topic 2 frequency distributions and graphs
TRANSCRIPT
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2-1
Topic 2Chapter 2: Frequency
Distributions and Graphs
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2-2 Outline
2-1 Introduction
2-2 Organizing Data
2-3 Histograms, Frequency
Polygons, and Ogives
2-4 Other Types of Graphs
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2-3 Objectives
Organize data using frequency
distributions. Represent data in frequency
distributions graphically using
histograms, frequency polygons,and ogives.
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2-4 Objectives
Represent data using Pareto
charts, time series graphs, andpie graphs.
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2-5 2-2 Organizing Data
When data are collected in original
form, they are called raw data. When the raw data is organized
into a frequency distribution, the
frequency will be the number ofvalues in a specific class of thedistribution.
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2-6 2-2 Organizing Data
A frequency distribution is the
organizing of raw data in tableform, using classes andfrequencies.
The following slide shows anexample of a frequencydistribution.
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2-7
2-2 Three Types of Frequency
Distributions
Categorical frequency distributions -
can be used for data that can beplaced in specific categories, suchas nominal- or ordinal-level data.
Examples - political affiliation,religious affiliation, blood type etc.
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2-8
2-2 Blood Type Frequency
Distribution - Example (pg 2-4)
A
Class
B
Tally
C
Frequency
D
Percent
A ||||| 5 20
B ||||||| 7 28
O ||||||||| 9 36
AB |||| 4 16
TOTAL 25 100
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Blood Type Frequency
Distribution - Example (pg 2-4)
Twenty-five army
inductees were given ablood test to determine
their blood type. The
data set is shown on
table. Construct afrequency distribution
for the data.
A B B AB O
O O B AB B
B B O A O
A O O O AB
AB A O B A
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Procedure For Constructing
Frequency Distribution
Step 1:
Make a table as shown.Step 2:
Tally the data and place
the results in Col B.
Step 3:
Count the tallies and place
the results in Col C.
A
Class
B
Tally
C
Freq
D
PercentA
B
O
AB
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Procedure For Constructing
Frequency Distribution
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). The completed table is shown on slide 2-8.
%100.%n
f
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2-9
2-2 Ungrouped Frequency
Distributions
Ungrouped frequency distributions -
can be used for data that can beenumerated and when the range ofvalues in the data set is not large.
Examples - number of miles your
instructors have to travel from home tocampus, number of girls in a 4-childfamily etc.
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2-10
2-2 Number of Miles Traveled -
Example
Class Frequency
5 24
10 16
15 10
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2-11
2-2 Grouped Frequency
Distributions
Grouped frequency distributions - can
be used when the range of values in thedata set is very large. The data must begrouped into classes that are more thanone unit in width.
Examples - the life of boat batteries inhours.
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2-12
2-2 Lifetimes of Boat Batteries -
Example
C l a s s l i m
i t s C l a s s
B o u
n
d
a r i e s F r e q u e n c y C u m u l a t i v e
f r e q
u
e n
c y
2 4 - 3 7 2 3 . 5 - 3 7 . 5 4 4 3 8 - 5 1 3 7 . 5 - 5 1 . 5 1 4 1 8 5 2 - 6 5 5 1 . 5 - 6 5 . 5 7 2 5
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2-13
2-2 Terms Associated with a
Grouped Frequency Distribution
Class limits represent the smallest and
largest data values that can be includedin a class.
In the lifetimes of boat batteriesexample, the values 24 and 37 of thefirst class are the class limits.
The lower class limit is 24 and the upper class limit is 37.
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2-14
The class boundaries are used toseparate the classes so that there
are no gaps in the frequency
distribution.
2-2 Terms Associated with a
Grouped Frequency Distribution
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2-15
The class width for a class in afrequency distribution is found by
subtracting the lower (or upper)
class limit of one class minus the
lower (or upper) class limit of theprevious class.
2-2 Terms Associated with a
Grouped Frequency Distribution
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2-16
2-2 Guidelines for Constructing
a Frequency Distribution
There should be between 5 and 20
classes. It is preferable but not absolutely
necessary that the class width be an
odd number. The classes must be mutually
exclusive.
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2-17
2-2 Guidelines for Constructing
a Frequency Distribution
The classes must be continuous.
The classes must be exhaustive.
(There should be enough classes to accommodate
all the data)
The class must be equal in width.
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2-18
2-2 Procedure for Constructing
a Grouped Frequency Distribution
Find the highest and lowest value.
Find the range. Select the number of classes
desired.
Find the width by dividing the rangeby the number of classes androunding up.
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2-19
Select a starting point (usually the
lowest value); add the width to get thelower limits.
Find the upper class limits.
Find the boundaries. Tally the data, find the frequencies, and
find the cumulative frequency.
2-2 Procedure for Constructing
a Grouped Frequency Distribution
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2-20
In a survey of 20 patients who smoked,
the following data were obtained. Eachvalue represents the number ofcigarettes the patient smoked per day.Construct a frequency distributionusing six classes. (The data is given onthe next slide.)
2-2 Grouped Frequency Distribution -
Example
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2-21
10 8 6 14
22 13 17 19
11 9 18 14
13 12 15 15
5 11 16 11
2-2 Grouped Frequency Distribution -
Example
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2-22
Step 1: Find the highest and lowest
values: H = 22 and L = 5. Step 2: Find the range:
R = H – L = 22 – 5 = 17.
Step 3: Select the number of classesdesired. In this case it isequal to 6.
2-2 Grouped Frequency Distribution -
Example
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2-23
Step 4: Find the class width by
dividing the range by the numberof classes. Width = 17/6 = 2.83.
This value is rounded up to 3.
2-2 Grouped Frequency Distribution -
Example
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2-24
Step 5: Select a starting point for
the lowest class limit. Forconvenience, this value is chosen
to be 5, the smallest data value.
The lower class limits will be 5, 8,11, 14, 17, and 20.
2-2 Grouped Frequency Distribution -
Example
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2-25
Step 6: The upper class limits will
be 7, 10, 13, 16, 19, and 22. Forexample, the upper limit for the
first class is computed as 8 - 1,
etc.
2-2 Grouped Frequency Distribution -
Example
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2-26
Step 7: Find the class boundaries
by subtracting 0.5 from each lowerclass limit and adding 0.5 to the
upper class limit.
2-2 Grouped Frequency Distribution -
Example
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2-27
Step 8: Tally the data, write the
numerical values for the tallies in thefrequency column, and find thecumulative frequencies.
The grouped frequency distributionis shown on the next slide.
2-2 Grouped Frequency Distribution -
Example
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2-28
C l a s s L i m i t s C l a s s B o u n d a r i e s F r e q u e n c y C u m u l a t i v e F r e q u e n c y 0 5 t o 0 7 4 . 5 - 7 . 5 2 2 0 8 t o 1 0 7 . 5 - 1 0 . 5 3 5 1 1 t o 1 3 1 0 . 5 - 1 3 . 5 6 1 1 1 4 t o 1 6 1 3 . 5 - 1 6 . 5 5 1 6 1 7 t o 1 9 1 6 . 5 - 1 9 . 5 3 1 9 2 0 t o 2 2 1 9 . 5 - 2 2 . 5 1 2 0
Note: The dash “ -” represents “to”.
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2-29
2-3 Histograms, Frequency
Polygons, and Ogives
The three most commonly used
graphs in research are: The histogram.
The frequency polygon.
The cumulative frequency graph,
or ogive (pronounced o-jive).
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2-30
The histogram is a graph that
displays the data by using verticalbars of various heights to
represent the frequencies.
2-3 Histograms, Frequency
Polygons, and Ogives
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2-31 Example of a Histogram
2 0171 41 185
6
5
4
3
2
1
0
N u m b e r o f C i g a r e t t e s S m o k e d p e r D a y
F r e q u e n c y
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2-32
A frequency polygon is a graph
that displays the data by usinglines that connect points plottedfor frequencies at the midpoint of
classes. The frequenciesrepresent the heights of themidpoints.
2-3 Histograms, Frequency
Polygons, and Ogives
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2-33 Example of a Frequency Polygon
Frequency Polygon
262320171411852
6
5
4
3
2
1
0
Number of Cigar ettes Smoked per Day
F r e q u e n c y
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2-34
A cumulative frequency graph or
ogive is a graph that representsthe cumulative frequencies for the
classes in a frequency distribution.
2-3 Histograms, Frequency
Polygons, and Ogives
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2-35 Example of an Ogive
262320171411852
20
10
0
Number of Cigarettes Smoked per Day
C u m
u l a t i v e F r e q u e n c y
Ogive
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Example 2-7 on page 2-22
Construct a histogram, frequency
polygon, and ogive using relativefrequencies for the distribution of the
miles that 20 randomly selected runners
ran during a given week.
2-36
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Example 2-7 on page 2-22Class
Boundaries
Frequency Cumulative
Frequency
5.5-10.5 1 110.5-15.5 2 3
15.5-20.5 3 6
20.5-25.5 5 11
25.5-30.5 4 15
30.5-35.5 3 18
35.5-40.5 2 20
Total 20
2-37
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Example 2-7 on page 2-22Step 1:
Convert each frequency
to a relative frequencyby dividing thefrequency for eachclass by the totalnumber of observations.
Step 2:Find the cumulative
relative frequencies.
Step 3:
Draw each graph.
Class
Boundaries
Midpoints Relative
Freq
C. Relative
Freq
5.5-10.5 8 0.05 0.05
10.5-15.5 13 0.10 0.15
15.5-20.5 18 0.15 0.30
20.5-25.5 23 0.25 0.55
25.5-30.5 28 0.20 0.7530.5-35.5 33 0.15 0.90
35.5-40.5 38 0.10 1.00
1.00
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Histogram for Runner's Times
0
0.05
0.1
0.15
0.2
0.25
5.5-
10.5
10.5-
15.5
15.5-
20.5
20.5-
25.5
25.5-
30.5
30.5-
35.5
35.5-
40.5
Miles
R e l a t i v e F r e
q u e n c y
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Frequency Polygon for Runner's Times
0
0.05
0.1
0.15
0.2
0.25
3 8 13 18 23 28 33 38 43 48
Miles (midpoint)
R e l a t i v e F r e q
u e n c y
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Ogive for Runner's Times
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5.5 11 16.5 22 27.5 33 38.5
Miles
C u m u l a t i v e R e l a t i v e F r e q u e n c y
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2-38
2-4 Other Types of Graphs
Pareto charts - a Pareto chart is
used to represent a frequencydistribution for a categorical
variable.
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2-392-4 Other Types of Graphs - Pareto
Chart
When constructing a Pareto chart -
Make the bars the same width. Arrange the data from largest to
smallest according to frequencies.
Make the units that are used for thefrequency equal in size.
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2-40
Example of a Pareto Chart
1329341645.412.114.268.3
100.094.682.568.3
250
200
150
100
50
0
100
80
60
40
20
0
D ef ec t
C ountPer c entC um %
P e r c e n t
C o u n t
Enforcement Officers in U.S. National Parks During 1995.Par eto C har t f or the num ber of C r im es Inves tigated by Law
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Example 2-8 on page 2-32
The table shown
here is the averagecost per mile for
passenger vehicles
on state turnpikes.
Construct andanalyze a Pareto
chart for the data.
State Number
Indiana 2.9 cents
Oklahoma 4.3 cents
Florida 6.0 cents
Maine 3.8 cents
Pennsylvania 5.8 cents
Source: Pittsburg Tribune Review.
2-41
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Example 2-8 on page 2-32
Step 1:
Arrange the data from the
largest to smallestaccording to frequency.
Step 2:
Draw and label the x and y
axes.Step 3:
Draw the barscorresponding to thefrequencies.
State Number
Florida 6.0 cents
Pennsylvania 5.8 cents
Oklahoma 4.3 cents
Maine 3.8 cents
Indiana 2.9 cents
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Average Cost per Mile on State Turnpikes
0
1
23
4
5
6
7
F l o r i d
a
P e n n
s y l v a n i a
O k l a h o
m a
M a
i n e
I n d i a n
a
State
C o s t ( c e n t s )
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2-42
2-4 Other Types of Graphs –
Time Series Graph
Time series graph - A time series
graph represents data that occurover a specific period of time.
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2-43
2-4 Other Types of Graphs -
Time Series Graph
1 9 9 41 9 9 31 9 9 21 9 9 11 9 9 0
8 9
8 7
8 5
8 3
8 1
7 9
7 7
7 5
Y e a r
R i d e
r s h i p ( i n m i l l i o
n s )
P O R T AU T H O R IT Y T R AN S IT R ID E R S H IP
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Example 2-9 on page 2-33
The number (inmillion) of vehicles,
both passenger andcommercial, that usedthe PennsylvaniaTurnpike for the years
1999 through 2003 isshown. Construct andanalyze a time seriesgraph for the data.
Year Number
1999 156.2
2000 160.1
2001 162.3
2002 172.8
2003 179.4
Source: Tribune Review.
2-44
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Example 2-9 on page 2-33
Step 1:
Draw and label the x and y axes.
Step 2:
Label the x axis for years and the y axis for the number of vehicles.
Step 3:
Plot each point according to table.
Step 4:
Draw line segments connecting adjacent points.
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Number of Vehicles Traveling on Pennsylvania
Turnpike
150
160
170
180
1999 2000 2001 2002 2003
Year
N
u m b e r o f v e h
i c l e s ( i n
m i l l i o n s )
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2-45
2-4 Other Types of Graphs – Pie
Graph
Pie graph - A pie graph is a circle
that is divided into sections orwedges according to the
percentage of frequencies in each
category of the distribution.
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2-4 Other Types of Graphs -
Pie GraphRobbery (29, 12.1%)
Rape (34, 14.2%)
Assaults
(164, 68.3%)
Homicide
(13, 5.4%)
Pie Chart of the
Number of Crimes
Investigated by
Law EnforcementOfficers In U.S.
National Parks
During 1995
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Example 2-10 on page 2-35
This frequency
distribution showsthe number of
pounds of each snack
food eaten during the
Super Bowl.Construct a pie graph
for the data.
Snack Pounds
(Frequency)
Potato chips 11.2 million
Tortilla chips 8.2 million
Pretzels 4.3 million
Popcorn 3.8 million
Snack nuts 2.5 million
Total 30.0 million
Source: USA TODAY Weekend.
2-47
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Example 2-10 on page 2-35
Step 1:
Since there are 360o in a circle, the frequency for each
class must be converted into a proportional part of thecircle.
Step 2:
Each frequency must also be converted to a percentage.
Step 3:Using a protractor and a compass, draw the graph using
the appropriate degree measures.
o
n
f Degree 360.
%100.% n f
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Example 2-10 on page 2-35
Snack Pounds
(frequency)
Degree Percentage
Potato chips 11.2 million (11.2/30)x
360o=134o
(11.2/30)x
100%=38%
Tortilla chips 8.2 million 98o 27%
Pretzels 4.3 million 52o 14%
Popcorn 3.8 million 46o 13%
Snack nuts 2.5 million 30o 8%
30.0 million
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Super Bowl Snacks
Potato chips
38%
Tortila chips
27%
Pretzels
14%
Popcorn
13%
Snack nuts
8%
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2-4 Other Types of Graphs –
Stem and Leaf Plots
A stem and leaf plot is a data that uses
part of the data value as the stem andpart of the data value as the leaf to
form groups or classes.
2-48
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Example 2-12 on page 2-41
At an outpatient
testing center, the
number of
cardiograms
performed each day
for 20 days is shown.Construct a stem and
leaf plot for the data.
25 14 36 32
31 43 32 52
20 02 33 44
32 57 32 51
13 23 44 45
2-49
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Example 2-12 on page 2-41
Step 1:
Arrange the data in order.02, 13, 14, 20, 23, 25, 31, 32, 32, 32,
32, 33, 36, 43, 44, 44, 45, 51, 52, 57
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Example 2-12 on page 2-41
Step 2:
Separate the data according to the first digit.Step 3:
A display can be made by using the leading
digit as the stem and the trailing digit asthe leaf.
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Example 2-12 on page 2-41
Leading digit (stem) Trailing digit (leaf)
0 | 21 | 3 4
2 | 0 3 5
3 | 1 2 2 2 2 3 64 | 3 4 4 5
5 | 1 2 7