Displaying Distributions Displaying Distributions – Qualitative Variables – – Qualitative Variables – Part 2Part 2

Lecture 16Lecture 16

Sec. 4.3.3Sec. 4.3.3

Wed, Feb 11, 2004Wed, Feb 11, 2004

Studies with Two Studies with Two Qualitative VariablesQualitative Variables Typically, the purpose of studying Typically, the purpose of studying

two variables is to see whether two variables is to see whether there is a relationship between there is a relationship between them.them.

Also, when working with Also, when working with qualitative data, percentages are qualitative data, percentages are the numerical measure of choice.the numerical measure of choice.

The next-most-common measure The next-most-common measure is frequency (or count).is frequency (or count).

Relationships between Relationships between Two Qualitative Two Qualitative VariablesVariables Frequency tableFrequency table – A table where – A table where

The rows represent values of one The rows represent values of one variable,variable,

The columns represent values of the The columns represent values of the other variable,other variable,

And the And the cellscells show the frequency of show the frequency of the row-column combinations of the row-column combinations of values.values.

A frequency table is also called a A frequency table is also called a contingency tablecontingency table..

ExampleExample

Let the row variable be the Let the row variable be the student’s year in college.student’s year in college.

Let the column variable be Let the column variable be whether the student is from whether the student is from Virginia or is from out of state.Virginia or is from out of state.

This will be a 4 x 2 frequency This will be a 4 x 2 frequency table.table.

ExampleExample

(frequency)(frequency) VirginiaVirginia Out of StateOut of State

FreshmanFreshman 00 00

SophomoreSophomore 00 00

JuniorJunior 00 00

SeniorSenior 00 00

Frequency TablesFrequency Tables

If there is a relationship between If there is a relationship between the variables, then perhaps it will the variables, then perhaps it will be apparent from the table.be apparent from the table.

Perhaps not.Perhaps not. Do we see any relationship Do we see any relationship

between year in college and state between year in college and state of residence?of residence?

ExampleExample

(percentage)(percentage) VirginiaVirginia Out of StateOut of State

FreshmanFreshman 00 00

SophomoreSophomore 00 00

JuniorJunior 00 00

SeniorSenior 00 00

ExampleExample

See example on page 199.See example on page 199.

Nutritional StatusNutritional Status

PoorPoor AdeqAdeq..

Exc.Exc.

AcademicAcademic

PerformanPerformancece

BelowBelow 7070 9595 3535

AveragAveragee

130130 450450 3030

AboveAbove 9090 3030 7070

ExampleExample

Is there any apparent relationship Is there any apparent relationship between academic performance between academic performance and nutritional status?and nutritional status?

It is hard to say (in my opinion).It is hard to say (in my opinion). A possible relationship is that A possible relationship is that

students with better nutrition students with better nutrition perform better academically.perform better academically.

Excel Bar GraphExcel Bar Graph

Academic Performance vs. Nutritional Status

0

100

200

300

400

500

Poor Adequate Excellent

Nutritional Status

Fre

qu

ency Below average

Average

Above average

Excel Bar GraphExcel Bar Graph

Academic Performance vs. Nutritional Status

0

100

200

300

400

500

Below average Average Above average

Academic Performance

Fre

qu

ency Poor

Adequate

Excellent

The Marginal The Marginal DistributionDistribution Each variable has a Each variable has a marginal marginal

distributiondistribution.. To find the marginal distribution To find the marginal distribution

of a variable, find the total of a variable, find the total frequency of the cells for each frequency of the cells for each value of that variable.value of that variable.

Then express each total Then express each total frequency as a percentage of the frequency as a percentage of the grand total for all cells.grand total for all cells.

The Marginal The Marginal Distribution of Distribution of Nutritional StatusNutritional Status

Nutritional StatusNutritional Status

PoorPoor AdeqAdeq..

Exc.Exc.

AcademicAcademic

PerformancPerformancee

BelowBelow 7070 9595 3535

AveragAveragee

130130 450450 3030

AboveAbove 9090 3030 7070

TotalTotal 290290 575575 135135

ExampleExample

The grand total of frequencies is The grand total of frequencies is 1000.1000.

The marginal distribution for The marginal distribution for nutritional status isnutritional status is

PoorPoor AdequatAdequatee

ExcellentExcellent

29%29% 57.5%57.5% 13.5%13.5%

The Marginal The Marginal Distribution of Distribution of Academic PerformanceAcademic Performance

Nutritional StatusNutritional StatusTotaTota

llPoorPoor AdeqAdeq..

Exc.Exc.

AcademicAcademic

PerformancPerformancee

BelowBelow 7070 9595 3535 200200

AveragAveragee

130130 450450 3030 610610

AboveAbove 9090 3030 7070 190190

ExampleExample

The marginal distribution for The marginal distribution for academic performance isacademic performance is

BelowBelow AverageAverage AboveAbove

?????? ?????? ??????

The Marginal The Marginal DistributionDistribution The marginal distribution shows The marginal distribution shows

us the distribution of one variable us the distribution of one variable independently of the other independently of the other variable.variable.

Conditional Conditional DistributionsDistributions In the example, In the example,

What percentage of all students are What percentage of all students are below average academically and below average academically and have poor nutrition?have poor nutrition?

What percentage of students What percentage of students who who are below average academicallyare below average academically have poor nutrition?have poor nutrition?

What percentage of students What percentage of students who who have poor nutritionhave poor nutrition are below are below average academically?average academically?

Conditional Conditional DistributionsDistributions The answers areThe answers are

70/1000 = 7%70/1000 = 7% 70/200 = 35%70/200 = 35% 70/290 = 24%70/290 = 24%

Conditional Conditional DistributionsDistributions To get the To get the conditional distributionconditional distribution

of academic performance of academic performance givengiven nutritional status,nutritional status, For each category of nutritional For each category of nutritional

status (i.e., for each column), divide status (i.e., for each column), divide the various frequencies in that the various frequencies in that category by the total for that category by the total for that category.category.

Conditional Conditional DistributionsDistributions The conditional distribution of The conditional distribution of

academic performance given academic performance given nutritional status isnutritional status is

Nutritional StatusNutritional Status

PoorPoor OkayOkay GoodGood

AcademicAcademic

PerformancPerformancee

BelowBelow 24%24% 17%17% 26%26%

AveragAveragee

45%45% 78%78% 22%22%

AboveAbove 31%31% 5%5% 52%52%

Conditional Conditional DistributionsDistributions The conditional distribution of The conditional distribution of

nutritional status given academic nutritional status given academic performance isperformance is

Nutritional StatusNutritional Status

PoorPoor OkayOkay GoodGood

AcademicAcademic

PerformancPerformancee

BelowBelow ?????? ?????? ??????

AveragAveragee

?????? ?????? ??????

AboveAbove ?????? ?????? ??????

Let's Do It!Let's Do It!

Let's do it! 4.8, p. 203 – Beer Let's do it! 4.8, p. 203 – Beer Tastes.Tastes.

Let’s do it! 4.9, p. 205 – About Let’s do it! 4.9, p. 205 – About Your Class.Your Class. Use the data concerning year in Use the data concerning year in

college vs. whether in or out of college vs. whether in or out of state.state.

AssignmentAssignment

Page 206: Exercises 12 – 17.Page 206: Exercises 12 – 17. Page 249: Exercises 62 – 66.Page 249: Exercises 62 – 66.