Topic 2 Frequency Distributions and Graphs

7/30/2019 Topic 2 Frequency Distributions and Graphs

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© The McGraw-Hill Companies, Inc., 2000

2-1

Topic 2Chapter 2: Frequency

Distributions and Graphs




2-2 Outline

2-1 Introduction

2-2 Organizing Data

2-3 Histograms, Frequency

Polygons, and Ogives

2-4 Other Types of Graphs




2-3 Objectives

Organize data using frequency

distributions. Represent data in frequency

distributions graphically using

histograms, frequency polygons,and ogives.




2-4 Objectives

Represent data using Pareto

charts, time series graphs, andpie graphs.




2-5 2-2 Organizing Data

When data are collected in original

form, they are called raw data. When the raw data is organized

into a frequency distribution, the

frequency will be the number ofvalues in a specific class of thedistribution.




2-6 2-2 Organizing Data

A frequency distribution is the

organizing of raw data in tableform, using classes andfrequencies.

The following slide shows anexample of a frequencydistribution.




2-7

2-2 Three Types of Frequency

Distributions

Categorical frequency distributions -

can be used for data that can beplaced in specific categories, suchas nominal- or ordinal-level data.

Examples - political affiliation,religious affiliation, blood type etc.




2-8

2-2 Blood Type Frequency

Distribution - Example (pg 2-4)

A

Class

B

Tally

C

Frequency

D

Percent

A ||||| 5 20

B ||||||| 7 28

O ||||||||| 9 36

AB |||| 4 16

TOTAL 25 100




Blood Type Frequency

Distribution - Example (pg 2-4)

Twenty-five army

inductees were given ablood test to determine

their blood type. The

data set is shown on

table. Construct afrequency distribution

for the data.

A B B AB O

O O B AB B

B B O A O

A O O O AB

AB A O B A




Procedure For Constructing

Frequency Distribution

Step 1:

Make a table as shown.Step 2:

Tally the data and place

the results in Col B.

Step 3:

Count the tallies and place

the results in Col C.

A

Class

B

Tally

C

Freq

D

PercentA

B

O

AB




Procedure For Constructing

Frequency Distribution

Step 4:

Find the percentage of values in each class by using theformula:

where f=frequency of the class and n=total number of values

Step 5:

Find the totals for columns C (Frequency) and D(Percent). The completed table is shown on slide 2-8.

%100.%n

f




2-9

2-2 Ungrouped Frequency

Distributions

Ungrouped frequency distributions -

can be used for data that can beenumerated and when the range ofvalues in the data set is not large.

Examples - number of miles your

instructors have to travel from home tocampus, number of girls in a 4-childfamily etc.




2-10

2-2 Number of Miles Traveled -

Example

Class Frequency

5 24

10 16

15 10




2-11

2-2 Grouped Frequency

Distributions

Grouped frequency distributions - can

be used when the range of values in thedata set is very large. The data must begrouped into classes that are more thanone unit in width.

Examples - the life of boat batteries inhours.




2-12

2-2 Lifetimes of Boat Batteries -

Example

C l a s s l i m

i t s C l a s s

B o u

n

d

a r i e s F r e q u e n c y C u m u l a t i v e

f r e q

u

e n

c y

2 4 - 3 7 2 3 . 5 - 3 7 . 5 4 4 3 8 - 5 1 3 7 . 5 - 5 1 . 5 1 4 1 8 5 2 - 6 5 5 1 . 5 - 6 5 . 5 7 2 5




2-13

2-2 Terms Associated with a

Grouped Frequency Distribution

Class limits represent the smallest and

largest data values that can be includedin a class.

In the lifetimes of boat batteriesexample, the values 24 and 37 of thefirst class are the class limits.

The lower class limit is 24 and the upper class limit is 37.




2-14

The class boundaries are used toseparate the classes so that there

are no gaps in the frequency

distribution.






2-15

The class width for a class in afrequency distribution is found by

subtracting the lower (or upper)

class limit of one class minus the

lower (or upper) class limit of theprevious class.






2-16

2-2 Guidelines for Constructing

a Frequency Distribution

There should be between 5 and 20

classes. It is preferable but not absolutely

necessary that the class width be an

odd number. The classes must be mutually

exclusive.




2-17

2-2 Guidelines for Constructing

a Frequency Distribution

The classes must be continuous.

The classes must be exhaustive.

(There should be enough classes to accommodate

all the data)

The class must be equal in width.




2-18

2-2 Procedure for Constructing

a Grouped Frequency Distribution

Find the highest and lowest value.

Find the range. Select the number of classes

desired.

Find the width by dividing the rangeby the number of classes androunding up.




2-19

Select a starting point (usually the

lowest value); add the width to get thelower limits.

Find the upper class limits.

Find the boundaries. Tally the data, find the frequencies, and

find the cumulative frequency.

2-2 Procedure for Constructing

a Grouped Frequency Distribution




2-20

In a survey of 20 patients who smoked,

the following data were obtained. Eachvalue represents the number ofcigarettes the patient smoked per day.Construct a frequency distributionusing six classes. (The data is given onthe next slide.)

2-2 Grouped Frequency Distribution -

Example




2-21

10 8 6 14

22 13 17 19

11 9 18 14

13 12 15 15

5 11 16 11


Example




2-22

Step 1: Find the highest and lowest

values: H = 22 and L = 5. Step 2: Find the range:

R = H – L = 22 – 5 = 17.

Step 3: Select the number of classesdesired. In this case it isequal to 6.


Example




2-23

Step 4: Find the class width by

dividing the range by the numberof classes. Width = 17/6 = 2.83.

This value is rounded up to 3.


Example




2-24

Step 5: Select a starting point for

the lowest class limit. Forconvenience, this value is chosen

to be 5, the smallest data value.

The lower class limits will be 5, 8,11, 14, 17, and 20.


Example




2-25

Step 6: The upper class limits will

be 7, 10, 13, 16, 19, and 22. Forexample, the upper limit for the

first class is computed as 8 - 1,

etc.


Example




2-26

Step 7: Find the class boundaries

by subtracting 0.5 from each lowerclass limit and adding 0.5 to the

upper class limit.


Example




2-27

Step 8: Tally the data, write the

numerical values for the tallies in thefrequency column, and find thecumulative frequencies.

The grouped frequency distributionis shown on the next slide.


Example




2-28

C l a s s L i m i t s C l a s s B o u n d a r i e s F r e q u e n c y C u m u l a t i v e F r e q u e n c y 0 5 t o 0 7 4 . 5 - 7 . 5 2 2 0 8 t o 1 0 7 . 5 - 1 0 . 5 3 5 1 1 t o 1 3 1 0 . 5 - 1 3 . 5 6 1 1 1 4 t o 1 6 1 3 . 5 - 1 6 . 5 5 1 6 1 7 t o 1 9 1 6 . 5 - 1 9 . 5 3 1 9 2 0 t o 2 2 1 9 . 5 - 2 2 . 5 1 2 0

Note: The dash “ -” represents “to”.




2-29



The three most commonly used

graphs in research are: The histogram.

The frequency polygon.

The cumulative frequency graph,

or ogive (pronounced o-jive).




2-30

The histogram is a graph that

displays the data by using verticalbars of various heights to

represent the frequencies.






2-31 Example of a Histogram

2 0171 41 185

6

5

4

3

2

1

0

N u m b e r o f C i g a r e t t e s S m o k e d p e r D a y

F r e q u e n c y




2-32

A frequency polygon is a graph

that displays the data by usinglines that connect points plottedfor frequencies at the midpoint of

classes. The frequenciesrepresent the heights of themidpoints.






2-33 Example of a Frequency Polygon

Frequency Polygon

262320171411852

6

5

4

3

2

1

0

Number of Cigar ettes Smoked per Day

F r e q u e n c y




2-34

A cumulative frequency graph or

ogive is a graph that representsthe cumulative frequencies for the

classes in a frequency distribution.






2-35 Example of an Ogive

262320171411852

20

10

0

Number of Cigarettes Smoked per Day

C u m

u l a t i v e F r e q u e n c y

Ogive




Example 2-7 on page 2-22

Construct a histogram, frequency

polygon, and ogive using relativefrequencies for the distribution of the

miles that 20 randomly selected runners

ran during a given week.

2-36




Example 2-7 on page 2-22Class

Boundaries

Frequency Cumulative

Frequency

5.5-10.5 1 110.5-15.5 2 3

15.5-20.5 3 6

20.5-25.5 5 11

25.5-30.5 4 15

30.5-35.5 3 18

35.5-40.5 2 20

Total 20

2-37




Example 2-7 on page 2-22Step 1:

Convert each frequency

to a relative frequencyby dividing thefrequency for eachclass by the totalnumber of observations.

Step 2:Find the cumulative

relative frequencies.

Step 3:

Draw each graph.

Class

Boundaries

Midpoints Relative

Freq

C. Relative

Freq

5.5-10.5 8 0.05 0.05

10.5-15.5 13 0.10 0.15

15.5-20.5 18 0.15 0.30

20.5-25.5 23 0.25 0.55

25.5-30.5 28 0.20 0.7530.5-35.5 33 0.15 0.90

35.5-40.5 38 0.10 1.00

1.00




Histogram for Runner's Times

0

0.05

0.1

0.15

0.2

0.25

5.5-

10.5

10.5-

15.5

15.5-

20.5

20.5-

25.5

25.5-

30.5

30.5-

35.5

35.5-

40.5

Miles

R e l a t i v e F r e

q u e n c y




Frequency Polygon for Runner's Times

0

0.05

0.1

0.15

0.2

0.25

3 8 13 18 23 28 33 38 43 48

Miles (midpoint)

R e l a t i v e F r e q

u e n c y




Ogive for Runner's Times

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5.5 11 16.5 22 27.5 33 38.5

Miles

C u m u l a t i v e R e l a t i v e F r e q u e n c y




2-38

2-4 Other Types of Graphs

Pareto charts - a Pareto chart is

used to represent a frequencydistribution for a categorical

variable.




2-392-4 Other Types of Graphs - Pareto

Chart

When constructing a Pareto chart -

Make the bars the same width. Arrange the data from largest to

smallest according to frequencies.

Make the units that are used for thefrequency equal in size.




2-40

Example of a Pareto Chart

1329341645.412.114.268.3

100.094.682.568.3

250

200

150

100

50

0

100

80

60

40

20

0

D ef ec t

C ountPer c entC um %

P e r c e n t

C o u n t

Enforcement Officers in U.S. National Parks During 1995.Par eto C har t f or the num ber of C r im es Inves tigated by Law





The table shown

here is the averagecost per mile for

passenger vehicles

on state turnpikes.

Construct andanalyze a Pareto

chart for the data.

State Number

Indiana 2.9 cents

Oklahoma 4.3 cents

Florida 6.0 cents

Maine 3.8 cents

Pennsylvania 5.8 cents

Source: Pittsburg Tribune Review.

2-41





Step 1:

Arrange the data from the

largest to smallestaccording to frequency.

Step 2:

Draw and label the x and y

axes.Step 3:

Draw the barscorresponding to thefrequencies.

State Number

Florida 6.0 cents

Pennsylvania 5.8 cents

Oklahoma 4.3 cents

Maine 3.8 cents

Indiana 2.9 cents




Average Cost per Mile on State Turnpikes

0

1

23

4

5

6

7

F l o r i d

a

P e n n

s y l v a n i a

O k l a h o

m a

M a

i n e

I n d i a n

a

State

C o s t ( c e n t s )




2-42

2-4 Other Types of Graphs –

Time Series Graph

Time series graph - A time series

graph represents data that occurover a specific period of time.




2-43

2-4 Other Types of Graphs -

Time Series Graph

1 9 9 41 9 9 31 9 9 21 9 9 11 9 9 0

8 9

8 7

8 5

8 3

8 1

7 9

7 7

7 5

Y e a r

R i d e

r s h i p ( i n m i l l i o

n s )

P O R T AU T H O R IT Y T R AN S IT R ID E R S H IP





The number (inmillion) of vehicles,

both passenger andcommercial, that usedthe PennsylvaniaTurnpike for the years

1999 through 2003 isshown. Construct andanalyze a time seriesgraph for the data.

Year Number

1999 156.2

2000 160.1

2001 162.3

2002 172.8

2003 179.4

Source: Tribune Review.

2-44





Step 1:

Draw and label the x and y axes.

Step 2:

Label the x axis for years and the y axis for the number of vehicles.

Step 3:

Plot each point according to table.

Step 4:

Draw line segments connecting adjacent points.




Number of Vehicles Traveling on Pennsylvania

Turnpike

150

160

170

180

1999 2000 2001 2002 2003

Year

N

u m b e r o f v e h

i c l e s ( i n

m i l l i o n s )




2-45

2-4 Other Types of Graphs – Pie

Graph

Pie graph - A pie graph is a circle

that is divided into sections orwedges according to the

percentage of frequencies in each

category of the distribution.




2-46

2-4 Other Types of Graphs -

Pie GraphRobbery (29, 12.1%)

Rape (34, 14.2%)

Assaults

(164, 68.3%)

Homicide

(13, 5.4%)

Pie Chart of the

Number of Crimes

Investigated by

Law EnforcementOfficers In U.S.

National Parks

During 1995





This frequency

distribution showsthe number of

pounds of each snack

food eaten during the

Super Bowl.Construct a pie graph

for the data.

Snack Pounds

(Frequency)

Potato chips 11.2 million

Tortilla chips 8.2 million

Pretzels 4.3 million

Popcorn 3.8 million

Snack nuts 2.5 million

Total 30.0 million

Source: USA TODAY Weekend.

2-47





Step 1:

Since there are 360o in a circle, the frequency for each

class must be converted into a proportional part of thecircle.

Step 2:

Each frequency must also be converted to a percentage.

Step 3:Using a protractor and a compass, draw the graph using

the appropriate degree measures.

o

n

f Degree 360.

%100.% n f





Snack Pounds

(frequency)

Degree Percentage

Potato chips 11.2 million (11.2/30)x

360o=134o

(11.2/30)x

100%=38%

Tortilla chips 8.2 million 98o 27%

Pretzels 4.3 million 52o 14%

Popcorn 3.8 million 46o 13%

Snack nuts 2.5 million 30o 8%

30.0 million




Super Bowl Snacks

Potato chips

38%

Tortila chips

27%

Pretzels

14%

Popcorn

13%

Snack nuts

8%




2-4 Other Types of Graphs –

Stem and Leaf Plots

A stem and leaf plot is a data that uses

part of the data value as the stem andpart of the data value as the leaf to

form groups or classes.

2-48





At an outpatient

testing center, the

number of

cardiograms

performed each day

for 20 days is shown.Construct a stem and

leaf plot for the data.

25 14 36 32

31 43 32 52

20 02 33 44

32 57 32 51

13 23 44 45

2-49





Step 1:

Arrange the data in order.02, 13, 14, 20, 23, 25, 31, 32, 32, 32,

32, 33, 36, 43, 44, 44, 45, 51, 52, 57





Step 2:

Separate the data according to the first digit.Step 3:

A display can be made by using the leading

digit as the stem and the trailing digit asthe leaf.




Leading digit (stem) Trailing digit (leaf)

0 | 21 | 3 4

2 | 0 3 5

3 | 1 2 2 2 2 3 64 | 3 4 4 5

5 | 1 2 7

Topic 2 Frequency Distributions and Graphs

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