Organization and Presentation of Data

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Organization and Presentation ofData

Chapter Two



Describing Data: Frequency Distributions and Grap

Presentation

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

1. Organize data in an array and into a frequency distribution.2. Portray a frequency distribution in a histogram, frequency

polygon,and cumulative frequency polygon. 3. Present data using such graphic techniques as line charts,

bar charts, and pie charts, in order to interpret thedata being graphed.



• Raw data , or data that have not been summarizedin any way, are sometimes referred to as

ungrouped data. • Collected data need to be organized in such a way

as to condense the information they contain in away that will show patterns of variation clearly.Precise methods of analysis can be decided upononly when the characteristics of the data areunderstood.

• Data that have been organized in a frequencydistribution are called grouped data.





Chap 2-5

2. Stem-and-Leaf Display

• A simple way to see how the data aredistributed and where concentrations ofdata exist

METHOD: Separate the sorted data seriesinto leading digits (the stems )



Organizing Numerical Data:Stem and Leaf DisplayA stem-and-leaf display organizes data into grou(called stems) so that the values within each group (theleaves) branch out to the right on each row.

Stem Leaf

1 67788899

2 0012257

3 28

4 2

Age of College Stude

Day Students NightS

Age ofSurveyedCollegeStudents

Day Students

16 17 17 18 18 18

19 19 20 20 21 2222 25 27 32 38 42

Night Students

18 18 19 19 20 21

23 28 32 33 41 45



The raw data are the numbers of Congressional billsvetoed during the administrations of seven U.S. presidentsfrom Johnson to Clinton.

Johnson Nixon Ford Carter Reagan Bush ClintonVetoes 30 43 66 31 78 44 3In stem-and-leaf terms, we could describe these data afollows:Stem (10’s Digit) Leaf (1’s Digit)

3/018 (represents 30, 31, and 384/34 (represents 43 and 44)6/6 (represents 66)7/8 (represents 78)



3. Tabular Presentation- a process of condensing classified data and arranging

them systematically in rows and columns.

A. Frequency Table for Categorical VariableThe figures found in the cells of the main body are the freqand the percentagesExample 1. Distribution of Employees in Terms of Civi l Statu

Civil Status FREQUENCY PERCENTAGE

Single 725 36.25%Married 250 12.50%

Widowed 375 18.75%

Separated 650 32.50%

TOTAL 2000 100%



Cross Tabulations:The Contingency Table

Importance of Brand

Name

Male Female Total

More 450 300 75

Equal or Less 3300 3450 6750

Total 3750 3750 7500

A survey was conducted to study the importance of brandname to consumers as compared to a few years ago. Theresults, classified by gender, were as follows:





Frequency Table

Rating FrequencyPoor 2Below Average 3

Average 5Above Average 9Excellent 1

Total 20

Example: Marada Inn



B. Contingency TableA contingency table (or a two-way frequency table) is a table in which

frequencies correspond to two variables.(One variable is used tocategorize rows and a second variable is used to categorized

columns)Example 2

Smoking Habit Gender

male female Total

Smoker 100 70 170

Non-smoker 50 80 130

Total 150 150 300

Distribution of Employees in Terms of Gender and Smoking Status



Questions

1. How many are males in the sample?2. How many are smokers in the sample?3. How many females are non-smokers?4. What percentage of the sample smokes?5. What percentage of the smokers are males?

6. What percentage of the females are smokers?



C. Frequency Distribution Table- refers to the tabular arrangement (grouping) of all observations into

intervals or classes together with the count of the number ofobservations that fall in each interval or class.

Remarks:1. There is no clear and definite method of finding the number of intervals.Finding the number of intervals depend upon the size of the data. It isgenerally accepted that the number of intervals to use is from 6 to 15intervals. If fewer than 6 intervals are used, much loss of informationwill be brought about due to lumping of many observations into class. Iftoo many classes are used, it could present some irregularities in thegraphical representation because many classes or intervals may

contain small frequencies.2. There is no definite rule in choosing the starting lower limit and theclass width. The starting lower limit could be the smallest observationor any number closest to the lowest observation, or any multiple of theclass size (c).

3. In constructing the class limits, one may get one more or one less thanthe suggested number of intervals.



Definitions:Class interval the numbers defining a classClass limits the smallest and largest values that can fall in a

given class

Class boundaries numbers that are halfway between theupper limit of a class and the lower limit of the next class

Class size length of the class interval; computed by takingthe difference between two successive upper/lower classboundaries or class limits.

Class mark midpoint of an interval; computed by taking theaverage of the lower and upper class limits of a given classinterval

Relative frequency obtained by dividing the class frequency bythe total number of observations

Relative percentage obtained by multiplying the relativefrequency the relative frequency by 100%



1 . Determine the range (R) of the observations.R = highest value – lowest value

2. Determine the number of class intervals (k)Suggested Rule : k must be an integer. If the computed k is notan integer, then round it off to the next higherinteger.

3. Determine the class width (c).Suggested Rule: c must have the same number of decimal places

as the original data.4. Determine the lower limit and the upper limit of the class

intervals.5. Determine the frequency for each interval, class marks, class

boundaries, cumulative frequencies (less than and greater than

cf), and the corresponding percentage.

How to Construct a Frequency Distribution Table

(Suggested Steps)

nk

k R

c



Additional Columns in the FDT

1 . Class boundaries(CB ). The CB’s are obtained by taking the midpointof the gaps between classes

LCB = LL – (0.5)(one unit of measure)UCB = UL + (0.5)(one unit of measure)

2.Class marks . It is the midpoint of a class .

3. Relative Frequency(RF ). This is the frequency of a class expressed inproportion to the total number of observations.

2

UL LL xi

.n

Frequency RF



4. Cumulative Frequency . This is the accumulated frequencyof a class.

The <CF (less than CF). It is the total number of observationswhose values do not exceed the upper limit of the class.

The >CF( greater than CF).It Corresponds to the totalnumber of observations whose values are not less than thelower limit of the class.

5 . Relative Cumulative Frequency . This is the cumulativfrequency of a class expressed in proportion to the totalnumber of observations.



Example 3 The following are the length of service (in months)of a sample of 50 employees in a certain shoe factory.

Organize the data using frequency distribution table.

EXCEL function: frequency(data_array, bins_array)Press: Crtl-Shift-Enter

70 87 99 112 127

78 88 99 114 132

80 89 100 115 13280 89 106 117 132

82 92 106 119 135

84 94 107 120 136

85 95 108 123 136

86 95 109 124 14087 97 110 125 140

87 98 112 125 146

Range = 146

=

k=sqrt(50

c=R/k

=76/8= 10

k



Output: Frequency Distribution Table

ClassIntervals

Freq Class Marks ClassBoundaries

<CF >CF




Class Intervals Freq

70-79 2

80-89 12

90-99 8

100-109 6

110-119 7120-129 6

130-139 6

140-149 3

Total 50



5.1 9.1 13

5.5 9.1 13

5.9 9.3 14

6.5 9.8 14

6.8 9.9

7 107.2 10.2 15.

7.3 10.3 1

8.3 11 1

8.4 11.5 16.

8.5 11.7 16.

8.5 12.3 1

8.7 12.7 17.

8.8 13.2 17.

Example 2

A recent report showed thefollowing data for percentages ofexecutives in 42 top UScorporations suffering from drugabuse problems.Construct the frequencydistribution.

1 H



1.Find the highest and lowest value2.Determine the range . It is the

difference between the highest andlowest values in the data set.

3.Determine the number of classes k. Thenumber of classes is somewhatarbitrary. In general your table shouldhave been between 5 and 20 classes.A simple rule you can follow toapproximate the number of classes is

4.Determine the width of the classintervals. (Round off result to thenearest value whose precision is thesame as those of the raw data.

5.Select a starting point for the lowerclass limit. (This can be the smallestdata or any convenient number lessthan the smallest data value .

1.Hv=

2.Range=Hv-Lv

3.n=sqrt(n) =

4. w=range/kw=

w=w ≈

5. Starting pointcan be

nk



Output: Frequency Distribution TableClass

IntervalsFreq Class Marks Class

Boundaries<CF >CF

5.1-6.8 5 5.95 5.05-6.9-8.6 7 7.758.7-10.4 10 9.5510.5-12.2

311.35

12.3-14.0

513.15

14.1-15.8

514.95

15.9-17.6

716.75



SEATWORK

• The following are the average weekly mortgage interest rates for a

40-week period.



1. In the following stem-and-leaf display for a set oftwo- digit integers, the stem is the 10’s digit, and each leafis the 1’s digit. What is the original set of data?

2/0022783/0113594/13445/47



SEATWORK• The accompanying data describe the hourly wage

rates (dollars per hour) for 30 employees of an electronics firm:22.66 24.39 17.31 21.02 21.61 20.97 18.58 19.74 21.57 20.56 22.16 20.16 18.97 22.64 122.05 22.03 17.09 24.60 23.82 17.80 16.28 19

22.22 19.49 22.27 18.20 19.29 20.43Construct a frequency distribution for these data.



2.58 In 2007, unemployment rates in the 50 U.S. states were reported asfollows.Source: Bureau of the Census, Statistical Abstract

of the United States 2009 , p. 373.



a. Construct a stem-and-leaf display for these data.b. Construct a frequency distribution for these data.c. Determine the interval width and the class mark foreach of the classes in your frequency distribution.d. Based on the frequency distribution obtained in part(b), draw a histogram and a relative frequency polygonto describe the data.

In a study of reaction times to a specific stimulus an animal trainer



Class limits Fre2.3 – 2.9 3.0 – 3.6 3.7 – 4.3 4.4 – 5.0 5.1 – 5.7 5.8 – 6.4

In a study of reaction times to a specific stimulus, an animal trainerobtained the following data, given in seconds.

With reference to the table, determine:a. the upper limit of the fourth class.b.The class midpoint of the third classc.The class boundaries of the second classd.The size of the fifth class interval.e.The number of animals which respondto the stimulus in 3.0 to 3.6 seconds.

f.The number of animals which respond to the stimulus after 4.35 secondsg.The percentage of animals which respond to the stimulus in 5.1 to 5.7 seconds.hh.The percentage of animals which respond to the stimulus in less that 3.65seconds.



GRAPHICALPRESENTATION



GRAPHICAL PRESENTATIONa method of presenting numericalvalues or relationships in pictorial form.

1.LINE GRAPHS2.BAR GRAPHS3. PIE CHARTS

Membership Growth of



Membership Growth ofFICCO

Year Total Number ofmembers

1980 987

1990 9186

2000 24026

2010 140128

2012 163722





Line graphs are typically used to show thechange or trend in a variable over time.

The line graph is capable of simultaneously showingvalues of two quantitative variables ( y , or vertical aand x , or horizontal axis); it consists of linear segmentsconnecting points observed or measured for eachvariable.



A B Ch t i t f i f t l b h th



A B ar Chart - consists of a series of rectangular bars where thelength of the bar represents the magnitude to be

demonstrated.It can be used to depict any of the levels of

measurement (nominal, ordinal, interval, or ratio).

0

1

23

4

5

6

7

8

9

43-48 49-54 55-60 61-66 67-72 73-78 79-84

F

r

e

q

u

e

n

c

y

Class Intervals

Bar Chart



Example 2.Construct a bar chart for the number



of unemployed per 100,000 population forselected cities during 2001

City Number of unemployedper 100,000 population

Atlanta, GA 7300Boston, MA 5400Chicago, IL 6700

Los Angeles, CA 8900New York, NY 8200

Washington, D.C. 8900



0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1

Atlanta, GA

Boston, MA

Chicago, IL

Los Angeles, CA

New York, NY

Washington, D.C.

Number of unemployed per 100,000 population

A Pie Chart is useful for displaying a relative frequency



A Pie Chart is useful for displaying a relative frequencydistribution. A circle is divided proportionally to the

relative frequency and portions of the circle areallocated for the different groups.

1. MONTHLY BUDGET OF A COLLEGE STUDENT

EXPENSES AMOUNTLodging P2,500

Food 5,000School

Supplies

500

Pocket Money 1,000Miscellaneous 1,000

Recreation 500

PIE CHART



Food62%

School Supplies6%

Pocket

Money13%

Miscellaneous13%

Recreation6%

Monthly Budget of s College Student

EXAMPLE



• A sample of 200 runnerswere asked to indicate

their favorite type ofrunning shoe. Draw a piechart based on thefollowing information.

Type ofshoe

# ofrunners

% of total

Nike 92 46.0

Adidas 49 24.5

Reebok 37 18.5

Asics 13 6.5

Other 9 4.5



GRAPHICAL REPRESENTATION of the



FREQUENCY DISTRIBUTION

• The three commonly used graphicforms are Histograms, Frequency

Polygons , and Ogives

A Histogram is a graph in which the class midpoints



g g p por limits are marked on the horizontal axis and theclass frequencies on the vertical axis.The class frequencies are represented by the heights

of the bars and the bars are drawn adjacent to eachother.

0

2

4

6

8

10

12

12.65 17.45 22.25 27.05 31.85

Hours spent studying

F r e q u e n c y

5.1 9.1 13

9 3



5.5 9.1 13

5.9 9.3 14

6.5 9.8 14

6.8 9.9

7 107.2 10.2 15.

7.3 10.3 1

8.3 11 1

8.4 11.5 16.

8.5 11.7 16.

8.5 12.3 18.7 12.7 17.

8.8 13.2 17.

Example 2

A recent report showed

the following data forpercentages ofexecutives in 42 top UScorporations sufferingfrom drug abuseproblems.Construct the frequencydistribution.




Output: Frequency Distribution TableClass

IntervalsFreq Class Marks Class

Boundaries<CF >CF

5.1-6.8 55.95

5.05-6.85 5 42

6.9-8.6 77.75

6.85-8.65 12 37

8.7-10.4 109.55

8.65-10.45 22 30

10.5-12.2 311.35

10.45-12.25 25 20

12.3-14.0 513.15

12.25-14.05 30 17

14.1-15.8 514.95

14.05-15.85 35 12

15.9-17.6 716.75

15.85-17.65 42 7

HISTOGRAM



5

7

10

3

5

0

2

4

6

8

10

12

5.95 7.75 9.55 11.35 13.15

NUMBER OFCORPORATIONS

CLASS MARKS

Percentage of Executives Suffering from Drug Abuse ProblemsHISTOGRAM

A Frequency Polygon consists of line



A Frequency Polygon consists of linesegments connecting the points formed bythe class midpoint and the class frequency.

0

2

4

6

8

10

12

4.15 5.95 7.75 9.55 11.35 13.15 14.95 16.75

NO. OF CORPORATIONS

CLASS MARKS

Percentage of Executives Suffering from Drug Abuse Problems

Ogive(Cumulative frequency polygon) A graph showing the cumulative frequency



A graph showing the cumulative frequencyplotted against the class boundary. The graphcan be a “less than or greater than” ogiv

0

5

12

2225

30

35

42

0

5

10

15

20

25

30

35

40

45

5.05 6.85 8.65 10.45 12.25 14.05 15.85 17.65

L

e

s

s

t

ha

n

c

f

Class Boundaries

Less Than Ogive

Greater Than Ogive



42

37

30

2017

12

7

00

5

10

15

20

25

30

35

40

45

5.05 6.85 8.65 10.45 12.25 14.05 15.85 17.65

g

r

e

a

t

e

r

t

h

a

n

c

f

Class Boundaries

Greater Than Ogive



The Abuse of Visual DisplaysRemember that visuals can be designed to be either emotionally charged or

purposely misleading to the unwary viewer. This capacity to mislead isshared by a great many statistical tests and descriptions, as well as visualdisplays. We will consider just a few of the many possible examples wheregraphical methods could be viewed as misleading.





Organization and Presentation of Data

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