questions of relationship

Is the problem you are working on focus on Questions of Relationship?

Questions of relationship focus on how two or more variables co-vary or co-relate with each other.

Or how increases or decreases in one variable are accompanied by increases or decreases in another variable.

Here is an equation to use as a guide


An Increase or decrease in


Variable 1An Increase or decrease in



is accompanied by an increaseor decrease in




Variable 2



Variable 2

By variable we mean something that varies or

changes, like temperature, speed, weight, test scores,

etc.

Let’s see an example:

Researchers hypothesize that as the temperature increases burglaries increase.


as




Variable 2


TemperatureAn Increase or decrease in


Variable 2




Burglaries

Researchers hypothesize that as the temperature increases burglaries increase. Test this hypothesis with the data set provided.



Burglaries

Therefore, this is a question of Relationship

Let’s see what the data might look like for this word problem:

Month Average Temperature

Number of Burglaries

Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100

Let’s see what the data might look like for this word problem:



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100

What do we mean when we say that a relationship exists between two variables – in this case – temperature and burglaries?



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100

What we mean is that the two variables vary (increase or decrease) in either the same or opposite directions.



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100

A simple way to illustrate this idea of covary-ing is to see the relative rank of the values of one variable and see if those ranks are similar to the relative rank of the values of the other variable



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100

Let’s begin by rank ordering the average temperature values.



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 This is the highest value so we’ll give it a #1 Rank



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

This is the 2nd highest value so we’ll give it a #2 Rank



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

This is the 3rd highest value so we’ll give it a #3 Rank



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

This is the 4th highest value so we’ll give it a #4 Rank

#3



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4


#7



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

Now, let’s do the same for the number of burglaries:



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

This is the highest value so we’ll give it a

#1 Rank



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

This is the 2nd highest value so we’ll give it a

#2 Rank



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

This is the 3rd highest value so we’ll give it a

#3 Rank



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

This is the 4th highest value so we’ll give it a

#4 Rank#3



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4


#7 Rank



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7


#5 Rank



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7

#5



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7

#5


#6 Rank



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7

#5

#6



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7

#5

#6

Notice that the relative rank order for temperature and burglaries across each month

is the SAME.



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7

#5

#6

The highest rank on one is the highest rank order on the other.



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7

#5

#6

The 2nd highest rank on one is the 2nd highest rank order on the other.



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7

#5

#6

Ect. Ect. Ect.



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7

#5

#6

This is a way of visualizing how an increase in one is accompanied by an increase in another.

What would an increase in one variable and an decrease in another variable look like?



Jan 30o 10

Feb 40o 25

Mar 20o 5

Apr 60o 40

May 70o 60

Jun 80o 80

Jul 90o 100 #1

#2

#3

#4

#7

#5

#6

#1

#2

#3

#4

#7

#5

#6


#1

#2

#3

#4

#7

#5

#6 #1

#2

#3

#4

#7

#5

#6



Jan 30o 100

Feb 40o 80

Mar 20o 60

Apr 60o 40

May 70o 5

Jun 80o 25

Jul 90o 10





Jan 30o 100

Feb 40o 80

Mar 20o 60

Apr 60o 40

May 70o 5

Jun 80o 25

Jul 90o 10#1

#2

#3

#4

#7

#5

#6 #1

#2

#3

#4

#7

#5

#6

Notice that in this case, as temperature increases burglaries decrease.



Jan 30o 100

Feb 40o 80

Mar 20o 60

Apr 60o 40

May 70o 5

Jun 80o 25

Jul 90o 10#1

#2

#3

#4

#7

#5

#6 #1

#2

#3

#4

#7

#5

#6

Let’s consider other word problems

An ice cream parlor owner wishes to know the degree to which ice cream sales are related to average monthly temperature.




Variable 2


Ice cream sales



Variable 2


Ice cream sales



Temperature

Some word problems will look for a relationship between variables that can take on unlimited values like speed, age, height or weight


E.g., a jet can fly as slow as 0 mph and as fast as 700 mph. It’s speed can take on any number of values in between 0 and 700 (e.g., 2.8 mph or 345.6 mph )


E.g., A person can be as young as zero or as old as 100+

(e.g., 6.32 years or 98.9 years)


with variables that can take on limited values like gender, year in school or whether a person has experienced something or not.



E.g., with gender, male can take on a value of 1and female a value of 2.



or



female can take on a value of 1 and male a value of 2.



Year in school can take on a value of 1 for freshmen, 2 for sophomore, 3 for junior

and 4 for senior.



Experience can mean I experienced it = 1 or I did not experience it = 2 (e.g., exposed to gamma

rays or not exposed to gamma rays)

Let’s see an example of this,

Researchers wish to know if there is a relationship between the average freeway driving speed and gender.

In this case the wording of our relationship equation will change from




Variable 2


to



Variable 2


to



Variable 2

Variable 1Higher and

lower scores in

tend to be related to

certain groups in Variable 2


to



Variable 2


lower scores in



Because variables like Gender can neither

increase nor decrease

Let’s see how this works in our word problem about gender and freeway speed.



lower scores in




Freeway Driving Speed

Higher and lower scores

in




Freeway Driving Speed

Higher and lower scores

in


certain groups in Gender

Here is what the data set would look like:


Driver

Mary

Bill

Sarah

Mike

Sally

Charles

Fred


Driver Average FreewaySpeed (mph)

Mary 66

Bill 73

Sarah 56

Mike 82

Sally 62

Charles 78

Fred 91



Gender 1= male

2 = female

Mary 66 2

Bill 73 1

Sarah 56 2

Mike 82 1

Sally 62 2

Charles 78 1

Fred 91 1


Gender 1= male

2 = female

Mary 66 2

Bill 73 1

Sarah 56 2

Mike 82 1

Sally 62 2

Charles 78 1

Fred 91 1

So, are higher speeds associated with one gender

and are lower speeds associated

with the other gender?


Gender 1= male

2 = female

Mary 66 2

Bill 73 1

Sarah 56 2

Mike 82 1

Sally 62 2

Charles 78 1

Fred 91 1

It appears that the number 2s (female) have lower average driving speeds than


Gender 1= male

2 = female

Mary 66 2

Bill 73 1

Sarah 56 2

Mike 82 1

Sally 62 2

Charles 78 1

Fred 91 1

. . . the number 1s (male)


Gender 1= male

2 = female

Mary 66 2

Bill 73 1

Sarah 56 2

Mike 82 1

Sally 62 2

Charles 78 1

Fred 91 1

Notice that this could

sound like a difference question.

Let’s see an example of this:

Difference question:

Difference question: Are women faster freeway drivers than men?


Same question expressed as a relationship:


Same question expressed as a relationship: Does a relationship exist between freeway speed and gender?


Same question expressed as a relationship: Does a relationship exist between freeway speed and gender?

Depending on how the question is asked it will either be a difference or a relationship question.

Some relationship questions focus on how well one or two variables predict another variable.

For example,

To what degree do ACT scores predict college freshmen grades in a introduction to writing course.




Variable 2


ACT ScoresAn Increase or decrease in


Variable 2



predict Variable 2



predictWriting Course Grades



predictWriting Course Grades

Prediction Questions are classified as Questions of Relationship

In Summary

Relationship questions ask about the degree to which a change (increase/decrease) in one variable is accompanied by a change (increase/decrease) in another variable.

Relationship questions ask about the degree to which a change (increase/decrease) in one variable is accompanied by a change (increase/decrease) in another variable.



Variable 2

Relationship questions can be between variables that take on unlimited values (e.g, age and weight, or age and weight),


Or between variables with limited values (e.g., gender and year in school)


Or between variables with limited values (e.g., gender and year in school)


lower scores in



Finally, relationship questions can focus on the degree to which one variable predicts another variable.

Finally, relationship questions can focus on the degree to which one variable predicts another variable.


Predicts Variable 2

Some of the words to look for in your problem to determine if it is a question of relationship are:

Some of the words to look for in your problem to determine if it is a question of relationship are:

• Increase

• Decrease

• Association

• Are associated with

• Relationship

• Relate to

• Predict

• Predictive Power

Examine the question or problem you are working on.

Is it a question of relationship?

If so, select RELATIONSHIP

questions of relationship

Education

temperaturean increase

guidean increase

word problem

questions of relationship

data set

test scores

variables co

goodness of fit