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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
2
2.5
3
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
l
Graphical
Histograms cont.
Draw a histogram for the following data set:
Example of Histograms:Graphica
l
Graphical
Histograms cont.Graphica
l
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
l
Graphical
Draw a histogram for the following data set:Example of Histograms:
Histograms cont.Graphica
l
Graphical
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
6
5
4
2
00
1
2
3
4
5
6
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
l
Graphical
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
l
Graphical
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
l
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
l
Graphical
Draw a Polygons for the following data set:
Polygons cont.
Example of Polygons:Graphica
l
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
l
Graphical
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
l
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
l
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
l
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
4
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
l
Graphical
• 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
l
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
3
4
5
6
7
5 15 25 36 45 55 More
Frequency
Class Midpoints
100
80
60
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