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Copyright 2009Pearson Education, Inc.

Chapter 3Displaying and DescribingCategorical Data

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Slide 3- 2Copyright 2009Pearson Education, Inc.

Frequency Tables: Making Piles

n We can pile the data by counting the number ofdata values in each category of interest.


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n

We can pile the data by countingthe number of data values ineach category of interest.

n

n

n

n This table is called a frequencytable. It records the totals andthe category names.


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n A relative frequency table is similar, but gives thepercentages (instead of counts) for eachcategory.


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Relative Frequency vs. Percentagen

The table on the previous page may be called arelative frequency table, but it is displayingpercentages, not relative frequencies.


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Relative Frequency vs. Percentagen

The table on the previous page may be called arelative frequency table, but it is displayingpercentages, not relative frequencies.

Class Frequency RelativeFrequency %First 325 0.1477 14.77

Second 285 0.1295 12.95

Third 706 0.3208

32.08

Crew 885 0.4021 40.21

Total 2201 1.000 100.0


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Whats Wrong

With ThisPicture?

n

You might thinkthat a goodway to showthe Titanic

data is withthis display:

n

n

Frequency



The Area Principle

n The ship displaymakes it looklike most ofthe people on

the Titanicwere crewmembers,with a few

passengersalong for theride.



The Area Principle

n When we look ateach ship, wesee the areataken up by the

ship, instead ofthe length ofthe ship.



The Area Principle

n The ship displayviolates the areaprinciple:



The Area Principle

n The ship displayviolates the areaprinciple:

n The areaoccupiedby a partof thegraph

shouldcorrespond to themagnitude

of thevalue it



Bar Charts

n A bar chart displays

the distribution of

a categorical

variable, showing

the counts foreach category

next to each

other for easy

comparison.

Watch my video on how t
http://web.utk.edu/~leon/stat201/fall09/Video/Bar%20Charts/Bar%20Charts.htmhttp://web.utk.edu/~leon/stat201/fall09/Video/Bar%20Charts/Bar%20Charts.htm



Bar Charts


the distribution of a

categorical variable,

showing the counts

for each categorynext to each other

for easy

comparison.

n A bar chart stays true

to the area

principle.



Bar Charts


the distribution of a

categorical variable,

showing the counts

for each categorynext to each other

for easy

comparison.

n A bar chart stays true

to the area

principle.

n

Thus, it is a betterdis la for the shi



Bar Charts

n A relative frequencybar chart displaysthe relativeproportion ofcounts for eachcategory.



Bar Charts

n A relative frequencybar chart displaysthe relativeproportion ofcounts for eachcategory.

n A relative frequencybar chart alsostays true to thearea principle.



Output from JMP Software

Bar Chart Relative Frequency Bar Chart



A Bar Chart with Sorted Categories168 Late Arrivals to Work

http://en.wikipedia.org/wiki/Pareto_chart
http://en.wikipedia.org/wiki/Pareto_charthttp://en.wikipedia.org/wiki/Pareto_charthttp://en.wikipedia.org/wiki/Pareto_chart




The

categoriesare orderedby theirfrequency




It is clear that thebest way to reducelate arrivals is byeliminating trafficrelated late arrival,

say, by leaving earlieror taking a fasterroute.


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It is clear that the bestway to reduce latearrivals is by eliminatingtraffic related late arrival,say, by leaving earlier ortaking a faster route.

The Pareto Chart isused in qualityimprovement effortsbecause it identifies

the high leverageactions that wouldimprove quality


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n When you are

interested in parts of

the whole, a pie chart

might be your display

of choice.n Pie charts show the

whole group of

cases as a circle.

Pie Charts


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n When you are interested

in parts of the whole, a

pie chart might be your

display of choice.

n Pie charts show the

whole group of cases

as a circle.

n They slice the circle into

pieces whose

size is

proportional to the

Pie Charts


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Pie Charts from JMP

Counts Percentages


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Contingency Tables


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Contingency Tables

A contingency table allows us to look at twocategorical variables together.


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Contingency Tables

It shows how individuals are distributed along eachvariable, contingent on the value of the other variable

Example: we can examine the class of ticket andwhether a person survived the Titanic:


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Contingency Tables

n The margins of thetable, both on the

right and on thebottom, givetotals and thefrequencydistributions foreach of the

variables.


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Contingency Tables

n The margins of the

table, both on theright and on thebottom, give totalsand the frequencydistributions foreach of the

variables.

The marginal distribution ofSurvivalis:


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Contingency Tables

n Each cell of the table gives the count for a combination ofvalues of the two values.


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Contingency Tables

n Each cell of the table gives the count for a combinationof values of the two values.

n For example, the second cell in the crew column tellsus that 673 crew members died when the Titanicsunk.

n


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Conditional Distributions

This is the conditional distribution of ticketClass, conditional on having survived


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n A conditional distribution shows the distribution ofone variable for just the individuals who satisfysome condition on another variable.

Conditional distribution of ticket Class, conditional on having survived


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Conditional Distributions (cont.)

n Conditional distribution of ticket Class,

conditional on having perished:n


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n The conditional distributions tell us that there is a

difference in class for those who survived andthose who perished.

n


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n The conditional distributions tell us that there is a

difference in class for those who survived andthose who perished.

n

Is this a goodway tocompare thetwo conditional

distributions?


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First

First

Second

Second

ThirdThirdCrew

Crew

The conditional distributions tell us that there is a difference in class forthose who survived and those who perished.

28.6% 8.2%


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The conditional distributions tell us that there is a difference in class forthose who survived and those who perished.

A pie chart of the two conditional distributions is better way to show that

the marginal distributions differ for the alive and the dead.


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n We see that the distribution ofClass for thesurvivors is different from that of the non-survivors.


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Conditional Distributions (cont.)n We see that the distribution ofClass for the survivors is

different from that of the non-survivors.n This leads us to believe that Class and Survivalare

associated, that they are not independent.

n


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The variables would be consideredindependentwhen the distribution ofone variable in a contingency table isthe same for all categories of the

other variable.

For example, X and Y would be independent


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The variables would be consideredindependent when the distribution ofone variable in a contingency table isthe same for all categories of the

other variable.



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The variables would be consideredindependentwhen the distribution ofone variable in a contingency table isthe same for all categories of the

other variable.


Actually the

percentages forX-A and X-B donot have to beexactly alikebecause if this

percentagescome from asample therewould be somesampling

variation


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Segmented Bar Charts

Second

SecondFirst

First

Third

Third

Crew

Crew


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Segmented Bar Charts

n A segmented barchart displaysthe sameinformation asa pie chart, butin the form ofbars instead ofcircles.

Second

SecondFirst

First

Third

Third

Crew

Crew


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JMPs Mosaic Plot

Unlike the graphic on the previous page, the MosaicPlot has bar widths proportional to the frequencies inthe category on the horizontal axis.


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Example: Spring 2009 Survey Data (from

100 randomly selected UT students)

n Gender:n female male

n Where do you sit in a classroom?:n near the frontn around the middlen

near the backn no preference


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Example: Spring 2009 Survey Data (from

100 randomly selected UT students)n Gender:

n female male

n

Where do you sit in a classroom?:n near the frontn around the middlen near the backn

no preference

1.Are this twovariables

independent?2.Do males and

females havedifferent sittingpreferences?


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Spring 2009 Survey Data

1.Are this twovariablesindependent?

2.Do males andfemales havedifferent sittingpreferences?


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Front Middle Back NoPref.

Total

Female 19 26 5 2 52

Male 15 18 10 5 48

Total 34 44 15 7 100

Where Do You Prefer to Sit?

Gender

1.Are this twovariablesindependent?

2.Do males andfemales havedifferentsittingpreferences?


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Total

Female 19 26 5 2 52

Male 15 18 10 5 48

Total 34 44 15 7 100


Gender

What percent of females who answered the surveysay that they prefer to sit near the back?


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Total

Female 19 26 5 2 52

Male 15 18 10 5 48

Total 34 44 15 7 100


Gender

Percentage of females who prefer to sit near theback = 5/52 = 9.6%


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Total

Female 19 26 5 2 52

Male 15 18 10 5 48

Total 34 44 15 7 100


Gender

What percent of the males who answered the surveysay that they prefer to sit near the back?


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Total

Female 19 26 5 2 52

Male 15 18 10 5 48

Total 34 44 15 7 100


Gender

Percentage of the males who prefer to sit near theback 10/48 =20.8%


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Spring 2009 Survey Data (Cont.)

n Percentage of females who prefer to sit near the

back = 15/100 = 9.6%n Percentage of males who prefer to sit near the

back 10/48 =20.8%

n


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n Percentage of students who prefer to sit near the

back = 15/100 = 15%n Percentage of males who prefer to sit near the

back 10/48 =21%

n

What doesthis differencesuggest?


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n Percentage of students who prefer to sit near the

back = 15/100 = 15%n Percentage of males who prefer to sit near the

back 10/48 =21%

n

Is this difference

big enough toconclude thatgender and sittingpreference aredependent?


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Foreshadowing Chapter 26 Methods


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Foreshadowing Chapter 26 MethodsTests of Hypotheses



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n Since the P-Values Prob>ChiSg are not smallerthan 0.05 we cannot conclude that there isdifference between males and females in sitting

preference



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n Since the P-Values Prob>ChiSg are not smallerthan 0.05 we cannot conclude that there isdifference between males and females in sitting

preferencen The differences we see could be the result of

sampling variation


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The Three Rules of Data Analysis

n The three rules of data analysis wont be difficult toremember:


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1. Make a picturethings may be revealed thatare not obvious in the raw data. These

will be things to thinkabout.


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1. Make a picturethings may be revealed thatare not obvious in the raw data. These

will be things to thinkabout.2. Make a pictureimportant features of and

patterns in the data will showup. Youmay also see things that you did notexpect.

3. Make a picturethe best way to tellothersabout your data is with a well-chosenpicture.


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Simpsons Paradox

n

Moe argues that hes a better pilot than Jilln Joe managed to land 83% of his last 120 flightn Jill managed to land 78% of her last 120

flights

Is Moe Rightin arguing

this way?

Simpsons Paradox - Another Example


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n Jill outperforms Moe for day flights (95% vs. 90%)

and

n She also outperforms Moe at night (75% vs. 50%).

Simpson s Paradox Another Example

Simpsons Paradox Example (Cont )


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Simpson s Paradox - Example (Cont.)

n Jill outperforms Moe for day flights (95% vs. 90%)and she outperforms Moe at night (75% vs.50%).

n But, Jill has a poorer on-time record than Moeoverall (78% vs. 83%)

Simpsons Paradox Example (Cont )


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Simpson s Paradox - Example (Cont.)

n Jill outperforms Moe for day flights (95% vs. 90%)and she outperforms Moe at night (75% vs.50%).

n But, Jill has a poorer on-time record than Moeoverall (78% vs. 83%).

n This seems to be a contradiction (or paradox).


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Simpsons Paradox - Example (Cont.)n

The explanation is thatn Jill has mostly night flights (more difficult)n Moe has mostly day flights (easier)n


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Simpsons Paradox - Example (Cont.)

Taking an overall average is misleading.

If you were Jill, what statistics would you report?


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Simpsons Paradox - Example (Cont.)

Taking an overall average is misleading.

If you were Jill, what statistics would you report?What about if you were Moe?


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Simpsons Paradox:Does It Matter in the Real World?

University of California sex-bias study in graduate admissions:

Is there a gender bias?

Si P d


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Simpsons Paradox:Does It Matter in the Real World?

University of California sex-bias study in graduate admissions:

Simpsons Paradox:


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pDoes It Matter in the Real World?

n Note that department by department if anythingthere seems to be discrimination againstmales!


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Resolving the Paradox

Women are applying to

departments that are harderto get in!!!


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What You Should Do?

Look at the variables separately

What Can Go


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Wrong?

n

Slight departuresfrom perfectindependencedo not provedependence

What Can Go


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Wrong?

n

Slight departuresfrom perfectindependencedo not provedependence

Since P-Values are not smaller than 0.05 we cannotconclude that the observed differences between males

and females is sufficient to disprove independence


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What Can Go Wrong?

n Dont use unfair or sillyaveragesthis couldlead to SimpsonsParadox


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n Do not violate the area principlen Which pie chart do you prefer?n

n

n

n

n

n

n

n

n

n

What Can Go Wrong?


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What Can Go Wrong? (cont.)

n Make sure your display shows what it says itshows

n

n

n

n

Percentage of high-school studentswho engage in specified dangerous behaviors


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What have we learned?

n We can summarize categorical data by countingthe number of cases in each category(expressing these as counts or percents).

n We can display the distribution in a bar chart or

pie chart.n We can examine two-way tables called

contingency tables, examining marginal and/orconditional distributions of the variables.

n If conditional distributions of one variable are thesame for every category of the other, thevariables are independent (i.e., not related).

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