Introduction to Statistics for Built Environment

Course Code: AED 1222

Compiled byDEPARTMENT OF ARCHITECTURE AND ENVIRONMENTAL DESIGN (AED)

CENTRE FOR FOUNDATION STUDIES (CFS)INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA

Lecture 6Summarizing Quantitative Data 2

Today’s Lecture: Summarizing Quantitative Data:

Histograms & Polygons The Stem-and-Leaf plot Ogives

Contingency Table

Contingency Table

Data

Qualitative Quantitative

TabularTabular GraphicalGraphical TabularTabular GraphicalGraphical

Frequency DistributionFrequency

Distribution

Rel. Freq. Dist.

Rel. Freq. Dist.

Bar GraphBar Graph

Pie ChartPie Chart

Frequency DistributionFrequency

Distribution

Rel. Freq. Dist.

Rel. Freq. Dist.

Cumulative Freq. Dist.

Cumulative Freq. Dist.

Histograms & PolygonsHistograms & Polygons

Stem and Leaf PlotStem and Leaf Plot

An overview

OgivesOgives

LECTURE 5

An overview of common data presentation:

LECTURE 4

HistogramsWhat is a Histograms?

• The histogram is a summary graph showing a count of the data points falling in various ranges.

• The groups of data are called classes, and in the context of a histogram they are known as bins, because we can think of them as containers that accumulate data and "fill up" at a rate equal to the frequency of that data class

• Consists of a set of rectangles• Bases at X axis,• Centers at the midpoints,• Lengths equals to the class interval size,• Areas proportional to the class frequencies.

Graphical

Graphical

Histograms cont.

• Unlike a bar graph, a histogram has no natural separation or gap between rectangles of adjacent classes.

• The class boundaries are marked on the horizontal axis (X Axis) and the frequency is marked on the vertical axis (Y Axis). Thus a rectangle is constructed on each class interval.

• If the intervals are equal, then the height of each rectangle is proportional to the corresponding class frequency.

• If the intervals are unequal, then the area of each rectangle is proportional to the corresponding frequency density.

Graphical

Graphical

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

How Many Class Intervals?• Many (Narrow class intervals)

• may yield a very jagged distribution with gaps from empty classes

• Can give a poor indication of how frequency varies across classes

• Few (Wide class intervals)• may compress variation too much

and yield a blocky distribution• can obscure important patterns of

variation.0

2

4

6

8

10

12

0 30 60 More

TemperatureFrequency

0

0.5

1

1.5

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2.5

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3.5

4 8

12

16

20

24

28

32

36

40

44

48

52

56

60

More

Temperature

Frequency

(X axis labels are upper class endpoints)

Histograms cont.Graphica

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Graphical

Histograms cont.

Draw a histogram for the following data set:

Example of Histograms:Graphica

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Graphical

Histograms cont.Graphica

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Graphical

Draw a histogram for the following data set:Example of Histograms:

Distribution of shops according to the number of wage - earners employed at a shopping complex

When the intervals are unequal, we construct each rectangle with the class intervals as base and frequency density as height.

Frequency Density

Histograms cont.Graphica

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Graphical

Draw a histogram for the following data set:Example of Histograms:

Histograms cont.Graphica

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Distribution of shops according to the number of wage - earners employed at a shopping complex

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Histogram

0

3

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2

00

1

2

3

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7

5 15 25 36 45 55 More

Frequency

Class Midpoints

0 10 20 30 40 50 60

Class Endpoints

Example (Cont.):

DATA ARRAY

12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

Sorted raw data from low to high:Insulation manufacturer 20 days high temperature record.

Histograms cont.Graphica

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No gaps between

bars, since continuous

data

(Note that we use the same example from Lecture 5)

Information conveyed by Histograms

Why use Histograms? -Histograms are useful data summaries that convey the

following information:

• The general shape of the frequency distribution

• Symmetry of the distribution and whether it is skewed

• Modality: unimodal, bimodal, or multimodal

Graphical

Graphical

-A histogram may become more appropriate as the data size increases.-The ease with which histograms can now be generated on computers.

Comparison between Histograms & Bar GraphsGraphica

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PolygonsWhat is a Polygons?

• A polygon is a line graph of the class frequency plotted against the class midpoint.

• Obtained by connecting the midpoints of the tops of the rectangles in the histogram.

• However, frequency Polygons can be drawn independently without drawing the histograms.

• In drawing a histogram/polygon of a given frequency distribution, we take the following steps:

Graphical

Graphical

Polygons cont.Graphica

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Graphical

Step 1. : If the frequency table is in the inclusive form, we first convert it into an exclusive form and make it a continuous interval.

Step 2. :To complete the polygon we assume a class interval with zero frequency preceding the first class interval and a class interval with zero frequency succeeding the last class interval.

Step 3. : Taking a suitable scale, we represent the class mid-points or (class marks) along X axis.

Step 4. : Taking a suitable scale, we represent frequency along Y axis.

Step 5. : We plot the corresponding points and join it with the help of line segment.

Procedure

Polygons cont.

Example of Polygons:Graphica

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Graphical

Draw a Polygons for the following data set:

Polygons cont.

Example of Polygons:Graphica

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Graphical

The Stem-and-Leaf plotWhat is a Stem-and-Leaf Plot?

• The Stem-and-leaf plot is a device for presenting quantitative data in a graphical format, similar to a histogram, to assist in visualizing the shape of a distribution.

• Unlike histograms, stemplots retain the original data.

• A basic stemplot contains two columns separated by a vertical line. The left column contains the stems and the right column contains the leaves.

• Consists of a set of a : Stem: Leading DigitsLeaf: Trailing digits

Graphical

Graphical

Step 1. : Separate the sorted data series into leading digits (the stem) and the trailing digits (the leaves).

Step 2. : List all stems in a column from low to high.

Step 3. : For each stem, list all associated leaves.

Procedure

The Stem-and-leaf plot cont.Graphica

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Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

• Here, use the 10’s digit for the stem unit:

Data sorted from low to high:12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

12 is shown as

35 is shown as

Stem Leaf

The Stem-and-leaf plot cont.Graphica

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Graphical

1 2

3 5

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Using other stem units

– Round off the 10’s digit to form the leaves

The Stem-and-leaf plot cont.Graphica

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Graphical

• Here, use the 100’s digit for the stem unit:

613 would become 776 would become 1224 would become

Stem Leaf

6 1

12 2

7 8

Example (Let’s do it together with the Class)

12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

Display the following data with a Stem and Leaf Plot

The Stem-and-leaf plot cont.Graphica

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Graphical

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

4 5

2 1 0 1 3

3

1 2

Stem Leaf

3

7 4

4 2

5 3

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6

7 7

8 6

8

7

. .

. .

.

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Why use Stem-and-Leaf Plot?

• A simple way to see distribution details from quantitative data

The Stem-and-leaf plot cont.Graphica

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• Stemplots are useful for giving the reader a quick overview of distribution, highlighting outliers and finding the mode.

OgivesWhat is an Ogives?

• An Ogive is a graph of the cumulative relative frequencies from a relative frequency distribution.

• Ogives are sometime shown in the same graph as a relative frequency histogram.

• Also known as Cumulative Frequencies Graph.

Graphical

Graphical

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Example of an Ogives:Draw an Ogives for the following data set:

Ogives cont.Graphica

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Graphical

Example (Cont.):

DATA ARRAY

12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

Constructing an Ogives table

Sorted raw data from low to high:

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Add a Cumulative Relative Frequency New column:

Graphical

Graphical

Insulation manufacturer 20 days high temperature record.

Example (Cont.):

Constructing an Ogives table cont.

Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.

Graphical

Graphical

Histogram

0

1

2

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5

6

7

5 15 25 36 45 55 More

Frequency

Class Midpoints

100

80

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40

20

0 Cum

ulati

ve F

requ

ency

(%)

/ Ogive

0 10 20 30 40 50 60

Class Endpoints

Insulation manufacturer 20 days high temperature record.

/ Construct an Ogive

PYRAMID CHART

LINE CHART

SCATTER DIAGRAM

RADAR CHART

Other Graphical Data Presentation 1

What is other types of Graphical Data Presentation?

Graphical

Graphical

PIE CHART

BUBBLE CHART

AREA CHART

DOUGHNUT

Other Graphical Data Presentation 2

More types of Graphical Data Presentation.

Graphical

Graphical