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Chapter TwoDescribing Data: Frequency Distributions and Graphic PresentationGOALS
When you have completed this chapter, you will be able to:ONEOrganize data into a frequency distribution.
TWO Portray a frequency distribution in a histogram, frequency polygon, and cumulative frequency polygon.
THREEDevelop a stem-and-leaf display.
FOURPresent data using such graphic techniques as line charts, bar charts, and pie charts.
Irwin/McGraw-Hill © The McGraw-Hill Companies, Inc., 1999
Data Matrix• First data must be put into data matrix
• One row is called profile of that spesific statistical unit
• One column is called distribution of that spesific variable
Statistical unit
Variable 1 Variable 2 Variable 3 Variable 4
1 value …
2 Value …
3 …
Example of data matrix
Id. Gender Age Height(cm) Eye color
1 Male 23 176 Blue
2 Male 20 169 Grey
3 Female 21 164 Brown
4 Male 19 181 Blue
Frequency Distribution
• Frequency distribution: A grouping of data into categories showing the number of observations in each mutually exclusive (one statistical unit belongs to only one category) category. ONE VARIABLE!!
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Construction of a Frequency Distribution
C h a rt T it le
q ue stion tob e ad ressed
co llec t d a ta(ra w d a ta )
fre q ue n cy d is trib u tion
o rga n ize da ta p rese n t da ta(g ra p h)
d ra wco n clu s ion
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Statistical research prosess steps
Frequency Distribution
• Class mark (midpoint): A point that divides a class into two equal parts. This is the average between the upper and lower class limits.(Example)
• Class interval: For a frequency distribution having classes of the same size, the class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class.(Example)
• How to make a classification for a given data?
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EXAMPLE 1
• Dr. Tillman is the dean of the school of business and wishes to determine the amount of studying business school students do. He selects a random sample of 30 students and determines the number of hours each student studies per week: 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6.
• Organize the data into a frequency distribution.
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EXAMPLE 1 continued
Hours studying
Frequency, f
8-12 1 13-17 12 18-22 10 23-27 5 28-32 1 33-37 1
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Consider the classes 8-12 and 13-17. The class marks are 10 and 15. The class interval is 5 (13-8).
Suggestions on Constructing a Frequency Distribution
• Normally The class intervals used in the frequency distribution should be equal.(Not allways!)
• Determine a suggested class interval by using the formula: i = (highest value-lowest value)/number of classes.
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Number of classes
• Suggested class interval based on the number of observations =
)_log(*322,31
__
sfrequencietotal
valuelowestvaluehighest
• Number of classes is to be determided by professional judgement of the researher
Suggestions on Constructing a Frequency Distribution
• Use the computed suggested class interval to construct the frequency distribution. Note: this is a suggested class interval; if the computed class interval is 97, it may be better to use 100. ( Round uppwards to the nearest even number)
• Count the number of values in each class.
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Relative Frequency Distribution• The relative frequency of a class is obtained by dividing the class
frequency by the total frequency.
Frequency,f
RelativeFrequency
8-12 1 1/30=.0333
13-17 12 12/30=.400
18-22 10 10/30=.333
23-27 5 5/30=.1667
28-32 1 1/30=.0333
33-37 1 1/30=.0333
TOTAL 30 30/30=1
T
Hours
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Frequency table for nominal or order scale variable
• For nominal or order scale variable the frequency table is done in similar way. We just don’t need to form the classes.
• Example: Size of Clothes
Size f %f
S 12 …
M 6 …
L 23 …
XL 32 …
Stem-and-Leaf Displays
• Stem-and-Leaf Display: A statistical technique for displaying a set of data. Each numerical value is divided into two parts: the leading digits become the stem and the trailing digits the leaf.
• Note: An advantage of the stem-and-leaf display over a frequency distribution is we do not lose the identity of each observation.
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EXAMPLE 2
• Colin achieved the following scores on his twelve accounting quizzes this semester: 86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85. Construct a stem-and-leaf chart for the data.
stem leaf
6 9
7 8 9
8 2 3 4 5 6 8
9 1 2 6
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Graphic Presentation of a Frequency Distribution
• The three commonly used graphic forms are bar charts, histograms, frequency polygons, and a cumulative frequency distribution (ogive).
• Histogram: A graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other.
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Graphic Presentation of a Frequency Distribution
• A frequency polygon consists of line segments connecting the points formed by the class midpoint and the class frequency.
• A cumulative frequency distribution (ogive) is used to determine how many or what proportion of the data values are below or above a certain value.
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Histogram for Hours Spent Studying
0
2
4
6
8
10
12
14
10 15 20 25 30 35
Hours spent studying
Fre
qu
ency
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Notice that there is no gap between bars!!
Frequency Polygon for Hours Spent Studying
0
2
4
6
8
10
12
14
10 15 20 25 30 35
Hours spent studying
Fre
qu
ency
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Frequencies are marked to class midpoints!!!
Less Than Cumulative Frequency Distribution For Hours Studying
0
5
10
15
20
25
30
35
10 15 20 25 30 35
Hours Spent Studying
Frequency
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Cumulative frequencies are marked to actual upper limits of the classes!
Bar Chart
• A bar chart can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio).
• EXAMPLE 3: Construct a bar chart for the number of unemployed people per 100,000 population for selected cities of 1995.
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EXAMPLE 3 continued
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
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Bar Chart for the Unemployment Data
7300
5400
6700
89008200
8900
0
2000
4000
6000
8000
10000
1 2 3 4 5 6
Cities
# u
nem
plo
yed
/100
,000
AtlantaBostonChicagoLos AngelesNew YorkWashington
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Pie Chart
• A pie chart is especially useful in displaying a relative frequency distribution. A circle is divided proportionally to the relative frequency and portions of the circle are allocated for the different groups.
• EXAMPLE 4: A sample of 200 runners were asked to indicate their favorite type of running shoe.
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EXAMPLE 4 continued
• Draw a pie chart based on the following information.
Type of shoe # of runners
Nike 92
Adidas 49
Reebok 37
Asics 13
Other 9
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Pie Chart for Running Shoes
Nike
Adidas
ReebokAsics
Other
Nike
Adidas
ReebokAsics
Other
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