Math 227 Elementary Statistics

Bluman 5th edition

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CHAPTER 2

Frequency Distributions and Graphs

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Objectives

• Organize data using frequency distributions.

• Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives.

• Represent data using Pareto charts, time series graphs, and pie graphs.

• Draw and interpret a stem and leaf plot.

• Draw and Interpret a scatter plot for a set of paired data.

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Introduction

This chapter will show how to organize data

and then construct appropriate graphs to

represent the data in a concise, easy-to-

understand form.

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Section 2.1 Organizing Data

Basic Vocabulary

• When data are collected in original form, they are called raw data.

• A frequency distribution is the organization of raw data in table form, using classes and frequencies.

• The two most common distributions are categorical frequency distribution and the grouped frequency distribution.

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Frequency Distributions

Categorical Frequency Distributions

count how many times each distinct

category has occurred and

summarize the results in a table

format

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Example 1: Letter grades for Math 227 Spring 2005:

C A B C D F B B A C C F C

B D A C C C F C C A A C

a) Construct a frequency distribution for the categorical data.

Answer: Class Tally Frequency Percent

A //// 5 20

B //// 4 16

D // 2 8

F /// 3 12

Total 25 100

/ //// //// / C / 11 44

/

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b) What percentage of the students pass the class with

the grade C or better?

Answer: Total number of letter grade = 25

Number of grade C or better = 20

Percentage =

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Frequency Distributions

Group Frequency Distributions -

When the range of the data is large, the

data must be grouped into classes

that are more than one unit in width

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Grouped Frequency Distributions

The lower class limit represents the smallest value

that can be included in the class.

The upper class limit represents the largest value

that can be included in the class.

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Lower and Upper Class Limit

Lower Class

Limit

Class

Limits Frequency

24 - 30 3

31 - 37 1

38 - 44 5

45 - 51 9

52 - 58 6

Class

Limits Frequency

24 - 30 3

31 - 37 1

38 - 44 5

45 - 51 9

52 - 58 6

Upper Class

Limit

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Grouped Frequency

Distributions (cont.)

Class

Limits Frequency 23.5

24 - 30 3 30.5

31 - 37 1 37.5

38 - 44 5 44.5

45 - 51 9 51.5

52 - 58 6 58.5

Class

Limits Frequency 23.5

24 - 30 3 30.5

31 - 37 1 37.5

38 - 44 5 44.5

45 - 51 9 51.5

52 - 58 6 58.5

Class

Boundaries

The class boundaries are used to separate the classes so

that there are no gaps in the frequency distribution.

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• Rule of Thumb: Class limits should have the same decimal place value as the data, but the class boundaries have one additional place value and end in a 5.

e.g. data were whole numbers

lower class boundary = lower class limit – 0.5

upper class boundary = upper class limit + 0.5

e.g. data were one decimal place

lower class boundary = lower class limit – 0.05

upper class boundary = upper class limit +0.05

Class Boundaries Significant Figures

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Class Midpoints

•The class midpoint (mark) is found by

adding the lower and upper boundaries (or

limits) and dividing by 2.

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Class Midpoints

Class

Limits Frequency

24 27 30 3

31 34 37 1

38 41 44 5

45 48 51 9

52 55 58 6

Class

Midpoints

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Class Width

The class width for a class in a frequency

distribution is found by subtracting the lower

(or upper) class limit of one class from the

the lower (or upper) class limit of the next

class.

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Class Width

Class

Limits Frequency

24 - 30 3

7 31 - 37 1

7 38 - 44 5

7 45 - 51 9

7 52 - 58 6

Class

Width

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Class Rules

• There should be between 5 and 20 classes.

• The class width should be an odd number but not absolutely necessary.

• The classes must be mutually exclusive.

• The classes must be continuous.

• The classes must be exhaustive.

• The classes must be equal width.

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Class width as an odd

number The class width being an odd number is preferable since it

ensures that the midpoint of each class has the sample place value as the data.

If the class width is an even number, the midpoint is in tenths. For example, if the class width is 6 and the class limits are 6 and 11, the midpoint is:

*This is only a suggestion, and it is not rigorously followed.

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Relative Frequency Relative Frequency is the frequency of each class

divided by the total number.

relative frequency = class frequency

sum of all frequencies

Class Relative

Limits Frequency Frequency

24 - 30 3 3/24 =0.125

31 - 37 1 1/24 =0.042

38 - 44 5 5/24 =0.208

45 - 51 9 9/24 =0.375

52 - 58 6 6/24 =o.25

Total 24 1

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Cumulative Frequency

Cumulative Frequency is the sum of the

frequencies accumulated up to the upper

boundary of a class.

Class Cumulative

Limits Frequency Frequency

24 - 30 3 3

31 - 37 1 4

38 - 44 5 9

45 - 51 9 18

52 - 58 6 24

Total 24

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Procedure for constructing a

grouped frequency distribution 1. Decide on the number of classes you want. ( 5 to 20 classes)

2. Calculate (round up) the class width

*Round the answer up to the nearest whole number if there is a remainder: ex) the class width of 14.167 –round up to 15.

*If there is no remainder, you will need to add an extra class to accommodate all the data.

3. Choose a number for the lower limit of the first class

4. Use the lower limit of the first class and the class width to list

the other lower class limits.

5. Enter the upper class limits.

6. Tally the frequency for each class

class width (highest value) – (lowest value)

number of classes

range

number of classes =

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Example 1 : Construct a grouped frequency table for the

following data values.

44, 32, 35, 38, 35, 39, 42, 36, 36, 40, 51, 58

58, 62, 63, 72, 78, 81, 25, 84, 20

Tip: Consider reordering the data.

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Answer: 20, 25, 32, 35, 35, 36, 36, 38, 39, 40, 42, 44,

51, 58, 58, 62, 63, 72, 78, 81, 84

1. Let number of classes be 5

2. Range = High – Low = 84 – 20 = 64

Class width = 64 / 5 = 12.8 ≈ 13 (Round-up)

Class Tally Frequency

20 –

+13

33 –

+13

46 –

+13

59 –

+13

72 –

///

//// ////

///

//

////

/

3

9

3

2

4

32

45

58

71

84

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Ex) Complete the table.

Class Class Midpt Tally Frequency Cumulative Relative

limit boundaries frequency frequency

Total 21 1

20+32

2

=26

20 – 32

33 – 45

46 – 58

59 – 71

72– 84

19.5 – 32.5

32.5 – 45.5

45.5 – 58.5

58.5 – 71.5

71.5 – 84.5

39

52

65

78

///

//// ////

///

//

////

/

3

9

3

2

4

3

12

15

17

21

3

21

=0.14

0.43

0.14

0.10

0.19

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Frequency Distributions

An ungrouped frequency distribution is used

for numerical data and when the range of

data is small.

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Example: The number of incoming telephone calls

per day over the first 25 days of business:

4, 4, 1, 10, 12, 6, 4, 6, 9, 12, 12, 1, 1, 1,

12, 10, 4, 6, 4, 8, 8, 9, 8, 4, 1

Construct an ungrouped frequency distribution

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Answer:

Class limits Class Cumulative

Number of Calls boundaries Tally Frequency frequency

1 0.5 – 1.5 //// 5 5

2 1.5 – 2.5 0 5

3 2.5 – 3.5 0 5

4 3.5 – 4.5 //// / 6 11

5 4.5 – 5.5 0 11

6 5.5 – 6.5 /// 3 14

7 6.5 – 7.5 0 14

8 7.5 – 8.5 /// 3 17

9 8.5 – 9.5 // 2 19

10 9.5 – 10.5 // 2 21

11 10.5 –11.5 0 21

12 11.5 –12.5 //// 4 25

/ /

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Types of Frequency Distributions

(summary)

A categorical frequency distribution is used when the data is nominal.

• A grouped frequency distribution is used when the range is large and classes of several units in width are needed.

• An ungrouped frequency distribution is used for numerical data and when the range of data is small.

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Why Construct Frequency

Distributions? • To organize the data in a meaningful, intelligible way.

• To enable the reader to make comparisons among

different data sets.

• To facilitate computational procedures for measures of

average and spread.

• To enable the reader to determine the nature or shape of

the distribution.

• To enable the researcher to draw charts and graphs for

the presentation of data.

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Section 2.2 Histogram, Frequency

Polygons, Ogives

This chapter will show how to organize data

and then construct appropriate graphs to

represent the data in a concise, easy-to-

understand form.

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The Role of Graphs

• The purpose of graphs in statistics is to convey

the data to the viewer in pictorial form.

• Graphs are useful in getting the audience’s

attention in a publication or a presentation.

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Three Most Common Graphs

• The histogram displays the data by using vertical

bars of various heights to represent the

frequencies.

0

2

4

6

8

0 10.5 20.5 30.5 40.5 50.5 60.5 70.5 80.5

Class Boundaries

Fre

quency x-axis: class boundaries

y-axis: frequency

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Three Most Common Graphs

(cont’d.) • The frequency polygon displays the data by using

lines that connect points plotted for the

frequencies at the midpoints of the classes.

x-axis: midpoints

y-axis: frequency

0

1

2

3

4

5

6

7

8

0.5 10.5 20.5 30.5 40.5 50.5 60.5 70.5 80.5

Class Midpoints

Frequency

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Three Most Common Graphs

(cont’d.)

• The cumulative frequency or ogive

represents the cumulative frequencies for the

classes in a frequency distribution.

x-axis: class boundaries

y-axis: cumulative frequency

0

10

20

30

0 10.5 21 31.5 42 52.5 63 73.5 84 94.5

Class Boundaries

Cum

ula

tive

Fre

quen

cy

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Relative Frequency Graphs

• A relative frequency graph is a graph that

uses proportions instead of frequencies.

Relative frequencies are used when the

proportion of data values that fall into a given

class is more important than the frequency.

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Example 1 :

The following data are the number of the English-language Sunday Newspaper per state in the United States as of February 1, 1996.

2 3 3 4 4 4 4 4 5 6 6 6 7

7 7 8 10 11 11 11 12 12 13 14 14 14

15 15 16 16 16 16 16 16 18 18 19 21 21

23 27 31 35 37 38 39 40 44 62 85

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(for part b) (for part e) (for part c) (for part d)

a) Using 1 as the starting value and a class width of 15, construct a grouped

frequency distribution.

Class

Limit Freq

Class

Boundaries

Relative

Frequency

Cumulative

Frequency

Class

Midpoint

Cumulative

Relative

Frequency

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b) Construct a histogram for the grouped frequency

distribution.

(x-axis: class boundaries; y-axis: frequency)

Answer:

0

5

10

15

20

25

30

15.530.5

45.560.5

75.590.5

More

Class Boundaries

Fre

qu

en

cy

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c) Construct a frequency polygon.

(x-axis: class midpoints(marks); y-axis: frequency)

Answer:

0

5

10

15

20

25

30

-7 8 23 38 53 68 83 98

Class Marks

Fre

qu

en

cy

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d) Construct an ogive.

(x-axis: class boundaries; y-axis: cumulative frequency)

Answer:

0

10

20

30

40

50

60

0.5 15.5 30.5 45.5 60.5 75.5 90.5

Class Boundaries

Cu

mu

lati

ve F

req

uen

cy

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e) Construct a (i) relative frequency histogram,

(ii) relative frequency polygon,

and (iii) relative cumulative frequency ogive.

Answer:

(i) relative frequency histogram

(x-axis: class boundaries; y-axis: relative frequency)

0

0.1

0.2

0.3

0.4

0.5

0.6

15.530.5

45.560.5

75.590.5

More

Class Boundaries

Re

lati

ve

Fre

qu

en

cy

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(ii) relative frequency polygon

(x-axis: class midpoints (marks); y-axis: relative

frequency)

0

0.1

0.2

0.3

0.4

0.5

0.6

-7 8 23 38 53 68 83 98

Class Marks

Rela

tive F

req

uen

cy

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(iii) Ogive using relative cumulative frequency

(x-axis: class boundaries; y-axis: relative cumulative

frequency)

0

0.2

0.4

0.6

0.8

1

1.2

0.5 15.5 30.5 45.5 60.5 75.5 90.5

Class Boundary

Re

lati

ve

Cu

mu

lati

ve

Fre

qu

en

cy

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Distribution

shapes

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Section 2.3 Other Types of Graphs

A Pareto chart is used to represent a frequency distribution for

categorical variable, and the frequencies are displayed by the

heights of vertical bars, which are arranged in order from highest to

lowest.

(x-axis: categorical variables; y-axis: frequencies, which are arranged in

order from highest to lowest)

How People Get to Work

0

10

20

30

Auto Bus Trolley Train Walk

Fre

quency

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Other Types of Graphs (cont) A pie graph is a circle

that is divided into

sections or wedges

according to the

percentage of

frequencies in each

category of the

distribution.

Favorite American Snacks

potato

chips

38%

tortilla

chips

27%

pretzels

14%

popcorn

13%

snack nuts

8%

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Example 1: Grade received for Math 227

C A B B D C C C C B B A F F

a) Construct a pareto chart.

Answer: Grade Frequency

A 2

B 4

C 5

D 1

F 2

Next, arrange the frequency in descending order

Grade Frequency

C 5

B 4

A 2

F 2

D 1

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0

1

2

3

4

5

6

C B A F D

Grade

Nu

mb

er

Pareto chart

50

Grade Frequency

Relative

Frequency Degree

A 2

B 4

C 5

D 1

F 2

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b) Construct a pie chart.

Answer:

51

F (14%) A (14%)

D (7%)

B(29%)

C(36%)

Pie chart

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Other Types of Graphs (cont.)

• A time series graph represents data that occur

over a specific period of time.

Temperature Over a 5-hour Period

35

40

45

50

55

12 1 2 3 4 5

Time

Tem

p.

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Example 1: The percentages of voters voting in the last 5 Presidential

elections are shown here. Construct a time series graph.

Answer:

Year 1984 1988 1992 1996 2000

% of voters voting 74.63% 72.48% 78.01% 65.97% 67.50%

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

1984 1988 1992 1996 2000

Year

% o

f vo

ters

vo

tin

g

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Stem-and-Leaf Plots

• A stem-and-leaf plot is a data plot that uses part

of a data value as the stem and part of the data

value as the leaf to form groups or classes.

• It has the advantage over grouped frequency

distribution of retaining the actual data while

showing them in graphic form.

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Stem-and-Leaf Plots (cont) Digits of each data to the left of a vertical bar are called the stems.

Digits of each data to the right of the appropriate stem are called the leaves.

Example 1: The test scores on a 100-point test were recorded for 20 students

61 93 91 86 55 63 86 82 76 57

94 89 67 62 72 87 68 65 75 84

Construct an ordered stem-and-leaf plot

Answer: Reorder the data:

55 57 61 62 63 65 67 68 72 75 76 82 84 86 86 87 89 91 93 94

Stem Leaf

5 5 7

6 1 2 3 5 7 8

7 2 5 6

8 2 4 6 6 7 9

9 1 3 4

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Example 2 :

0 – 4 → belongs to the first stem 5 – 9 → belongs to the second stem

5

5 5 7

6 1 2 3

6 5 7 8

7 2

7 5 6

8 2 4

8 6 6 7 9

9 1 3 4

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Stem Leaf

Use the data in example 1 to construct a double stem and leaf plot.

e.g. split each stem into two parts, with leaves 0 – 4 on one part

and 5 – 9 on the other.

Answer:

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Stem-and-Leaf Plots (cont)

A stem-and-leaf plot portrays the shape of a

distribution and restores the original data values.

It is also useful for spotting outliers.

Outliers are data values that are extremely large or

extremely small in comparison to the norm.

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Misleading Graphs

Example: Graph of Automaker’s Claim

Using a Scale from 95% to 100% Using a Scale from 0% to 100%

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2.4 Paired Data and Scatter Plots

• Many times researchers are interested in determining if a relationship between two variables exist.

• To do this, the researcher collects data consisting of two measures that are paired with another.

• The variable first mentioned is called the independent variable; the second variable is the dependent variable.

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• Scatter Plot – is a graph of order pairs

values that is used to determine

if a relationship exists between two

variables.

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Analyzing the Scatter Plot

• A positive linear relationship exists when the points fall approximately in an ascending straight line and both the x and y values increase at the same time.

• A negative linear relationship exists when the points fall approximately in a straight line descending from left to right.

• A nonlinear relationship exists when the points fall along a curve.

• No relationship exists when there is no discernable pattern of the points.

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Examples of Scatter Plots and Relationships

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Example 1: A researcher wishes to determine if there is a relationship

between the number of days an employee missed a year and the

person’s age. Draw a scatter plot and comment on the nature of

the relationship.

Age (x) 22 30 25 35 65 50 27 53 42 58

Days missed (y) 0 4 1 2 14 7 3 8 6 4

0

2

4

6

8

10

12

14

16

0 20 40 60 80Age (x)

Da

ys

Mis

se

d (

y)

The relationship of the data

shows a positive linear

relationship.

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Summary of Graphs and Uses

• Histograms, frequency polygons, and

ogives are used when the data are

contained in a grouped frequency

distribution.

• Pareto charts are used to show

frequencies for nominal variables.

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Summary of Graphs and Uses

(cont.)

• Time series graphs are used to show a

pattern or trend that occurs over time.

• Pie graphs are used to show the

relationship between the parts and the

whole.

• When data are collected in pairs, the

relationship, if one exists, can be determined by

looking at a scatter plot.

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Conclusions

• Data can be organized in some

meaningful way using frequency

distributions. Once the frequency

distribution is constructed, the

representation of the data by graphs is a

simple task.