
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
-73 / 6
We sum the cross products and then divide it by the number of students
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Cross ProductsBambi 98 2 16 - 2 -32Belle 92 3 10 - 1 -10Billy 84 4 2 0 0Boston 77 4 - 5 0 0Bryne 73 6 - 9 2 -18Bubba 68 5 - 14 1 -14
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
-73 / 6 = -12.2
This is the covariance!
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Cross ProductsBambi 98 2 16 - 2 -32Belle 92 3 10 - 1 -10Billy 84 4 2 0 0Boston 77 4 - 5 0 0Bryne 73 6 - 9 2 -18Bubba 68 5 - 14 1 -14
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
-73 / 6 = -12.2
This is the covariance!
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Cross ProductsBambi 98 2 16 - 2 -32Belle 92 3 10 - 1 -10Billy 84 4 2 0 0Boston 77 4 - 5 0 0Bryne 73 6 - 9 2 -18Bubba 68 5 - 14 1 -14
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
-73 / 6 = -12.2
Notice that when there is a negative relationship between two variables
This is the covariance!
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Cross ProductsBambi 98 2 16 - 2 -32Belle 92 3 10 - 1 -10Billy 84 4 2 0 0Boston 77 4 - 5 0 0Bryne 73 6 - 9 2 -18Bubba 68 5 - 14 1 -14
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
-73 / 6 = -12.2
Notice that when there is a negative relationship between two variables
This is the covariance!
Student
StudentTest
ScoresMath
Anxiety Test
ScoresMath
Anxiety Cross ProductsBambi 98 2 16 - 2 -32Belle 92 3 10 - 1 -10Billy 84 4 2 0 0Boston 77 4 - 5 0 0Bryne 73 6 - 9 2 -18Bubba 68 5 - 14 1 -14
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
-73 / 6 = -12.2
Notice that when there is a negative relationship between two variables
The covariance is negative
On the other hand, when the relationship between two variables is positive . . .
On the other hand, when the relationship between two variables is positive . . . The covariance will be positive.
On the other hand, when the relationship between two variables is positive . . . The covariance will be positive.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
73 / 6 = 12.2
On the other hand, when the relationship between two variables is positive . . . The covariance will be positive.
Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2
Sum 492 24Mean 82 4
iX iY i XX i YY Student
StudentTest
ScoresMath
Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14
Sum 492 24Mean 82 4
iX iY i XX
1620009
28
Student
Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28
Sum 492 24Mean 82 4
iX iY i XX i YY i iX YX Y
73 / 6 = 12.2
Why is this important to know?
Why is this important to know?Especially in light of the question we are trying to answer?
Why is this important to know?Especially in light of the question we are trying to answer?
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Computing covariance helps us-
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Computing covariance helps us-
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Computing covariance helps us-
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Meaning that we want to know what the difference between the three groups would be if we took away all of the covariance between pizza preference and amount of ounces of pizza eaten.
Computing covariance helps us-
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
If we did not take out the covariance, then a bunch of soccer players may like pizza not because they are soccer players (our research question) but because they just LOVE PIZZA (not our research question).
Computing covariance helps us-
The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.
Because their love of pizza is not what we are testing, we will control for it by computing covariance and see how much of the fact that they are soccer players really affects the amount of ounces of pizza they eat.
So, let’s begin by running a One-way ANOVA without removing the covariance between pizza preference and ounces eaten by athlete type.
Here’s the data
Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
Here are the results of the one-way ANOVA for this data set:
Sums of Squares df Mean Square F-Ratio
Between Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3
Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
Here are the results of the one-way ANOVA for this data set:
Sums of Squares df Mean Square F-Ratio
Between Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3
Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
As you will recall, an F-ratio 1 or lower with any ANOVA method is not significant.
Here are the results of the one-way ANOVA for this data set:
Sums of Squares df Mean Square F-Ratio
Between Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3
Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
As you will recall, an F-ratio 1 or lower with any ANOVA method is not significant.
Then
After calculating the covariance between pizza preference and ounces of pizza eaten in one sitting, we find that there is a positive relationship.
After calculating the covariance between pizza preference and ounces of pizza eaten in one sitting, we find that there is a positive relationship.
Pizza Preference (scale 1-10)Football Basketball Soccer
7.0 3.0 7.55.0 8.0 4.53.5 4.5 3.59.0 9.5 6.07.0 6.5 6.08.0 7.0 4.56.5 7.5 6.07.5 9.0 1.52.5 8.5 6.59.0 4.0 5.08.0 7.5 5.55.0 8.0 4.0
Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
After calculating the covariance between pizza preference and ounces of pizza eaten in one sitting, we find that there is a positive relationship.
Pizza Preference (scale 1-10)Football Basketball Soccer
7.0 3.0 7.55.0 8.0 4.53.5 4.5 3.59.0 9.5 6.07.0 6.5 6.08.0 7.0 4.56.5 7.5 6.07.5 9.0 1.52.5 8.5 6.59.0 4.0 5.08.0 7.5 5.55.0 8.0 4.0
Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten
24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten
14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten
27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten
27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten
28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten
27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten
32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten
13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten
35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten
32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten
17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten
Covariance = 12.1
After running the Analysis of Covariance on the data and partialling out pizza reference, here is the resulting ANOVA table:
After running the Analysis of Covariance on the data and partialling out pizza reference, here is the resulting ANOVA table:
SS df MS FAdjusted means (BG) 74.5 2 37.2 3.8Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34
After running the Analysis of Covariance on the data and partialling out pizza reference, here is the resulting ANOVA table:
SS df MS FAdjusted means (BG) 74.5 2 37.2 3.8Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34
Adjusted means – after we took out the covariance between the two variables: Type of Athlete and Pizza Preference (the covariate)
After running the Analysis of Covariance on the data and partialling out pizza reference, here is the resulting ANOVA table:
SS df MS FAdjusted means (BG) 74.5 2 37.2 3.8Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34
Notice the F-ratio is larger making it more likely to be
significant.
Let’s compare the F-ratio for just the ANOVA
Let’s compare the F-ratio for just the ANOVA
SS df MS F-RatioBetween Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3
Before
Let’s compare the F-ratio for just the ANOVA
With the ANCOVA
SS df MS F-RatioBetween Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3
Before
Let’s compare the F-ratio for just the ANOVA
With the ANCOVA
SS df MS F-RatioBetween Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3
Before
SS df MS F-RatioAdjusted means (BG) 74.5 2 37.2 3.8Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34
After
Let’s compare the F-ratio for just the ANOVA
With the ANCOVA
SS df MS F-RatioBetween Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3
Before
SS df MS F-RatioAdjusted means (BG) 74.5 2 37.2 3.8Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34
After
So we would conclude that there is a significant difference between football, basketball & soccer players in terms of the ounces of pizza they eat, that is, when we control for pizza preference.
Let’s compare the F-ratio for just the ANOVA
With the ANCOVA
SS df MS F-RatioBetween Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3
Before
SS df MS F-RatioAdjusted means (BG) 74.5 2 37.2 3.8Adjusted error (WG) 314.1 32 9.8
Adjusted total 388.6 34
After
So we would conclude that there is a significant difference between football, basketball & soccer players in terms of the ounces of pizza they eat, that is, when we control for pizza preference.
We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.
We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Adjusted Means (after controlling for the covariance)
We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Adjusted Means (after controlling for the covariance)
Notice that after controlling for pizza preference, the mean for
Basketball players drops
We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Adjusted Means (after controlling for the covariance)
Notice that after controlling for pizza preference, the mean for
Basketball players drops
We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Adjusted Means (after controlling for the covariance)
Notice that after controlling for pizza preference, the mean for
Basketball players drops
We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Adjusted Means (after controlling for the covariance)
And the mean for Soccer players
INCREASES!
We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Adjusted Means (after controlling for the covariance)
And the mean for Soccer players
INCREASES!
We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Adjusted Means (after controlling for the covariance)
And the mean for Soccer players
INCREASES!
We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Adjusted Means (after controlling for the covariance)
That’s the Power of ANCOVA!
Important note,The more the covariate (pizza preference) covaries with the independent variable (type of athlete) . .
Important note,The more the covariate (pizza preference) covaries with the independent variable (type of athlete) . . . the bigger the adjustment will be between original and adjusted means.
Important note,The more the covariate (pizza preference) covaries with the independent variable (type of athlete) . . . the bigger the adjustment will be between original and adjusted means.
Meaning they share a larger covariance value (either positive or negative).
Important note,The more the covariate (pizza preference) covaries with the independent variable (type of athlete) . . . the bigger the adjustment will be between original and adjusted means.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Adjusted Means (after controlling for the covariance)
Important note,The more the covariate (pizza preference) covaries with the independent variable (type of athlete) . . . the bigger the adjustment will be between original and adjusted means.
Athlete Football Basketball Soccer
Means 25.4 26.3 23.8
Original Means
Athlete Football Basketball Soccer
Means 24.3 23.8 27.3
Adjusted Means (after controlling for the covariance)
Big Adjustments
One final note
In this case the covariate (the thing we were controlling for) was a continuous variable like
• Ounces of pizza eaten• Time it takes to eat pizza• The weight of each athlete.
In this case the covariate (the thing we were controlling for) was a continuous variable like • Ounces of pizza eaten• Time it takes to eat pizza• The weight of each athlete.
In this case the covariate (the thing we were controlling for) was a continuous variable like • Ounces of pizza eaten• Time it takes to eat pizza• The weight of each athlete.But it also can be categorical (one or the other)
In this case the covariate (the thing we were controlling for) was a continuous variable like • Ounces of pizza eaten• Time it takes to eat pizza• The weight of each athlete.But it also can be categorical (one or the other)• Year in School (Sophomores, Juniors, or Seniors)• Gender (Male or Female)• Religious Affiliation (Muslim, Catholic, etc.)
In summary
Analysis of Covariance is a powerful tool that makes it possible to control for any variable that is not of interest (eg. pizza preference)
Analysis of Covariance is a powerful tool that makes it possible to control for any variable that is not of interest (eg. pizza preference) in order to see the true effect of the variable of interest (type of athlete) on a dependent variable of interest (ounces of pizza eaten)
There are more complex methods such as Factorial ANCOVA, Repeated measures ANCOVA and Multivariate ANCOVA.
There are more complex methods such as Factorial ANCOVA, Repeated measures ANCOVA and Multivariate ANCOVA.
This presentation gives you the conceptual foundation necessary to understand the Analysis of Covariance elements of these methods.
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