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3/31/201

DATA PRESENTATION

SEEING IS BELIEVING!

PREPARED BY

SANIZAH AHMAD

CHAPTER 3

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LEARNING OUTCOMES

Construct a frequency table from raw data

Organize and graph qualitative data using pie,

bar and component bar charts

Use information contained in various charts to

make decisions

Organize and graph quantitative data such as

stem-and-leaf plot, histogram, ogive and use

these graphs to understand the problem and

make decisions

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INTRODUCTION

Data can be summarized in tabular forms and

presented in pictorial form using graphs so that

important features can be grasped quickly and

effectively.

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ORGANIZING AND GRAPHING DATA

QUALITATIVE DATA

Frequency distribution

Pie chart

Bar chart

Vertical (or horizontal)

bar chart Cluster bar chart

Stacked bar chart

Contingency table

QUANTITATIVE DATA

Stem-and Leaf plots

Frequency distribution for

ungrouped data

Frequency distribution for

grouped data

Histogram/polygon

Cumulative frequency

distribution and Ogive

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ORGANIZING & GRAPHING

QUALITATIVE DATA

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Nominal and

Ordinal Data

ORGANIZING & GRAPHING QUALITATIVE DATA

After data is collected, it will be processed,

organized and presented.

In order to enhance the presentation, some

charts, tables and graphs can be used.

Some considerations in drawing charts/graphs:

a. Indicate the title

b. Draw the axes properly

c. Use proper size and scale

d. Use colours/shading if needed

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1 FREQUENCY DISTRIBUTION (TABLE)

Twenty-five army inductees were given a blood test todetermine their blood type. The data set is

A B B AB O

O O B AB BB B O A O

A O O O AB

AB A O B A

Construct a frequency distribution for the data

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Table consisting of columns and rows.

Example 1

PROCEDURE FOR CONSTRUCTING FREQUENCY TABLE

Step 1 Make a table with four columns ( Column A = Class,Column B = Tally, Column C = Frequency, D = Percent).

Step 2 Tally the data and place the result in column B.

Step 3 Count the tallies and place the results in column C.

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).

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%100%

n f



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Class Tally Frequency Percent

AB

O

AB

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2. PIE CHART

Pie chart can be used to represent categorical

data.

It is a circle that is divided into sectors.

The sectors show the percentage of

frequencies of each category of the

distribution.

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PROCEDURE FOR CONSTRUCTING A PIE CHART

• Step 1: Find the number of degrees for each class,

using formula

• Step 2 : Find the percentages.

• Step 3: Using a protractor, graph each section and

write its name and corresponding percentage.

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Pie Chart using data in Example 1

Note: If possible, construct the pie chart so that

%s are either in ascending or descending order

(helps in the interpretation of the data).



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3. BAR CHART

A graph of bars whose heights represent the

frequencies of respective categories.

Types of Bar Charts:

i) Vertical/horizontal bar chart (single/simple)

ii) Cluster bar chart (multiple)

iii) Stacked bar chart (component)

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I) BAR CHART One chart present only one subject

Using the data in Example 1

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II) CLUSTER BAR CHART

No College Four-year degree Advanced Degree

Urban 15 12 8

Suburban 8 15 9

Rural 6 8 7

0

2

4

6

8

10

12

14

16

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• One graph presents more than one subject

• Colour/shading needed

III) STACKED BAR CHART

0

5

10

15

20

25

30

35

40

No College Four-year degree Advanced Degree

Rural

Suburban

Urban

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Each bar contains more than one information

Shading is needed



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EXAMPLE 2From the following table, construct:

i. Single(simple) bar chart for the year 2000

ii. Cluster(multiple) bar chart for the year 2000 and 2001

iii. Stacked(component) bar chart for the year 2000 and 2001

iv. Pie chart for the year 2001

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Program Number of Students

Year 2000 Year 2001

A 450 600

B 1200 1500

C 800 1100

D 300 400

E 650 800

4. CROSS TABULATION/CONTINGENCY TABLE

A cross tabulation (often abbreviated as cross

tab) or cross-classification table is often used

to examine the categorical response in terms oftwo qualitative variables simultaneously.

Some data can be grouped according to two or

more criteria of classification or variables.

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Cross tabs are frequently used because:

They are easy to understand. They appeal topeople who do not want to use moresophisticated measures.

They can be used with any level of data:nominal, ordinal, interval, or ratio

- cross tabs treat all data as if it is nominal.

A table can provide greater insight than singlestatistic.

It solves the problem of empty or sparse cells

They are simple to conduct.

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EXAMPLE 3

Location No College Four-year

degree

Advanced

Degree

Total

Urban 5 12 8 35

Suburban 8 15 9 32

Rural 6 8 7 21

Total 29 35 24 88

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Cross tabulation between location and education level

http://en.wikipedia.org/wiki/Variable

http://en.wikipedia.org/wiki/Variable



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EXAMPLE 4

A group of researchers surveyed 530 staff working

with Company Y. Out of 145 professional staff, 40 are

women whereas 140 non-professional staff are men.

Present this data in the form ofa 2 x 2 table.

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ORGANIZING & GRAPHING

QUANTITATIVE DATA

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Interval and

Ratio Data

ORGANIZING AND GRAPHING

QUANTITATIVE DATA

Normally summarized in tabular forms.

Quantitative data can be divided into

ungrouped and grouped data.

Display of data:

Stem-and leaf plot Frequency Distribution (table)

Histogram

Frequency polygon

Ogive

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1. 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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PROCEDURE OF CONSTRUCTING STEM-AND-

LEAF PLOTSTEP 1

Split each score or value into two sets of digits. The first (or

leading) set of digits is the stem, and the second (or trailing) set

of digits is the leaf .

STEP 2

List all the possible stem digits

from the lowest to highest.

STEP 3

For each score in the dataset,

write down the leaf numbers on

the line labeled by the

appropriate number.

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EXAMPLE 5 (UNGROUPED DATA)

At an outpatient testing center, the number ofcardiograms performed each day for 20 days is

shown. Construct a stem-and-leaf plot for the

data.

25 31 20 32 13

14 43 02 57 23

36 32 33 32 44

32 52 44 51 45

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EXAMPLE 6 (GROUPED DATA)An insurance company researcher conducted a survey onthe number of car theft in a large city for a period of 30 dayslast summer. The raw data are shown.

Construct a stem-and-leaf plot by using classes 50-54,

55-59, 60-64, 65-69, 70-74 and 75-79.

52 62 51 50 69

58 77 66 53 57

75 56 55 67 73

79 59 68 65 72

57 51 63 69 75

65 53 78 66 55

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EXAMPLE 7

The IQs of 30 students are listed below.

Construct a stem-and-leaf plot, using two lines

per stem and stems of 11, 12 and 13.

110 122 119 114 135

134 130 138 124 127

123 120 114 128 125113 131 117 128 116

123 117 114 132 128

121 132 137 117 126

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2. UNGROUPED FREQUENCY DISTRIBUTION The frequency distribution is a table that contains a list of data

values and its f requency.

Frequency is the number of times a value occurs.

Example 8: The following data record the number of children in

20 families chosen at random.

1 4 2 0 2

3 3 2 1 4

5 2 1 2 0

1 2 3 1 2

This set of ungrouped data can be summarized in tabular form

known as the frequency distribution.

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2. GROUPED FREQUENCY DISTRIBUTION When the data set contains many different and repetitive

values, the data can be grouped into class intervals before thefrequency distribution is constructed.

TERMINOLOGIES OF FREQUENCY DISTRIBUTION

i. Class limit

The end values of each class interval.

Example: 80 – 90

Lower limit is 80 and upper limit is 90

ii. Class boundary

Value that falls mid/half way between the upper limit of one class

and the lower limit of the next class.

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CLASS BOUNDARY

Class interval/class limit Class boundary

Type 1

30 – <50 30 -

50 – <70 or 50 -

70 - <90 70 -

30 – 50

50 – 70

70 – 90

30 and less than 50

50 and less than 70

70 and less than 90

30 – 50

50 – 70

70 – 90

Type 2 30 – 4950 – 69

70 – 89

29.5 – 49.549.5 – 69.5

69.5 – 89.5

Type 3 30 – 50

50 – 70

70 – 90

30 – 50

50 – 70

70 – 90

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TERMINOLOGIES OF FREQUENCY DISTRIBUTION

iii. Class midpoint

The middle value of a class interval; averaging the

upper limit and lower limit or upper boundary and

class boundary.

e.g.

iv. Cumulative frequency

Total frequency for the particular class and all the

prior classes.

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5.392

49.529.5or5.39

2

4930



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CONSTRUCTING A FREQUENCY DISTRIBUTION TABLE

Step 1 Determine the classes

a)Find the highest value, H and lowest value, L.

b) Find the range, R.

R = H Lc) Find the number of classes, k.

k = 1 + 3.3 log n

d) Find the class width:

e) Select a starting point for the lowest class limit

f) Add the width to the lowest class limit. Keep addinguntil k classes.

g) Find the class boundaries.

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classesof Number

RWidth

CONSTRUCTING A FREQUENCY DISTRIBUTION

TABLE…CONT. Step 2

Tally the data

Step 3Find the numerical frequencies from

the tallies

Step 4

Find the cumulative frequencies(Extra step for cumulative frequency distribution ogive)


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EXAMPLE 9:

These data represent the record of high temperatures in F for

each of the 50 states in the United States. Construct a groupedfrequency distribution for the data.

112 100 127 120 134 118 105 110 109 112

110 118 117 116 118 122 114 114 105 109

107 112 114 115 118 117 118 122 106 110

116 108 110 121 113 120 119 111 104 111

120 113 120 117 105 110 118 112 114 114

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SOLUTION

Range : R = H L

= 134 – 100 = 34

Number of class interval:

k = 1 + 3.3 log n

= 1 + 3.3 log 50 = 6.6 7

Width = R/k

= 34/7 = 4.9 5

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TABLE: FREQUENCY DISTRIBUTION OF TEMPERATURE


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Temperature in F

(Class interval)

Frequency

(No of States)

100 – 104

105 – 109

110 – 114

115 – 119

120 – 124

125 – 129

130 – 134

NOTE:

1. Both the smallest and largest observations must be included in a class interval.2. Each observationmust be assigned to one and only one class.

Class

frequency

GRAPHIC PRESENTATION OF FREQUENCY

DISTRIBUTION

Refer textbook pg. 53–56.

i. Histogram

Y-axis: frequency

X-axis: class boundary

ii. Frequency polygon

Y-axis: frequency

X-axis: class midpoint

iii. Ogive/Cumulative frequency curve(less than)

Y-axis: cumulative frequency

X-axis: less than class boundary

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3. HISTOGRAM

Histogram graph

that displays the data

by using continuous

vertical bars (unless

the frequency of a

class is 0) of variousheights to represent

the frequencies of

the classes.

0

2

4

6

8

0 10.5 20.5 30.5 40.5 50.5 60.5 70.5 80.5

Class Boundaries

F r e q u e n c y

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PROCEDURE FOR CONSTRUCTING HISTOGRAM

Step 1

Draw and label the x-axis(horizontal) and y-axis

(vertical).

Step 2

Represent the frequency on the y-axis and the classboundaries on the x-axis.

Step 3

Using the frequencies as the heights, draw vertical

bars for each class.

****Exercises: i) Example 7 pg. 53

ii) pg. 58 Q1 Q2

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4. FREQUENCY POLYGON

Frequency Polygon

graph that

displays the data by

using lines that

connect points

plotted for the

frequencies at the

midpoints of the

classes.

0

1

2

3

45

6

7

8

0.5 10.5 20.5 30.5 40.5 50.5 60.5 70.5 80.5

Class Midpoints

F r e q u e n

c y

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PROCEDURE FOR CONSTRUCTING FREQUENCY POLYGON

Step 1

Find the midpoints of each class.

Step 2

Draw the x and y axis. Label the x axis with the midpoint of each

class and then use a suitable scale on the y axis for thefrequencies.

Step 3

Using the midpoints for the x values and the frequencies as the

y values, plot the points

Step 4

Connect adjacent points with line segments. Draw a line back to

the x axis at the beginning and end of the graph, at the samedistance that the previous and next midpoints would be located.

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5. CUMULATIVE FREQUENCY DISTRIBUTION

& OGIVE

Ogive graph

that represents

the cumulative

frequencies for

the classes in afrequency

distribution

0

10

20

30

0 1 0. 5 2 1 3 1. 5 4 2 5 2.5 6 3 7 3 .5 8 4 9 4 .5

Class Boundaries

C u m u l a t i v e

F r e q u e n c y

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PROCEDURE FOR CONSTRUCTING OGIVE Step 1

Find the cumulative frequency for each class.

Step 2

Draw the x- and y-axis. Label the x-axis with the classboundaries. Use an appropriate scale for the y-axis to representthe cumulative frequencies.

Step 3

Plot the cumulative frequency at each upper class boundary.Upper boundaries are used since the cumulative frequenciesrepresent the number of data values accumulated up to the

upper boundary of each class. Step 4

Starting from the first upper class boundary, connect adjacentpoints with line segments. Then extend the graph to the firstlower class boundary on the x-axis.

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1

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0

2

4

6

8

0 10.5 20. 5 30. 5 40.5 50. 5 60. 5 70.5 80. 5

Class Boundaries

F r e q u e n c y

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

F r e q u e n c

y

0

10

20

30

0 10.5 21 31.5 42 52.5 63 73.5 84 94.5

Class Boundaries

C u m u l a t i v e

F r e q u e n c y

Histogram

Frequency

polygon

Ogive

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TABLE: CUMULATIVE FREQUENCY DISTRIBUTION

Class Limit

(Temp, F)

Frequency Class

Boundary

ClassMidpoint (X)

Cumulativefrequency

100 - 104

105 - 109

110 - 114

115 - 119

120 - 124

125 -129

130 - 134

EXAMPLE 10

Given the following data, construct the:

i. Histogram

ii. Frequency Polygon

iii. ‘Less than’ Ogive

Amount received (RM) by 50 children for Hari Raya.

95 101 126 114 134 117 148 103 110 125

144 112 83 136 116 129 114 132 105 118122 110 136 124 91 148 125 89 133 95

105 135 108 123 108 137 114 124 118 119

117 93 115 117 100 106 104 115 128 105


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EXAMPLE 10: HISTOGRAM & FREQUENCY POLYGON

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/ /



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EXAMPLE 10: HISTOGRAM & FREQUENCY POLYGON

Amount received (RM)

(class limit)

No. of children

(frequency, f)

C lass boundary Class

midpoint

(x)

Cumulative

frequency

80 and less than 90

90 and less than 100

Total

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‘LESS THAN’ OGIVE

Amount received (RM) Cumulative frequency

Less than 79.5 0

Less than 89.5

Less than 99.5

Less than 109.5

Less than 119.5

Less than 129.5

Less than 139.5

Less than 149.5

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HOMEWORK

Do Review Questions 3 pg. 61-64

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