Chapter 3

Graphical Methods for Describing Data

Graphs for categorical data

Double Bar Charts

When to Use Categorical data

How to construct– Constructed like bar charts, but with two (or

more) groups being compared– MUST use relative frequencies on the

vertical axis– MUST include a key to denote the different

barsWhy MUSTMUST we use relative frequencies?

Each year the Princeton Review conducts a survey of students applying to college and of parents of college applicants. In 2009, 12,715 high school students responded to the question “Ideally how far from home would you like the college you attend to be?” Also, 3007 parents of students applying to college responded to the question “how far from home would you like the college your child attends to be?” Data is displayed in the frequency table below.

Frequency

Ideal Distance Students Parents

Less than 250 miles 4450 1594

250 to 500 miles 3942 902

500 to 1000 miles 2416 331

More than 1000 miles 1907 180

Create a comparative bar chart with these data.

What should you do first?

Relative Frequency

Ideal Distance Students Parents

Less than 250 miles .35 .53

250 to 500 miles .31 .30

500 to 1000 miles .19 .11

More than 1000 miles .15 .06

Found by dividing the frequency by the total number of students

Found by dividing the frequency by the total number of parents

What does this graph show about the ideal distance college should be from home?

Segmented (or Stacked) Bar Charts

When to Use Categorical data

How to construct– MUST first calculate relative frequencies– Draw a bar representing 100% of the group– Divide the bar into segments corresponding

to the relative frequencies of the categories

Relative Frequency

Ideal Distance Students Parents

Less than 250 miles .35 .53

250 to 500 miles .31 .30

500 to 1000 miles .19 .11

More than 1000 miles .15 .06

Remember the Princeton survey . . .

Create a segmented bar graph with these data.

First draw a bar that

represents 100% of the

students who answered

the survey.

Less than 250 miles

250 to 500 miles

500 to 1000 miles

More than 1000 miles

Relative Frequency

Ideal Distance Students Parents

Less than 250 miles .35 .53

250 to 500 miles .31 .30

500 to 1000 miles .19 .11

More than 1000 miles .15 .06

First draw a bar that

represents 100% of the

students who answered

the survey. 0.2

0.4

0.6

0.8

1.0

Rela

tive f

requency

Students

Next, divide the bar into segments.

Do the same thing for

parents – don’t forget a key

denoting each category

Parents

Notice that this segmented bar chart displays the same relationship

between the opinions of students and parents concerning the ideal

distance that college is from home as the double bar chart does.

Pie (Circle) ChartWhen to Use Categorical data

How to construct– Draw a circle to represent the entire data

set– Calculate the size of each “slice”:

Relative frequency × 360° – Using a protractor, mark off each slice

To describe – comment on which category had the largest

proportion or smallest proportion

Typos on a résumé do not make a very good impression when applying for a job. Senior executives were asked how many typos in a résumé would make them not consider a job candidate. The resulting data are summarized in the table below.

Number of Typos

Frequency

Relative Frequency

1 60 .40

2 54 .36

3 21 .14

4 or more 10 .07

Don’t know 5 .03

Create a pie chart for these data.

Number of Typos

Frequency

Relative Frequency

1 60 .40

2 54 .36

3 21 .14

4 or more 10 .07

Don’t know 5 .03First draw a

circle to represent the entire data set.

Next, calculate the size of the

slice for “1 typo”

.40×360º =144º

Draw that slice.

Repeat for each slice.

Here is the completed pie chart created

using Minitab.

What does this pie chart tell us about the number of typos occurring in

résumés before the applicant would not be considered for a job?

How to describe a numerical, univariate

graph

What strikes you as the most distinctive difference among the distributions of exam scores in

classes A, B, & C ?

1. Center

• discuss where the middle of the data falls

• three measures of central tendency–mean, median, & mode

The mean and/or median is typically reported rather than

the mode.

What strikes you as the most distinctive difference among the

distributions of scores in classes D,

E, & F?

2. Spread

• discuss how spread out the data is

• refers to the variability in the data

• Measure of spread are–Range, standard deviation, IQR

Remember,Range = maximum value – minimum

value

Standard deviation & IQR will be discussed in Chapter 4

What strikes you as the most distinctive difference among the distributions of exam scores in

classes G, H, & I ?

3. Shape

• refers to the overall shape of the distribution

• symmetrical, uniform, skewed, or bimodal

The following slides will discuss these shapes.

Symmetrical

• refers to data in which both sides are (more or less) the same when the graph is folded vertically down the middle

•bell-shaped is a special type–has a center mound with two sloping tails

1. Collect data by rolling two dice and recording the sum of the two dice. Repeat three times.

2. Plot your sums on the dotplot on the board.

3. What shape does this distribution have?

Uniform

• refers to data in which every class has equal or approximately equal frequency

1. Collect data by rolling a single die and recording the number rolled. Repeat five times.

2. Plot your numbers on the dotplot on the board.

3. What shape does this distribution have?

To help remember the name for this shape,

picture soldier standing in

straight lines. What are they

wearing?

Skewed

• refers to data in which one side (tail) is longer than the other side

• the direction of skewness is on the side of the longer tail

1. Collect data finding the age of five coins in circulation (current year minus year of coin) and record

2. Plot the ages on the dotplot on the board.

3. What shape does this distribution have?The directions are positively (or right) skewed or negatively (or

left) skewed.

Name a variable with a distribution that is negatively skewed.

Bimodal (multi-modal)

• refers to the number of peaks in the shape of the distribution

• Bimodal would have two peaks• Multi-modal would have more

than two peaksBimodal distributions can occur when the data set consist of observations

from two different kinds of individuals or objects.

Suppose collect data on the time it takes to drive from San Luis Obispo, California to Monterey, California. Some people may take the inland route (approximately 2.5 hours) while others may take the coastal route (between 3.5 and 4 hours).

What shape would this distribution have?

What would a distribution be called if it had ONLY one peak?Unimodal

What strikes you as the most distinctive difference among the

distributions of exam scores in class J ?

4. Unusual occurrences

• Outlier - value that lies away from the rest of the data

• Gaps

• Clusters

5. In context• You must write your answer in

reference to the context in the problem, using correct statistical vocabulary and using complete sentences!

Graphs for numerical data

Stem-and-Leaf DisplaysWhen to Use Univariate numerical data

How to construct– Select one or more of the leading digits for the

stem– List the possible stem values in a vertical

column– Record the leaf for each observation beside

each corresponding stem value– Indicate the units for stems and leaves in a key

or legend

To describe – comment on the center, spread, and shape of the

distribution and if there are any unusual features

Each number is split into two parts:

Stem – consists of the first digit(s)Leaf - consists of the final digit(s)

Use for small to moderate sized

data sets. Doesn’t work well for large data sets.

Be sure to list every stem

from the smallest to the largest value

If you have a long lists of leaves behind a few

stems, you can split stems in order to

spread out the distribution.

Can also create comparative stem-and-leaf displays

Remember the data set collected in Chapter 1 – how many piercings do you have? Would a stem-and-leaf display be a good graph for this distribution? Why or

why not?

The following data are price per ounce for various brands of different brands of dandruff shampoo at a local grocery store.

0.32 0.21 0.29 0.54 0.17 0.28 0.36 0.23

Create a stem-and-leaf display with this data? Stem Leaf

1

2

3

4

5

What would an appropriate stem be?

List the stems vertically

For the observation of “0.32”, write the

2 behind the “3” stem.

2

Continue recording each leaf with the

corresponding stem

1 9

4

7

8

6

3

Describe this distribution.

The median price per ounce for dandruff

shampoo is $0.285, with a range of $0.37. The

distribution is positively skewed with an outlier

at $0.54.

The Census Bureau projects the median age in 2030 for the 50 states and Washington D.C. A stem-and-leaf display is shown below.

Notice that you really cannot see a distinctive

shape for this distribution due to the long list of

leaves

We can split the stems in order to better see the

shape of the distribution.

Notice that now you can see the

shape of this distribution.

We use L for lower leaf values (0-4) and H for higher leaf

values (5-9).

The following is data on the percentage of primary-school-aged children who are enrolled in school for 19 countries in Northern Africa and for 23 countries in Central African.

Northern Africa54.6 34.3 48.9 77.8 59.6 88.5 97.4 92.5 83.9 98.891.6 97.8 96.1 92.2 94.9 98.6 86.6 96.9 88.9

Central Africa58.3 34.6 35.5 45.4 38.6 63.8 53.9 61.9 69.9 43.085.0 63.4 58.4 61.9 40.9 73.9 34.8 74.4 97.461.0 66.7 79.6

Create a comparative stem-and-leaf display. What is an appropriate

stem?

Let’s truncate the leaves to the unit place.

“4.6” becomes “4”

Be sure to use comparative language when describing

these distributions!

The median percentage of primary-school-aged children enrolled in school is larger for countries in Northern Africa than in Central Africa, but the ranges are the same. The

distribution for countries in Northern Africa is strongly negatively skewed, but the

distribution for countries in Central Africa is approximately symmetrical.

HistogramsWhen to Use Univariate numerical

data

How to construct Discrete data―Draw a horizontal scale and mark it with the

possible values for the variable―Draw a vertical scale and mark it with frequency

or relative frequency―Above each possible value, draw a rectangle

centered at that value with a height corresponding to its frequency or relative frequency

To describe – comment on the center, spread, and shape of the

distribution and if there are any unusual features

Constructed differently for

discrete versus continuous data

For comparative histograms – use two separate graphs with the same scale on the horizontal axis

Queen honey bees mate shortly after they become adults. During a mating flight, the queen usually takes several partners, collecting sperm that she will store and use throughout the rest of her life. A study on honey bees provided the following data on the number of partners for 30 queen bees.

12 2 4 6 6 7 8 7 8 11 8 3 5 6 7 10 1 9 7 6 9 7 5 4 7 4 6 7 8 10

Create a histogram for the number of partners of the queen bees.

First draw a horizontal

axis, scaled with

the possible values of

the variable of interest.

Next draw a vertical

axis, scaled

with frequency or relative frequency.

Draw a rectangle

above each value with a

height correspondin

g to the frequency.

Suppose we use relative frequency instead of frequency

on the vertical axis.What do you notice about the shapes of these two histograms?

0 1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

HistogramsWhen to Use Univariate numerical data

How to construct Continuous data―Mark the boundaries of the class intervals on the

horizontal axis―Draw a vertical scale and mark it with frequency or

relative frequency―Draw a rectangle directly above each class interval

with a height corresponding to its frequency or relative frequency

To describe – comment on the center, spread, and shape of the

distribution and if there are any unusual features

This is the type of histogram that most students are familiar

with.

A study examined the length of hours spent watching TV per day for a sample of children age 1 and for a sample of children age 3. Below are comparative histograms.

Children Age 1 Children Age 3

Notice the common scale on the horizontal axis

Write a few sentences comparing the distributions.

The median number of hours spent watching TV per day was greater for the 1-year-olds than for the 3-year-olds. The distribution for the 3-year-olds was more

strongly skewed right than the distribution for the 1-year-olds, but the two distributions had similar ranges.

Histograms with unequal intervals

When to use- when you have a concentration of data in the middle with some extreme values

How to construct- construct similar to histograms with continuous data, but with density on the vertical axis

interval of widthinterval for frequency relative

density

Cumulative Relative Frequency Plot

When to use- used to answer questions about percentiles.

How to construct- Mark the boundaries of the intervals on the horizontal axis- Draw a vertical scale and mark it with relative frequency- Plot the point corresponding to the upper end of each interval with its cumulative relative frequency, including the beginning point- Connect the points.

Percentiles are a value with a given percent of

observations at or below that value.

The National Climatic Center has been collecting weather data for many years. The annual rainfall amounts for Albuquerque, New Mexico from 1950 to 2008 were used to create the frequency distribution below.

Annual Rainfall

(in inches)

Relative frequency

Cumulative relative frequency

4 to <5 0.052

5 to <6 0.103

6 to <7 0.086

7 to <8 0.103

8 to <9 0.172

9 to <10 0.069

10 to < 11 0.207

11 to <12 0.103

12 to <13 0.052

13 to <14 0.052

Find the cumulative relative frequency for

each interval

0.052

0.155

0.241

+

+

Continue this pattern to

complete the table

The National Climatic Center has been collecting weather data for many years. The annual rainfall amounts for Albuquerque, New Mexico from 1950 to 2008 were used to create the frequency distribution below.

Annual Rainfall

(in inches)

Relative frequency

Cumulative relative frequency

4 to <5 0.052 0.052

5 to <6 0.103 0.155

6 to <7 0.086 0.241

7 to <8 0.103 0.344

8 to <9 0.172 0.516

9 to <10 0.069 0.585

10 to < 11 0.207 0.792

11 to <12 0.103 0.895

12 to <13 0.052 0.947

13 to <14 0.052 0.999

In the context of this problem, explain the meaning of

this value.

In the context of this problem, explain the meaning of

this value.

Why isn’t this value one (1)?

To create a cumulative relative frequency plot, graph a point for the upper value of the interval and the

cumulative relative frequency

Plot a point for each interval. Plot a starting point at (4,0).

Connect the points.

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1.0

Rainfall

Cum

ula

tive r

ela

tive f

requen

cy

What proportion of years had rainfall amounts that were 9.5 inches or less?

Approximately 0.55

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1.0

Rainfall

Cum

ula

tive r

ela

tive f

requen

cy

Approximately 30% of the years had annual rainfall less than what amount?

Approximately 7.5 inches

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1.0

Rainfall

Cum

ula

tive r

ela

tive f

requen

cy

Which interval of rainfall amounts

had a larger proportion of

years –9 to 10 inches or 10 to 11 inches?

Explain The interval 10 to 11 inches, because its slope is steeper, indicating a larger proportion occurred.

Displaying Bivariate Numerical Data

ScatterplotsWhen to Use Bivariate numerical data

How to construct - Draw a horizontal scale and mark it with

appropriate values of the independent variable - Draw a vertical scale and mark it appropriate

values of the dependent variable - Plot each point corresponding to the

observations

To describe - comment the relationship between the variables

Scatterplots are discussed in much greater depth in

Chapter 5.

Time Series PlotsWhen to Use

- measurements collected over time at regular intervalsHow to construct

- Draw a horizontal scale and mark it with appropriate values of time

- Draw a vertical scale and mark it appropriate values of the observed variable

- Plot each point corresponding to the observations and connect

To describe - comment on any trends or patterns over time

Can be considered bivariate data where the y-variable is the variable measured

and the x-variable is time

The accompanying time-series plot of movie box office totals (in millions of dollars) over 18 weeks in the summer for 2001 and 2002 appeared in USA Today (September 3, 2002).

Describe any trends or patterns that you see.