Transcript Ch2-Sec2.1

Chapter 2
Descriptive Statistics
1
Chapter Outline
 2.1 Frequency Distributions and Their Graphs
 2.2 More Graphs and Displays
 2.3 Measures of Central Tendency
 2.4 Measures of Variation
 2.5 Measures of Position
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Section 2.1
Frequency Distributions
and Their Graphs
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Section 2.1 Objectives
 Construct frequency distributions
 Construct frequency histograms, frequency polygons,
relative frequency histograms, and ogives
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Summarizing data/Frequency tables
 When data is collected from a survey or designed
experiment, they must be organized into a manageable
form. Data that is not organized is referred to as raw
data.
 Ways to Organize Data: Tables; Graphs; and
Numerical Summaries.
 A frequency distribution lists classes (or categories) of
values, along with frequencies (or counts) of the number
of values that fall into each class.
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Frequency tables: Definitions
 Lower Class Limits: are the smallest numbers that can
actually belong to different classes.
 Upper Class Limits: are the largest numbers that can
actually belong to different classes.
 Class Width: is the difference between two consecutive
lower class limits or two consecutive upper class limits.
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Frequency Distribution
Frequency
Class
Frequency, f
Distribution
1–5
5
 A table that shows Class width
6 – 10
8
classes or intervals 6 – 1 = 5
of data with a count
11 – 15
6
of the number of
entries in each class.
16 – 20
8
 The frequency, f, of
21 – 25
5
a class is the number
26 – 30
4
of data entries in the
class.
Lower class
Upper class
limits
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limits
Constructing a Frequency Distribution
1.
Decide on the number of classes.

2.
Find the class width.



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Usually between 5 and 20; otherwise, it may be difficult to
detect any patterns.
Determine the range of the data.
Divide the range by the number of classes.
Round up to the next convenient number.
Constructing a Frequency Distribution
3.
Find the class limits.




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You can use the minimum data entry as the lower limit of the
first class.
Find the remaining lower limits (add the class width to the lower
limit of the preceding class).
Find the upper limit of the first class. Remember that classes
cannot overlap.
Find the remaining upper class limits.
Constructing a Frequency Distribution
Make a tally mark for each data entry in the row of the
appropriate class.
5. Count the tally marks to find the total frequency f for each
class.
4.
1
10
0
Guidelines for Frequency tables
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11
1
1.
Be sure that the classes are mutually exclusive.
2.
Include all classes, even if the frequency is zero.
3.
Use the same width for all classes.
4.
Select convenient numbers for class limits.
5.
Use between 5 and 20 classes.
6.
The sum of the class frequencies must equal the number of
original data values.
Example: Constructing a Frequency
Distribution
The following sample data set lists the number of minutes 50
Internet subscribers spent on the Internet during their most recent
session. Construct a frequency distribution that has seven classes.
50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86
41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20
18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44
1
12
2
Solution: Constructing a Frequency
Distribution
50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86
41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20
18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44
Number of classes = 7 (given)
2. Find the class width
1.
max  min 86  7

 11.29
#classes
7
Round up to 12
1
13
3
Solution: Constructing a Frequency
Distribution
Lower
3. Use 7 (minimum value)
limit
as first lower limit. Add
7
the class width of 12 to Class
width = 12
19
get the lower limit of the
31
next class.
43
7 + 12 = 19
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Find the remaining
67
lower limits.
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1
14
4
Upper
limit
Solution: Constructing a Frequency
Distribution
The upper limit of the first
class is 18 (one less than the
lower limit of the second
class).
Add the class width of 12 to
get the upper limit of the next
class.
18 + 12 = 30
Find the remaining upper
limits.
1
15
5
Lower
limit
Upper limit
7
19
31
43
18
30
42
54
66
55
67
79
78
90
Class
width = 12
Solution: Constructing a Frequency
Distribution
4. Make a tally mark for each data entry in the row of the
appropriate class.
5. Count the tally marks to find the total frequency f for
each class.
Class
Tally
Frequency, f
7 – 18
1
16
6
IIII I
6
19 – 30
IIII IIII
10
31 – 42
IIII IIII III
13
43 – 54
IIII III
8
55 – 66
IIII
5
67 – 78
IIII I
6
79 – 90
II
2
Σf = 50
Determining the Midpoint
Midpoint of a class
(Lower class limit)  (Upper class limit)
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1
17
7
Class
Midpoint
Frequency, f
7 – 18
7  18
 12.5
2
6
19 – 30
19  30
 24.5
2
31 – 42
31  42
 36.5
2
Class width = 12
10
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Determining the Relative Frequency
Relative Frequency of a class
 Portion or percentage of the data that falls in a particular class.
class frequency
f

• relative frequency 
Sample size
n
1
18
8
Class
Frequency, f
7 – 18
6
19 – 30
10
31 – 42
13
Relative Frequency
6
 0.12
50
10
 0.20
50
13
 0.26
50
Determining the Cumulative Frequency
Cumulative frequency of a class
 The sum of the frequency for that class and all previous classes.
1
19
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Class
Frequency, f
Cumulative frequency
7 – 18
6
6
19 – 30
+ 10
16
31 – 42
+ 13
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Expanded Frequency Distribution
Class
Frequency, f
Midpoint
Relative
frequency
7 – 18
6
12.5
0.12
6
19 – 30
10
24.5
0.20
16
31 – 42
13
36.5
0.26
29
43 – 54
8
48.5
0.16
37
55 – 66
5
60.5
0.10
42
67 – 78
6
72.5
0.12
48
79 – 90
2
84.5
0.04
50
Σf = 50
2
20
0

f
1
n
Cumulative
frequency
Graphs of Frequency Distributions
frequency
Frequency Histogram
 A bar graph that represents the frequency distribution.
 The horizontal scale is quantitative and measures the data
values.
 The vertical scale measures the frequencies of the classes.
 Consecutive bars must touch.
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1
data values
Class Boundaries
Class boundaries
 The numbers that separate classes without forming gaps between
them.
Class
Frequency, f
• The distance from the upper
limit of the first class to the
lower limit of the second class
is 19 – 18 = 1.
• Half this distance is 0.5.
Class
Boundaries
7 – 18
6.5 – 18.5
19 – 30
10
31 – 42
13
• First class lower boundary = 7 – 0.5 = 6.5
• First class upper boundary = 18 + 0.5 = 18.5
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2
6
Class Boundaries
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3
Frequency, f
Class
7 – 18
19 – 30
31 – 42
Class
boundaries
6.5 – 18.5
18.5 – 30.5
30.5 – 42.5
43 – 54
55 – 66
67 – 78
79 – 90
42.5 – 54.5
54.5 – 66.5
66.5 – 78.5
78.5 – 90.5
8
5
6
2
6
10
13
Example: Frequency Histogram
Construct a frequency histogram for the Internet usage frequency
distribution.
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24
4
Class
Class
boundaries
Frequency, f
Midpoint
7 – 18
6.5 – 18.5
12.5
6
19 – 30
18.5 – 30.5
24.5
10
31 – 42
30.5 – 42.5
36.5
13
43 – 54
42.5 – 54.5
48.5
8
55 – 66
54.5 – 66.5
60.5
5
67 – 78
66.5 – 78.5
72.5
6
79 – 90
78.5 – 90.5
84.5
2
Solution: Frequency Histogram
(using Midpoints)
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25
5
Solution: Frequency Histogram
(using class boundaries)
6.5
18.5
30.5
42.5
54.5
66.5
78.5
90.5
You can see that more than half of the subscribers spent between 19 and
54 minutes on the Internet during their most recent session.
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Graphs of Frequency Distributions
frequency
Frequency Polygon
 A line graph that emphasizes the continuous change in
frequencies.
 The line segments are extended to the right and left so that the
graph begins and ends on the x-axis.
data values
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7
Example: Frequency Polygon
Construct a frequency polygon for the Internet usage frequency
distribution.
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8
Class
Midpoint
Frequency, f
7 – 18
12.5
6
19 – 30
24.5
10
31 – 42
36.5
13
43 – 54
48.5
8
55 – 66
60.5
5
67 – 78
72.5
6
79 – 90
84.5
2
Solution: Frequency Polygon
Internet Usage
Frequency
The graph should
begin and end on the
horizontal axis, so
extend the left side to
one class width before
the first class
midpoint and extend
the right side to one
class width after the
last class midpoint.
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12
10
8
6
4
2
0
0.5
12.5
24.5
36.5
48.5
60.5
72.5
84.5
Time online (in minutes)
You can see that the frequency of subscribers increases up to 36.5
minutes and then decreases.
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96.5
Graphs of Frequency Distributions
relative
frequency
Relative Frequency Histogram
 Has the same shape and the same horizontal scale as the
corresponding frequency histogram.
 The vertical scale measures the relative frequencies, not
frequencies.
data values
3
30
0
Example: Relative Frequency
Histogram
Construct a relative frequency histogram for the Internet usage
frequency distribution.
3
31
1
Class
Class
boundaries
Frequency,
f
Relative
frequency
7 – 18
6.5 – 18.5
6
0.12
19 – 30
18.5 – 30.5
10
0.20
31 – 42
30.5 – 42.5
13
0.26
43 – 54
42.5 – 54.5
8
0.16
55 – 66
54.5 – 66.5
5
0.10
67 – 78
66.5 – 78.5
6
0.12
79 – 90
78.5 – 90.5
2
0.04
Solution: Relative Frequency Histogram
6.5
18.5
30.5
42.5
54.5
66.5
78.5
90.5
From this graph you can see that 20% of Internet subscribers spent
between 18.5 minutes and 30.5 minutes online.
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2
Graphs of Frequency Distributions
cumulative
frequency
Cumulative Frequency Graph or Ogive
 A line graph that displays the cumulative frequency of each class
at its upper class boundary.
 The upper boundaries are marked on the horizontal axis.
 The cumulative frequencies are marked on the vertical axis.
 Ogives are useful for determining the number of values less than
some particular class boundary.
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3
data values
Constructing an Ogive
Construct a frequency distribution that includes cumulative
frequencies as one of the columns.
2. Specify the horizontal and vertical scales.
1.


3.
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4
The horizontal scale consists of the upper class boundaries.
The vertical scale measures cumulative frequencies.
Plot points that represent the upper class boundaries and
their corresponding cumulative frequencies.
Constructing an Ogive
Connect the points in order from left to right.
5. The graph should start at the lower boundary of the first class
(cumulative frequency is zero) and should end at the upper
boundary of the last class (cumulative frequency is equal to the
sample size).
4.
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5
Example: Ogive
Construct an ogive for the Internet usage frequency distribution.
3
36
6
Class
Class
boundaries
Frequency,
f
Cumulative
frequency
7 – 18
6.5 – 18.5
6
6
19 – 30
18.5 – 30.5
10
16
31 – 42
30.5 – 42.5
13
29
43 – 54
42.5 – 54.5
8
37
55 – 66
54.5 – 66.5
5
42
67 – 78
66.5 – 78.5
6
48
79 – 90
78.5 – 90.5
2
50
Solution: Ogive
Internet Usage
Cumulative frequency
60
50
40
30
20
10
0
6.5
18.5
30.5
42.5
54.5
66.5 78.5
90.5
Time online (in minutes)
From the ogive, you can see that about 40 subscribers spent 60 minutes
or less online during their last session. The greatest increase in usage
occurs between 30.5 minutes and 42.5 minutes.
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Practice Questions
Q. (2.1)
The average quantitative GRE scores for the top 30
graduate schools of engineering are listed below.
Construct a frequency distribution with six classes.
767 770 761 760 771 768 776 771 756 770
763 760 747 766 754 771 771 778 766 762
780 750 746 764 769 759 757 753 758 746
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Practice Questions
Q. (2.2)
The ages of the signers of the Declaration of Independence
are shown below (age is approximate since only the birth
year appeared in the source and one has been omitted since
his birth year is unknown). Construct a frequency
distribution for the data using seven classes.
3
39
9
41
54
47
40
39
35
50
37
49
42
70
32
44
52
39
50
40
30
34
69
39
45
33
42
44
63
60
27
42
34
50
42
52
38
36
45
35
43
48
46
31
27
55
63
46
33
60
62
35
46
45
34
53
50
50
Practice Questions
Q. (2.3)
For 108 randomly selected college applicants, the
following frequency distribution for entrance exam scores
was obtained. Construct a histogram, frequency polygon,
and ogive for the data.
4
40
0
Class limits
Frequency
90 – 98
6
99 – 107
22
108 – 116
43
117 – 125
28
126 – 134
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Practice Questions
Q. (2.4)
The number of calories per serving for selected ready –
to - eat cereals is listed here. Construct a frequency
distribution using seven classes. Draw a histogram,
frequency polygon, and ogive for the data, using relative
frequencies. Describe the shape of the histogram.
4
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1
130
190
140
80
100
120
220
220
110
100
210
130
100
90
210
120
200
120
180
120
190
210
120
200
130
180
260
270
100
160
190
240
80
120
90
190
200
210
190
180
115
210
110
225
190
130
Section 2.1 Summary
 Constructed frequency distributions
 Constructed frequency histograms, frequency polygons,
relative frequency histograms and ogives
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