# Long Island University Applied business Statistics Worksheet

Create bar chart, pie chart, and Pareto diagram for “How do you spend the holidays” data.

Chapter 2

Organizing and

Visualizing Variables

Learning Objectives

In this chapter, you will learn:

▪ To develop tables and charts for categorical

data

▪ To develop tables and charts for numerical

data

▪ The principles of properly presenting graphs

Chap 2-2

Organizing Categorical Data:

Summary Table

▪

A summary table indicates the frequency, amount, or

percentage of items in a set of categories so that you can see

differences between categories.

How do you spend the holidays?

Percent

At home with family

45%

Travel to visit family

38%

Vacation

5%

Catching up on work

5%

Other

7%

Chap 2-3

Visualizing Categorical Data:

Bar Chart

▪

In a bar chart, a bar shows each category, the length of

which represents the amount, frequency or percentage of

values falling into a category.

How Do You Spend the Holidays?

Other

7%

Catching up on w ork

5%

Vacation

5%

Travel to visit family

38%

At home w ith family

45%

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

Chap 2-4

Visualizing Categorical Data:

Pie Chart

▪

The pie chart is a circle broken up into slices that represent

categories. The size of each slice of the pie varies according

to the percentage in each category.

How Do You Spend the Holiday’s

5%

7%

5%

At home with family

45%

Travel to visit family

Vacation

Catching up on work

Other

38%

Chap 2-5

Visualizing Categorical Data:

Pareto Diagram

▪ Used to portray categorical data

▪ A bar chart, where categories are shown in

descending order of frequency

▪ A cumulative polygon is shown in the same graph

▪ Used to separate the “vital few” from the “trivial

many”

Chap 2-6

Visualizing Categorical Data:

Pareto Diagram

How Do You Spend the Holidays?

50%

100%

100%

95%

83%

40%

Percentage

90%

90%

80%

35%

70%

30%

60%

25%

50%

45%

20%

40%

38%

15%

30%

10%

20%

5%

7%

Cumulative Percentage

45%

10%

5%

5%

Vacation

Catching up on work

0%

0%

At home with family

Travel to visit family

Other

Chap 2-7

Organizing Numerical Data:

Ordered Array

▪

An ordered array is a sequence of data, in rank order, from

the smallest value to the largest value.

Age of

Surveyed

College

Students

Day Students

16

17

17

18

18

18

19

19

20

20

21

22

22

25

27

32

38

42

Night Students

18

18

19

19

20

21

23

28

32

33

41

45

Chap 2-8

Organizing Numerical Data:

Stem and Leaf Display

▪ A simple way to see how the data are distributed

and where concentrations of data exist.

METHOD: Separate the sorted data series

into leading digits (the stems) and

the trailing digits (the leaves).

Organizing Numerical Data:

Stem and Leaf Display

▪

A stem-and-leaf display organizes data into groups (called

stems) so that the values within each group (the leaves)

branch out to the right on each row.

Age of College Students

Day Students

Night Students

Stem Leaf

Stem Leaf

1

67788899

1

8899

2

0012257

2

0138

3

28

3

23

4

2

4

15

Chap 2-10

Organizing Numerical Data:

Frequency Distribution

▪

The frequency distribution is a summary table in which the

data are arranged into numerically ordered class groupings.

▪

You must give attention to selecting the appropriate number of

class groupings for the table, determining a suitable width of a

class grouping, and establishing the boundaries of each class

grouping to avoid overlapping.

▪

To determine the width of a class interval, you divide the

range (Highest value–Lowest value) of the data by the number

of class groupings desired.

Chap 2-11

Organizing Numerical Data:

Frequency Distribution Example

Example: A manufacturer of insulation randomly selects 20

winter days and records the daily high temperature

24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27

Chap 2-12

Organizing Numerical Data:

Frequency Distribution Example

▪ Sort raw data in ascending order:

12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

▪ Find range: 58 – 12 = 46

▪ Select number of classes: 5 (usually between 5 and 15)

▪ Compute class interval (width): 10 (46/5 then round up)

▪ Determine class boundaries (limits): 10, 20, 30, 40, 50, 60

▪ Compute class midpoints: 15, 25, 35, 45, 55

▪ Count observations & assign to classes

Chap 2-13

Organizing Numerical Data:

Frequency Distribution Example

Class

10 but less than 20

20 but less than 30

30 but less than 40

40 but less than 50

50 but less than 60

Total

Frequency

Relative

Frequency

Percentage

3

6

5

4

2

20

.15

.30

.25

.20

.10

1.00

15

30

25

20

10

100

Chap 2-14

Organizing Numerical Data:

Cumulative percentage distribution

Chap 2-15

Why Use a Frequency Distribution?

▪ It condenses the raw data into a more

useful form.

▪ It allows for a quick visual interpretation of

the data.

▪ It enables the determination of the major

characteristics of the data set including

where the data are concentrated /

clustered.

Frequency Distributions: Some Tips

▪

Different class boundaries may provide different pictures

for the same data (especially for smaller data sets).

▪

Shifts in data concentration may show up when different

class boundaries are chosen.

▪

As the size of the data set increases, the impact of

alterations in the selection of class boundaries is greatly

reduced.

▪

When comparing two or more groups with different

sample sizes, you must use either a relative frequency or

a percentage distribution.

Visualizing Numerical Data:

The Histogram

▪

A graph of the data in a frequency distribution is called a

histogram.

▪

The class boundaries (or class midpoints) are shown on the

horizontal axis.

▪

The vertical axis is either frequency, relative frequency, or

percentage.

▪

The horizontal axis display the variable of interest.

▪

Bars of the appropriate heights are used to represent the

number of observations within each class.

Chap 2-18

Visualizing Numerical Data:

The Histogram

10 but less than 20

20 but less than 30

30 but less than 40

40 but less than 50

50 but less than 60

Total

Frequency

3

6

5

4

2

20

Relative

Frequency

Percentage

.15

.30

.25

.20

.10

1.00

15

30

25

20

10

100

Histogram: Daily High

Temperature

7

6

6

Frequency

Class

5

5

4

4

3

3

2

2

1

0

0

0

5

15

25

35

45

55

More

Chap 2-19

Visualizing Numerical Data:

The Polygon

▪ A percentage polygon is formed by having the

midpoint of each class represent the data in that class

and then connecting the sequence of midpoints at

their respective class percentages.

▪ The cumulative percentage polygon, or ogive,

displays the variable of interest along the X axis, and

the cumulative percentages along the Y axis.

Chap 2-20

Visualizing Numerical Data:

The Polygon

Class

10 but less than 20

20 but less than 30

30 but less than 40

40 but less than 50

50 but less than 60

Total

Frequency

Relative

Frequency

Percentage

.15

.30

.25

.20

.10

1.00

15

30

25

20

10

100

3

6

5

4

2

20

Frequency Polygon: Daily High Tem perature

7

(In a percentage polygon

the vertical axis would

be defined to show the

percentage of

observations per class)

Frequency

6

5

4

3

2

1

0

5

15

25

35

45

55

More

Chap 2-21

Visualizing Numerical Data:

The Frequency Polygon

Useful When Comparing Two or More Groups

Visualizing Numerical Data:

The Percentage Polygon

Visualizing Numerical Data:

The Cumulative Percentage Polygon

Lower

Boundary

% Less Than

Lower Boundary

10

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