Statistics (Stat 101)
Sameh Saadeldin Ahmed
Associate Professor of Environmental Eng.
Civil Engineering Department
Engineering College
Almajma’ah University
smohamed1@ksu.edu.sa
http://faculty.ksu.edu.sa/SaMeH
Stat 101
2010/2011
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2.2 Organizing and Graphing
Qualitative Data
2.2.1 Frequency Distribution
2.2.2 Relative Frequency & Percentage
2.2.3 Graphical Presentation of Qualitative
Bar Graphs
Pie Chart.
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2.3 Organizing and Graphing
Quantitative Data
2.3.1
Frequency Distribution
2.3.2
Constructing Freq. Distribution Tables
2.3.3
Relative Freq. & Percentage Distribut.
2.3.4
Graphing Grouped Data.
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2.3.2
Constructing Frequency
Distribution Tables
• To construct the frequency distribution
table, you have to make the following
three steps:
Number of Classes
Class Width
Lower Limit of the First Class or the Starting
Point.
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Number of Classes
Usually the number of classes for a
frequency distribution table varies from
5 to 20. The decision of the number of
classes is arbitrarily made by the data
organizer.
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Class Width
It is preferable to have the same width for
all classes. To do so, find the difference
between the largest and smallest values in
the data. Then, the approximate width of a
class is obtained by dividing this
difference by the number of desired
classes.
Calculating of Class Width
Approximate class width =
[Largest value - Smallest value] / Number of classes
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Lower Limit of the First Class or
the Starting Point
Any convenient number that is equal to or
less than the smallest value in the data
set can be used as the lower limit of the
first class.
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2.3.3
Relative Frequency and
Percentage Distributions
Calculating Relative Frequency and
Percentage
Relative frequency of a class = Frequency of
that class / Sum of all frequencies
= f / ∑f
Percentage = (Relative frequency) x 100
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Exercise
Calculate the relative frequencies and percentage for
the following table:
Total goals scored
124 – 145
146 – 167
168 – 189
190 – 211
212 - 233
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6
13
4
4
3
9
2.3.4
Graphing Grouped Data
Grouped quantitative data can be displayed in a histogram
or a polygon.
Histogram
A histogram is a graph in which classes are
marked on the horizontal axis and the
frequencies, relative frequencies, or
percentages are marked on the vertical
axis. The frequencies, relative frequencies,
or percentages are represented by the
heights of the bars. In a histogram the
bars are drawn adjacent to each other.
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A histogram for the frequency distribution
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A histogram for the relative
frequency
A histogram for the percentages
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Polygons
A graph formed by joining the midpoint of the
tops of successive bars in a histogram with
straight lines is called a polygon.
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2.3.5
More on Classes and
Frequency Distribution
• Less than Method for Writing Classes
•
Single-Valued Classes
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Example 2.6: less than.....
The following data give the average travel time from
home to work (in minutes) for 50 cities.
22.4
18.2 23.7 19.8 26.7 23.4 23.5 22.5 24.3 26.7
24.2
26.1
19.9
15.6
19.7
22.7
31.2
22.7
27.0
21.6
22.6
23.6
21.7
21.9
15.4
20.8
17.6
23.2
22.1
21.1
17.7
16.0
19.6
25.4
22.5
16.1
21.4
24.9
23.7
22.3
23.8
25.5
21.2
24.4
21.9
20.1
29.2
28.7
21.9
17.1
1.Construct a frequency distribution table.
2.Calculate the relative frequencies and percentages for all classes.
3.Plot the histograms and polygons for the frequencies &
percentages.
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Solution
The min. value is 15.4 and the max. value is 31.2
Suppose we decide to group these data using six classes
of equal width. Then,
Approximate width of each class = [31.2 – 15.4] / 6
= 2.63
We round this number to a more convenient number to 3
Let us start the first class at 15, the classes are written
as 15 to less than 18, and so on.
Sturge’s formula to decide on the no. of classes:
c = 1 + 3.3 log n
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Average travel time Frequency Relative Percentage
to work (minutes)
(f)
Frequency
15 to less than 18
18 to less than 21
21 to less than 24
24 to less than 27
7
7
23
9
0.14
0.14
0.46
0.18
14
14
46
18
27 to less than 30
30 to less than 33
SUM
3
1
0.06
0.02
1.00
6
2
100%
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Example 2.7: Single value......
In this case we use classes that are made of single
values and not of intervals. It is useful in cases of
discrete data with only a few possible values. See the
following example.
The governorate of Almajmaa’h city wanted to know
the distribution of the computer sets owned by
families in the city. A sample of 40 randomly selected
houses from the city produced the following data on
the number of computers owned.
5
1
1
1
1
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1
1
2
2
1
1
1
3
2
2
2
3
4
1
1
0
3
2
1
1
1
0
1
1
4
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2
2
4
1
2
5
2
2
3
5
1
1
1
1
1
1
2
2
1
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Construct the frequency distribution table for these
data using single-valued classes.
Solution:
The observations in this data set assumes only 6 distinct
values: 0,1,2,3,4 and 5. Each of these values is used as a
class in the frequency distribution.
Computers Owned
Relative
Frequency
0.05
0.45
0.275
0.10
Percentage
0
1
2
3
Frequency
f
2
18
11
4
4
5
SUM
3
2
40
0.075
0.05
1.00
7.50
5
100%
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45
27.5
10
20
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2.4 Cumulative Frequency Distribution
2.1
Raw Data
2.2
Organizing Qualitative Data
2.3
Organizing Quantitative Data
2.4
Cumulative Frequency Distribution
2.5
Seam-and –Leaf Display
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2.4 Cumulative Frequency
Distribution
Cumulative Frequency Distribution
A cumulative frequency distribution gives the total
number of values that fall below the upper boundary
of each class.
In the cumulative frequency distribution table, each
class has the same lower limit but a different upper
limit. Example 2.8 illustrates the procedure to prepare
cumulative frequency distribution.
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Example 2.8:
Using the frequency distribution of the following
table, prepare a cumulative frequency distribution for
the total goals.
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Total goals scored
F
124 – 145
146 – 167
168 – 189
190 – 211
212 - 233
6
13
4
4
3
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Solution:
Class Limits Class Boundaries
124 – 145
124 – 167
124 – 189
124 – 211
124 - 233
123.5 to less than 145.5
123.5 to less than 167.5
123.5 to less than 189.5
123.5 to less than 211.5
123.5 to less than 233.5
Cumulative
frequency
6
6 + 13 = 19
6 + 13 + 4 = 23
6 + 13 + 4 + 4 = 27
6 + 13 + 4 + 4 + 3 = 30
From the above table we can determine the number of
observations that fall below the upper limit or boundary
of each class. For example, 23 football teams scored a
total of 189 goals or fewer.
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Calculating the Relative Frequency and
Cumulative Percentage
A cumulative frequency distribution gives the total
number of values that fall below the upper boundary
of each class.
Cumulative frequency of a class
Cumulative relative frequency = --------------------------------------Total observations in the data set
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The following table contains both the cumulative
relative frequencies and the cumulative percentages
for the data given in example 8.
Class Limits
124 – 145
124 – 167
124 – 189
124 – 211
124 - 233
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Cumulative
Relative Frequency
6/30 = 0.200
19/30 = 0.633
23/30 = 0.767
27/30 = 0.900
30/30 = 1.000
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Cumulative
Percentage
20.0
63.3
76.7
90.0
100.0
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Ogives
Ogive
An ogive is a curve drawn for the cumulative frequency
distribution by joining with straight lines the dots
marked above the upper boundaries of classes at
heights equal to the cumulative frequencies of
respective classes.
When plotted on a diagram, the cumulative frequencies
give that is called ogive. The next figure gives an ogive
for the cumulative frequency distribution of ex. 8.
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End of Part 3
Get ready for a quiz (2)……
next lecture!!
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