Chapter 3 3 Notes - Displaying and... · Copyright © 2010, 2007, 2004 Pearson Education, Inc....

Copyright © 2010, 2007, 2004 Pearson Education, Inc.

Chapter 3

Displaying and

Describing

Categorical Data

Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 3 - 3

The Three Rules of Data Analysis

The three rules of data analysis won’t be difficult to

remember:

1. Make a picture—things may be revealed that are

not obvious in the raw data. These will be things to

think about.

2. Make a picture—important features of and

patterns in the data will show up. You may also

see things that you did not expect like

extraordinary (possibly wrong) data values or

unexpected patterns.

3. Make a picture—the best way to tell others about

your data is with a well-chosen picture.


Frequency Tables: Making Piles

We can “pile” the data by counting the number of data values in each category of interest.

We can organize these counts into a frequency table, which records the totals and the category names.


Frequency Tables: Making Piles (cont.)

A relative frequency table is similar, but gives the percentages (instead of counts) for each category.


Frequency Tables: Making Piles (cont.)

Both types of tables show how cases are distributed across the categories.

They describe the distribution of a categorical variable because they name the possible categories and tell how frequently each occurs.


What’s Wrong With This Picture?

You might think that

a good way to show

the Titanic data is

with this display:


The Area Principle

The ship display makes it look like most of the people

on the Titanic were crew members, with a few

passengers along for the ride.

When we look at each ship, we see the area taken up

by the ship, instead of the length of the ship.

The ship display violates the area principle:

The area occupied by a part of the graph should

correspond to the magnitude of the value it

represents.


Bar Charts

A bar chart displays the distribution of a categorical variable,

showing the counts for each category next to each other for

easy comparison.

A bar chart obeys the

area principle.

Thus, a better display

for the ship data is:


Bar Charts (cont.)

A relative frequency bar chart displays the relative proportion of

counts for each category.

A relative frequency bar chart also obeys the area principle.

Replacing counts

with percentages

in the ship data:


When you are interested in parts of the whole, a pie chart

might be your display of choice.

Pie charts show the whole

group of cases as a circle.

They slice the circle into

pieces whose size is

proportional to the

fraction of the whole

in each category.

Pie Charts


Make sure to check…

If you want to make a relative frequency bar chart or a pie chart,

make sure the categories don’t overlap so that no individual is

counted twice.

Overlapping categories: Data is collected on students who

participated in high school sports. Students are asked what

sport(s) they play. One student may answer golf and tennis so

they are counted in two categories.

If the categories do overlap, you can still make a bar chart, but the

percentages won’t add up to 100%.

Try not to use a pie chart if the percentages are very close to each

other.

Do not violate the area principle!

Slide 3 - 12


Example (pg. 38 #6)

The pie chart shows

the ratings assigned to

120 first-run movies

released in 2005.

a) Is this an

appropriate display for

the genres? Why/why

not?

b) Which was the most

common rating?

Slide 3 - 13


Example (p. 38 #12)

An investigation compiled

information about recent

non military plane crashes.

The causes, to the extent

that they could be

determined, are

summarized in the table.

a) Is it reasonable to

conclude that the

weather or mechanical

failures caused only

about 20% of recent

plane crashes?

Cause Percent

Pilot Error 40

Other Human

Error

5

Weather 6

Mechanical

Failure

14

Sabotage 6


Example (p. 38 #12 cont)

An investigation compiled

information about recent

non military plane crashes.

The causes, to the extent

that they could be

determined, are

summarized in the table.

b) In what percent of

crashes were the causes

not determined?

c) Create an appropriate

display for these data.

Cause Percent

Pilot Error 40

Other Human

Error

5

Weather 6

Mechanical

Failure

14

Sabotage 6


Homework

Read p. 24-end

Chapter homework: p. 38 #5, 7, 11, 15

*11.c. just explain what chart you would create and why


Contingency Tables

A contingency table allows us to look at two categorical variables together.

It shows how individuals are distributed along each variable, contingent on (depending on) the value of the other variable.

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


Contingency Tables (cont.)

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

For example, the second cell in the crew column tells us that 673 crew members did not survive when the Titanicsunk.


Contingency Tables (cont.)

The margins of the table, both on the right and on the bottom, give totals and the frequency distributions for each of the variables.

Each frequency distribution is called a marginal distribution of its respective variable.

The marginal distribution of Survival is:


Example

What percent of the passengers were in second

class?

What percent of the second-class passengers

survived?

What percent of survivors were second-class

passengers ?

Slide 3 - 20


Conditional Distributions

A conditional distribution shows the distribution of

one variable for just the individuals who satisfy

some condition on another variable.

The following is the conditional distribution of

ticket Class, conditional on having survived:


Conditional Distributions (cont.)

The following is the conditional distribution of

ticket Class, conditional on having perished:



The conditional distributions tell us that there is a

difference in class for those who survived and those

who perished.

This is better

shown with

pie charts of

the two

distributions:



We see that the distribution of Class for the

survivors is different from that of the nonsurvivors.

This leads us to believe that Class and Survival

are associated (related), that they are not

independent. (Survival depended on ticket class)

The variables would be considered independent

when the distribution of one variable in a

contingency table is the same for all categories of

the other variable.


Example (p. 30 Just Checking)

A statistics

class reports

the following

data on Sex

and Eye

Color for

students in

the class.

1. What

percent of

females are

brown-eyed?

Eye Color

S

E

X

Blue Brown Green/

Hazel/

Other

Male 6 20 6

Female 4 16 12


Example (cont)

2. What percent of

brown-eyed

students are

female?

3. What percent of

students are

brown-eyed

females?

4. What’s the

marginal

distribution of Eye

Color?

Eye Color

S

E

X

Blue Brown Green

/Hazel

/Other

Total

Male 6 20 6 32

Female 4 16 12 32

Total 10 36 18 64


Example (cont)

5. What’s the conditional distribution of Eye Color for the males?

6. Compare the percent who are female among the blue-eyed students to the percent of all students who are female.

Eye Color

S

E

X

Blue Brown Green

/Hazel

/Other

Total

Male 6 20 6 32

Female 4 16 12 32

Total 10 36 18 64


Example (cont)

7. Does it seem

that Eye Color

and Sex are

independent?

Explain.

Eye Color

S

E

X

Blue Brown Green

/Hazel

/Other

Total

Male 6 20 6 32

Female 4 16 12 32

Total 10 36 18 64


Segmented Bar Charts

A segmented bar chartdisplays the same information as a pie chart, but in the form of bars instead of circles.

Each bar is treated as the “whole” and is divided proportionally into segments corresponding o the percentage in each group.

Here is the segmented bar chart for ticket Class by Survival status:


Don’t violate the area principle.

While some people might like the pie chart on the left better, it is harder to compare fractions of the whole, which a well-done pie chart does.

What Can Go Wrong?


What Can Go Wrong? (cont.)

Keep it honest—make sure your display shows what it says it shows.

This plot of the percentage of high-school students who engage in specified dangerous behaviors has a problem. Can you see it?



Don’t confuse similar-sounding percentages—pay particular attention to the wording of the context.

The percent of females who have brown eyes means something different than the percent of people who have brown eyes that are female.

Don’t forget to look at the variables separately too—examine the marginal distributions, since it is important to know how many cases are in each category.



Be sure to use enough individuals!

Do not make a report like “We found that

66.67% of the rats improved their

performance with training. The other rat died.”


What Can Go Wrong? (cont.) Don’t overstate your case—don’t claim something you

can’t.

We could say that it appears that there is a relationship

but we may not be able to say for sure.

Don’t use unfair or silly averages—this could lead to

Simpson’s Paradox, so be careful when you average one

variable across different levels of a second variable.

Make sure the groups are

comparable.


What have we learned?

We can summarize categorical data by counting

the number of cases in each category

(expressing these as counts or percents).

We can display the distribution in a bar chart or

pie chart.

And, we can examine two-way tables called

contingency tables, examining marginal and/or

conditional distributions of the variables.


Homework

p. 38 #20, 21, 23


Classwork

Worksheet

Homework: Read p. 44-49

Slide 3 - 37

Chapter 3 3 Notes - Displaying and... · Copyright © 2010, 2007, 2004 Pearson Education, Inc....

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