copyright © 2004 by the mcgraw-hill companies, inc. all rights reserved. 2 - 1 describing data...

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

2 - 1

Describing DataDescribing DataDescribing DataDescribing Datarequency requency DistributionsDistributionsf

Graphic PresentationsGraphic Presentations



2 - 2

Organize raw data into frequency distribution

Produce a histogram, a frequency polygon, and a cumulative frequency polygon from quantitative data

Develop and interpret a stem-and-leaf display

When you have completed this chapter, you will be able to:


2 - 3

Present qualitative data using such graphical techniques such as a clustered bar chart, a stacked bar chart, and a pie chart

Detect graphic deceptions and use a graph to present data with clarity, precision, and efficiency


2 - 4


2 - 5

A Frequency Distribution is a grouping of data into

non-overlapping classes (mutually exclusive)…

showing the number of observations

in each category or

class.

The range of categories includes all values in the data set (collectively exhaustive classes).

The range of categories includes all values in the data set (collectively exhaustive classes).


2 - 6

Class Midpoint or Class Mark:A point that divides a class into two equal parts, i.e. the

average of the upper and lower class limits.

12.5

Class frequency:The number of observations in each class.

Class interval:The class interval is obtained by subtracting the lower limit of

a class from the lower limit of the next class, e.g.

517.522.527.532.5


2 - 7

Dr. Tillman is Dean of the School of Business. He wishes to prepare a report showing the number of hours per week students spend studying.

He selects a random sample of 30 students and determines the number of hours

each student studied last week.

15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.

Organize the data into a frequency distribution.


2 - 8

Decide how many classes you wish to use.

Frequency Distributions

by hand

Frequency Distributions

by hand

Determine the class width.

There are five steps that can be used to

Construct a Frequency Distribution:

Set up the individual class limits.

Tally the items into the classes.

Count the number of items in each class.


2 - 9Decide how many classes you wish to use

Use the 2 to the K rule.

Choose k so that 2 raised to the power of k is greater than the number of data points (n) or 30.

Rule of Thumb:

For most data sets, you would want

between 3 and 12 classes!

Rule of Thumb:

For most data sets, you would want

between 3 and 12 classes!

2k = 30 students25 = 32, so use k = about 5 classes

In this case…

In this case…


2 - 10Determine the class width

Generally, the class width should be the same size for all classes.

Class width >= Max - Min K

(33.8 – 10.3)/ 5 = 4.7

Therefore, use class size of 5 hours

Therefore, use class size of 5 hours

15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.

15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.

10.3,33.8,

Max Min

K=5K=5


2 - 11

Minimum Value is 10.3, therefore,

classes should start at 10 hours

Minimum Value is 10.3, therefore,

classes should start at 10 hours

10.0 – 14.915.0 – 19.920.0 – 24.925.0 – 29.930.0 – 34.9

10.0 – 14.915.0 – 19.920.0 – 24.925.0 – 29.930.0 – 34.9

Lower class limits

will be: 10, 15, 20, etc.

Lower class limits

will be: 10, 15, 20, etc.

Classes oror10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35

10.0 to under 1515.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35

Classes

Set up the individual class limits

15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.

15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.

10.3,33.8,

Class Width 5 hoursClass Width 5 hours


2 - 12

15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.

15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.

Tally the items into the classes



Classes TallyTally

…and so on with the remaining

hours

10.3,

13.514.213.7

14.0

12.9

12.9FindFind


2 - 13Count the number of items in each class



Hours Studying x Frequency f

7

12

7

3

1

30


2 - 14

Using different limitsUsing different limits

7.5 to under 12.512.5 to under 17.517.5 to under 22.522.5 to under 27.527.5 to under 32.532.5 to under 37.5


Hours Studying x Frequency f

11210

11

305

…will give you a different distribution, e.g.


2 - 15Construct a Frequency Distribution

Using Excel


2 - 16

Click on MegaStat

SeeSee

Click on Frequency

Distributions

Click on Frequency

Distributions

See…See…

Using

Click on Quantitative

Click on Quantitative


2 - 17Using

SeeSee

INPUT NEEDS INPUT NEEDS

$A:$A

5

10

See…See…


2 - 18

SeeSee

Using


2 - 19

Relative Frequency Distribution



2 - 20

RRelativeelative FFrequency requency DDistributionistributionRRelativeelative FFrequency requency DDistributionistribution

…shows the percent of observations in each class!

Hours Studying x f7

12

7

3

1

Relative f

3030



TotalTotal

7/30 = 0.2333

12/30 = 0.40

7/30 = 0.2333

3/30 = 0.10

1/30 = 0.0333

30/30 =130/30 =1


2 - 21

Using different limitsUsing different limits



Hours Studying x f Relative f

3030TotalTotal

1/30 = 0.0333 12/30 = 0.40 10/30 = 0.3333

1/30 = 0.0333 1/30 = 0.0333

30/30 =130/30 =1

11210

11

5 5/30 = 0.1666


2 - 22

Stem-and-leaf DisplaysStem-and-leaf DisplaysStem-and-leaf DisplaysStem-and-leaf Displays

Each numerical value is divided into two parts:

1. the leading digits become the stem and

2. the trailing digits become the leaf.

…an advantage of the stem-and-leaf display over a frequency distribution is

that we retain the value of each observation!

…an advantage of the stem-and-leaf display over a frequency distribution is

that we retain the value of each observation!

A statistical technique for displaying a set of data.


2 - 23

A student achieved the following scores on the twelve accounting

quizzes this semester: 86, 79, 92, 84, 69, 88, 91,

83, 96, 78, 82, 85.

Construct a stem-and-leaf chart to illustrate the results.



2 - 24


First, find the lowest score

86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85.

Now list the next scores with the highest leading digits.

You should now have the following STEMS:

69, 78, 82, 916 7 8 9


2 - 25

Stem

69

78

82

91

7

8

9

6

Split Leaf

6 9

7 8

8 2

9 1

Now, list the remaining ‘leaf’ scores!

Now, list the remaining ‘leaf’ scores!

9

3 4 5 8

2 6

6

All 12 Scores


86, 79, 92, 84, 69, 88,

91, 83, 96, 78, 82, 85.


2 - 26

The grades on a statistics exam for a sample of 40 students are as

follows:

The grades on a statistics exam for a sample of 40 students are as

follows:Stem Leaf

3 6 8

4 1 2 7 8

5 0 1 2 5 5 8 9

6 0 1 1 1 2 5 7 8 8 8 9

7 0 0 2 5 6 6 7

8 4 6 8 8 9

9 0 2 4 6

Stem Leaf

3 6 8

4 1 2 7 8

5 0 1 2 5 5 8 9

6 0 1 1 1 2 5 7 8 8 8 9

7 0 0 2 5 6 6 7

8 4 6 8 8 9

9 0 2 4 6

How many students

earned an A on this test?

How many students

earned an A on this test?

55

What is the most common

letter grade earned?

What is the most common

letter grade earned?

FF

A+ = 90%-100%

A = 80%-89%

B+ = 75%-79%

B = 70%-74%

C+ = 65%-69%

C = 60%-64%

D = 55%-59%

F = 0%-54%

A+ = 90%-100%

A = 80%-89%

B+ = 75%-79%

B = 70%-74%

C+ = 65%-69%

C = 60%-64%

D = 55%-59%

F = 0%-54%

Alpha-NumericGrading

Alpha-NumericGrading


2 - 27

Graphic Graphic Presentation of a Presentation of a

Frequency Frequency DistributionDistribution

Graphic Graphic Presentation of a Presentation of a

Frequency Frequency DistributionDistribution


2 - 28Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution

Graphic Presentation of a Graphic Presentation of a Frequency DistributionFrequency Distribution

The three commonly used graphic forms are:

Histograms

Frequency Polygons or Line Charts

Cumulative Frequency Distributions

The three commonly used graphic forms are:

Histograms

Frequency Polygons or Line Charts

Cumulative Frequency Distributions


2 - 29

The class frequencies are represented by the heights of the bars and

the bars are drawn adjacent to each other.

A Histogram is a graph in which the classes are marked on the horizontal axis

and the class frequencies on the

vertical axis

Fre

quen

cyClass




2 - 30



Hours Studying x f

712731



0 10 15 20 25 30 35 Hours spent studying

14

12

10

8

6

4

2

Fre

quen

cy

HistogramHistogram


2 - 31

A frequency polygon consists of

line segments connecting the points formed by the class midpoint and

the class frequency.

A frequency polygon consists of

line segments connecting the points formed by the class midpoint and

the class frequency.0

2

4

6

8

10

12

14

7.5 12.5 17.5 22.5 27.5

0

5

10

15

20

25

30

35

10 15 20 25 30 35

A cumulative frequency distribution is used to determine

how many or what proportion of the data

values are below or above a certain

value.

A cumulative frequency distribution is used to determine

how many or what proportion of the data

values are below or above a certain

value.




2 - 32

Making a Making a Histogram Histogram

in Excel in Excel

Making a Making a Histogram Histogram

in Excel in Excel


2 - 33Using

Click on DATA ANALYSIS

SeeSee

Click on HISTOGRAM


2 - 34

The upper limits of the classes you have determined

The upper limits of the classes you have determined

Using

Complete INPUTTING of DATAComplete INPUTTING of DATA

must now be entered from Column B (Excel calls these “bins”)


2 - 35

To remove the Legend on the right side… Right mouse click and Click on Clear

Using

To remove the spaces between the bars… Right mouse click on one of the bars and

Click on Format Data Series


2 - 36Using

Now, Click on the Options tab;

To reduce/remove the spaces between the barsAdjust the Gap width down to 0 and Click on OK.


2 - 37Using

Edit the size of the histogram, titles, etc

as appropriate.

Note that the upper limit values are included in each class – this explains the difference between this

Excel Frequency Distribution

and the one we did by hand.


2 - 38


14

12

10

8

6

4

2F

requ

ency

Frequency Polygon Frequency Polygon or or Line Chart Line Chart for Hours Spent Studyingfor Hours Spent Studying

Frequency Polygon Frequency Polygon or or Line Chart Line Chart for Hours Spent Studyingfor Hours Spent Studying


14

12

10

8

6

4

2F

requ

ency10.0 to under 15

15.0 to under 2020.0 to under 2525.0 to under 3030.0 to under 35


Hours Studying x f

712731

Notice that the class midpoints (the plotted points) aren’t as “user friendly” in this distribution choice.

Notice that the class midpoints (the plotted points) aren’t as “user friendly” in this distribution choice.


2 - 39



Hours Studying x f

712731

Cumulative Frequency Distribution Cumulative Frequency Distribution For Hours StudyingFor Hours Studying


under 15under 20under 25under 30under 35


Hours StudyingCumulative

f

19262930

7

Graph…..Graph…..


2 - 40Cumulative Frequency Distribution Cumulative Frequency Distribution For Hours StudyingFor Hours Studying



35

30

25

20

15

10

5

Fre

quen

cy



19262930

7

Hours StudyingHours Studying

Cumulative f

Cumulative f

Notice that the limits are the plotted points.Notice that the limits are the plotted points.


2 - 41

Pie Bar Line

Pie Bar Line… used primarily for Qualitative Data


2 - 42

…is useful for displaying a


PiePie

A circle is divided proportionally to the

relative frequency and portions of the circle are allocated for the

different groups.

A circle is divided proportionally to the

relative frequency and portions of the circle are allocated for the

different groups.


2 - 43

PiePie

200 runners were asked to indicate their favourite type of running

shoe. TypeType

Nike 92Adidas 49Reebok 37Asics 13Other 9

# of runners selecting:# of runners selecting:

Draw a pie chart based on this information.Draw a pie chart based on this information.


2 - 44


TypeType ##

200200

%46.024.518.56.54.5

100

Adidas 24.5% Nike46.0%

Reebok

18.5%

Asics6.5%

Other

4.5%

Relative Frequency Distribution for the running shoes

PiePie


2 - 45


TypeType ##

200200

%46.024.518.56.54.5

100

Using Excel, follow the steps in the Chart Wizard to construct a Pie Chart!

Using Excel, follow the steps in the Chart Wizard to construct a Pie Chart!

PiePie


2 - 46BarBar

…can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio).

(also known as a ‘column chart’)

Examples of…

3-D


2 - 47BarBar

Use bar charts also when the order

in which qualitative data are

presented is meaningful.

Use bar charts also when the order

in which qualitative data are

presented is meaningful.


2 - 48

How could we chart this data?

BarBar


2 - 49BarBar

Using Excel we can produce this…

Using Excel we can produce this…

Other formats…Other formats…


2 - 50

Employment Rate

Canadian City

Victoria 57.7

Halifax 60.5

Montreal 60.4

Sherbrooke 59.2

Quebec 59.7

Toronto 65.1

Hamilton 63.2

Kitchener 66.0

London 63.3

Thunder Bay 61.0

Regina 67.4

Saskatoon 63.7

Edmonton 67.1

Vancouver 61.4

Winnipeg 66.7

BarBar

Halifax

Montreal

Sherbro

oke

Quebec

Toronto

Hamilt

on

Kitchener

London

Thunder Bay

Regina

Saskatoon

Win

nipeg

Edmonto

n

Vancouver

Victoria

% e

mp

loym

ent

52

54

56

58

60

62

64

66

68

70

Employment Rate in Canadian Cities


2 - 51

Employment Rate

Canadian City

Victoria 57.7

Halifax 60.5

Montreal 60.4

Sherbrooke 59.2

Quebec 59.7

Toronto 65.1

Hamilton 63.2

Kitchener 66.0

London 63.3

Thunder Bay 61.0

Regina 67.4

Saskatoon 63.7

Edmonton 67.1

Vancouver 61.4

Winnipeg 66.7

BarBar


% e

mp

loym

ent

52

54

56

58

60

62

64

66

68

70

Halifax

Montreal

Sherbro

oke

Quebec

Toronto

Hamilt

on

Kitchener

London

Thunder Bay

Regina

Saskatoon

Win

nipeg

Edmonto

n

Vancouver

Victoria

- by Province


2 - 52BarBar

Did any of the previous Bar Charts adequately display

all the information that was provided?

Did any of the previous Bar Charts adequately display

all the information that was provided?

The following has been modified from that data found by Statistics Canada.

Does it do an effective job of displaying the StatCan data?


2 - 53

Real estate and rental and leasingProfessional, scientific and technical servicesManagement of companies and enterprisesEducational services (private sector)

Health care and social assistance (private sector)Administration and support, waste management and

remediation servicesArts, entertainment and recreationAccommodation and food services

All private sector

Information and cultural industriesFinance and insurance

ManufacturingWholesale trade

Retail trade

0

20

40

60

80

100

% o

f e

nte

rpri

ses

Clustered Bar

Comparison of Internet Use in 2000 and 2001Comparison of Internet Use in 2000 and 2001

% of enterprises thatuse the Internet 2000

% of enterprises thatuse the Internet 2001

% of enterprises with aWeb site 2000

% of enterprises with aWeb site 2001

Data Source: Statistics Canada


2 - 54

Full-Time University Faculty By Gender,Canada and Jurisdictions, 1987-88 and 1997-98

Stacked Bar

Canadian Full Time University Faculty

020406080

100120

1987-88 1997-98

% o

f T

ota

l

% males

% females

Data Source: Statistics Canada

Total

34,651 33,925 12,829 13,910 12,650 12,095 9,172 7,817

Full Professor Associate Professor

1987-88 1997-98 1987-88 1997-98 1987-88 1997-98

Other

1987-88 1997-98

% Male% Female 17

832575

793

1387

1783

2872

3268

4456


2 - 55

Make sure that your charts are

not overly cluttered

Make sure that your charts are

not overly cluttered


2 - 56

There are four typical shape characteristics

Shapesof

Histograms Modal

Class


2 - 57

…a balanced effect!

Both ‘balanced’ or ‘have symmetry’

Both ‘balanced’ or ‘have symmetry’


2 - 58

… occurs when the observations are graphed as being skewed or tilted more to one side of the centre

of the observations than the other.

… occurs when the observations are graphed as being skewed or tilted more to one side of the centre

of the observations than the other.

The skewness, if on the right side is said to be

‘positive’.

The skewness, if on the left side is said to be

‘negative’.


2 - 59

ClassModal

A modal class is the one with the largest number of observations

A modal class is the one with the largest number of observations

This is a uniModal HistogrambiModal


2 - 60

ClassModal

biModalbiModal

This is a biModal Histogram


2 - 61

Population distributions are often bell shaped. Drawing a histogram helps verify the shape

of the population in question.


2 - 62LineLine

Line charts are particularly useful when the trend over time is to be

emphasizedExamples …

3-D In combination


2 - 63

Time PlotTime PlotTime PlotTime Plot

LineLine

OSAJJMAMFJDNOSAJJMAMFJDNOSAJJMAMFJ

8.5

7.5

6.5

5.5Mo n th

M o n th ly S te e l P ro d u c tio n

Mil

lion

s of

Ton

s

2000 2001 2002

M o n t h l y S t e e l P r o d u c t i o n


2 - 64


52

54

56

58

60

62

64

66

68

70%

em

plo

ymen

t

Halifax

Montreal

Sherbro

oke

Quebec

Toronto

Hamilt

on

Kitchener

London

Thunder Bay

Regina

Saskatoon

Win

nipeg

Edmonto

n

Vancouver

Victoria

LineLine

Preparing a Line Chart for this type of data is not overly useful!Preparing a Line Chart for this type of data is not overly useful!


2 - 65


52

54

56

58

60

62

64

66

68

70%

em

plo

ymen

t

Halifax

Montreal

Sherbro

oke

Quebec

Toronto

Hamilt

on

Kitchener

London

Thunder Bay

Regina

Saskatoon

Win

nipeg

Edmonto

n

Vancouver

Victoria

LineLine

Is this combination any better for displaying the data?Is this combination any better for displaying the data?


2 - 66

ffrequency requency PolygonPolygon and and OgiveOgiveffrequency requency PolygonPolygon and and OgiveOgive

frequency Polygon

50403020100

0.3

0.2

0.1

0.0

Re

lati

ve

Fre

qu

enc

y

Sales

Ogive

50403020100

1.0

0.5

0.0

Cu

mu

lati

ve R

ela

tiv

e F

req

ue

ncy

Sales

LineLine


2 - 67

Test your learning…Test your learning…

www.mcgrawhill.ca/college/lindClick on…Click on…

Online Learning Centrefor quizzes

extra contentdata setssearchable glossaryaccess to Statistics Canada’s E-Stat data…and much more!


2 - 68

This completes Chapter 2

copyright © 2004 by the mcgraw-hill companies, inc. all rights reserved. 2 - 1 describing data...

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