Statistics
101Displaying
Quantitative Data
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101Displaying
Quantitative Data
Displaying Quantitative Data
We begin with the following table of data providing the percentage of weekly visitors visiting one1 of the summary web pages of a web-page-rich website.
Percentage of Weekly Visitors Visiting one of the Summary Web Pages of a Web-Page-Rich Website between the Weeks of 17-23 April 2005 and 19-25 February 2006 Week Percentage 4/17/2005-4/23/2005
4/24/2005-4/30/2005
5/01/2005-5/07/2005
5/08/2005-5/14/2005
5/15/2005-5/21/2005
5/22/2005-5/28/2005
5/29/2005-6/04/2005
6/05/2005-6/11/2005
6/12/2005-6/18/2005
6/19/2005-6/25/2005
6/26/2005-7/02/2005
7/03/2005-7/09/2005
7/10/2005-7/16/2005
7/17/2005-7/23/2005
7/24/2005-7/30/2005
7/31/2005-8/06/2005
8/07/2005-8/13/2005
8/14/2005-8/20/2005
8/21/2005-8/27/2005
8/28/2005-9/03/2005
9/04/2005-9/10/2005
9/11/2005-9/17/2005
9/18/2005-9/24/2005
9/25/205-10/01/2005
10/02/2005-10/08/2005
10/09/2005-10/15/2005
10/16/2005-10/22/2005
10/23/2005-10/29/2005
10/30/2005-11/05/2005
11/06/2005-11/12/2005
11/13/2005-11/19/2005
11/20/2005-11/26/2005
11/27/2005-12/03/2005
12/04/2005-12/10/2005
12/11/2005-12/17/2005
12/18/2005-12/24/2005
12/25/2005-12/31/2005
1/01/2006-1/07/2006
1/08/2006-1/14/2006
1/15/2006-1/21/2006
1/22/2006-1/28/2006
1/29/2006-2/04/2006
2/05/2006-2/11/2006
2/12/2006-2/18/2006
2/19/2006-2/25/20061.26
1.05
1.02
0.92
1.03
0.89
0.99
0.93
1.03
0.75
0.75
0.86
*
0.82
0.81
0.82
0.84
0.62
0.83
0.82
0.65
0.52
0.58
0.81
0.72
0.64
0.70
0.75
0.77
0.71
0.76
1.09
0.94
0.66
0.85
1.01
0.98
1.05
0.86
0.80
0.76
0.61
0.99
0.77
0.91
From this table, we first create a dot plot of the data in the above table. To view the mechanics of creating such using Minitab, click the plot.
In the dot plot, each dot represents one occurrence of the percentage below, or rather so. The qualifications are: (1) there is not perfect alignment and (2) in the dot plot there appears to be two occurrences of 0.61%, but the table shows one occurrence of 0.61% and one occurrence of 0.62%. (The discrepancy appears to arise from a rounding by Minitab's worksheet (from 0.61844% to 0.62%) and a lopping (from 0.61844% to 0.61%) in Minitab's processing of the data for plots, displays, and charts.) Nonetheless, from the dot plot we easily see that (1) there is a heavy concentration of dots in the range of 0.80% to 0.85% and (2) as one moves to the right or left of this range the dots generally become more sparse.
Next, we create a stem-and-leaf display of the data within the table. Again, to view the mechanics of creating such, click the display.
Below, the stem of the stem-and-leaf display is highlighted in a bluish tone and the leaf - in a reddish tone.
From a stem-and-leaf display, one can quickly visualize both the data in the original table and their concentration(s). For example, from the first row of our stem-and-leaf display, we have on the right side 5 2, immediately above this row we see "Leaf Unit = 0.01", and consequently one of the entries in our original table is 52 x .01 = .52, the smallest table entry value. On the third line, we have the stem 6 and four leafs (1,1,3,4), which give us the four table entries 61 x .01 = .61, .61, 63 x .01 = .63, and 64 x .01 =.64. On the bottom line, we have the stem 12 and one leaf (6) which gives us the table entry .01 x 126 = 1.26, the largest table entry value.
From the pattern, it follows rather immediately that each leaf corresponds to a table entry and where there are no leafs (only a stem value), there is no corresponding data entry in the original table.
Note that in our stem-and-leaf display, each stem value occurs twice - two 5s, two 6s, two 7s, etc. When this occurs, the first stem entry may only have leafs 0, 1, 2, 3, or 4 and the second stem entry may only have leafs 5, 6, 7, 8, or 9. This may easily be seen from the pattern in our stem-and-leaf display.
Also, from the first column of a stem-and-leaf display, one can quickly determine the middle data value or the middle-two data values. As one descends from the top row toward the row containing the middle data value(s), the first-column row entry provides the cumulative number of leafs (data values) by the end of that row. Similarly, as one ascends from the bottom row toward the row containing the middle data value(s), the first-column row entry provides the cumulative number of leafs (data values) by the end of that row. Here, by the end of the 6th row from the top, the number of leaf entries total 18, and by the end of the 9th row from the bottom, the number of leaf entries total 18. Consequently, the 7th row from the top (the 10th row from the bottom), contains the middle data value or middle-two data values. The first-column entry for this row, unlike the others, provides the number of leafs in that particular row and is highlighted with parentheses; this permits a quick determination of the table's middle value(s). As the number of table entries is even, there are two middle data value entries and from visual inspection (and a little arithmetic and counting) they are .82 and .82; hence, the middle data value is .82.
Next, we create a histogram for the data of our table. Again, to view the mechanics of creating such, click the histogram.
The width of each blue bar marks off a range of visitation percentages and the height of each blue bar denotes the number of table entries falling within that range. For example, the first blue bar covers the percentage range .44-to-.56 and from the height of the bar we see that one table entry falls in this range. From the stem-and-leaf display, we see that this table entry is .52.
The frequency histogram provides a very visually friendly display of the data, although detail is lost from the stem-and-leaf display.
Histograms commonly come in slightly varying formats. One such gives the tic marks at the midpoint of each bar.
Another gives the tic marks at the midpoint and grid lines.
Another is a relative frequency histogram.
In a relative frequency histogram, each bar height gives the percent (or the proportion) of the table entries falling within the bar's horizontal data range.
Also commonly used is a cumulative relative frequency histogram.
In a cumulative relative frequency histogram, each bar height gives the percent (or the proportion) of the table entries falling within the bar's horizontal data range and all preceding data ranges. Consequently, the height of the last bar is 100 (or 1.00, when proportions are used).
Frequency and relative frequency histograms are likely the most commonly used histograms; however, for experienced chart readers, a cumulative frequency or cumulative relative frequency histogram may be the chart of choice.
1 The summary web page is Auto on Info's "Ten Best and Ten Worst Motor Vehicle Models."
Problem
Problem 1.
Below is a table providing the December 2005 book sales of an Amazon.com associate.
Table of Prices of Books Sold via an Amazon.com Associate in December 2005 Book
Book 1
Book 2
Book 3
Book 4
Book 5
Book 6
Book 7
Book 8
Book 9
Book 10
Book 11
Book 12
Book 13
Book 14
Book 15
Book 16
Book 17
Book 18
Book 19
Book 20
Book 21
Book 22
Book 23
Book 24
Book 25
Book 26
Book 27
Book 28
Book 29
Book 30
Book 31
Book 32
Book 33
Book 34
Book 35
Book 36
Book 37
Book 38
Book 39
Book 40
Book 41
Book 42
Book 43
Book 44
Book 45
Book 46
Book 47
Book 48
Book 49
Book 50
Book 51
Book 52
Book 53
Book 54
Book 55
Book 56
Book 57
Book 58
Book 59
Book 60
Book 61
Book 62
Book 63
Book 64
Book 65
Book 66
Book 67
Book 68
Book 69
Book 70
Book 71
Book 72
Book 73
Book 74
Book 75
Book 76
Book 77
Book 78
Book 79
Book 80
Book 81
Book 82
Book 83
Book 84
Book 85
Book 86
Book 87
Book 88
Book 89
Book 90
Book 91
Book 92
Book 93
Book 94
Book 95
Book 96
Book 97
Book 98
Book 99
Book 100
Book 101
Book 102
Book 103
Book 104
Book 105
Book 106
Book 107
Book 108
Book 109
Book 110
Book 111
Book 112
Book 113
Book 114
Book 115
Book 116
Book 117
Book 118
Book 119
Book 120
Book 121
Book 122
Book 123
Book 124
Book 125
Book 126
Book 127
Book 128
Book 129
Book 130
Book 131
Book 132
Book 133
Book 134
Book 135
Book 136
Book 137
Book 138
Book 139
Book 140
Book 141
Book 142
Book 143
Book 144
Book 145
Book 146
Book 147
Book 148
Book 149
Book 150
Book 151
Book 152
Book 153
Book 154
Book 155
Book 156
Book 157
Book 158
Book 159
Book 160
Book 161
Book 162
Book 163
Book 164
Book 165
Book 166
Book 167
Book 168
Book 169
Book 170
Book 171
Book 172
Book 173
Book 174Book Price
127.50
122.85
59.50
38.50
37.80
32.99
32.09
30.57
30.00
29.82
28.35
26.37
24.95
23.10
23.10
23.10
23.09
23.07
21.00
20.13
19.95
19.80
19.77
19.77
19.77
19.77
18.51
18.51
18.00
17.79
17.79
17.79
17.79
17.55
17.32
17.16
17.13
17.13
17.13
16.62
16.50
16.50
16.49
16.47
16.47
16.47
16.47
16.47
16.32
16.29
16.29
16.17
16.00
15.72
15.63
15.63
15.61
15.60
15.57
15.57
15.57
15.11
15.00
15.00
14.96
14.96
14.93
14.37
13.99
13.60
13.59
13.59
13.57
13.57
13.57
13.57
13.57
13.57
12.95
12.95
12.91
12.89
12.89
12.24
12.24
12.21
12.21
12.21
12.00
11.55
11.55
11.55
11.55
11.55
11.55
11.55
11.55
11.53
11.53
11.53
11.53
11.53
11.53
11.20
11.20
11.19
11.16
11.16
11.16
10.88
10.85
10.85
10.85
10.85
10.70
10.40
10.40
10.40
10.39
10.36
10.36
10.36
10.36
10.36
10.20
10.20
10.20
10.17
10.17
10.17
10.17
10.17
10.17
10.17
10.12
10.05
9.99
9.95
9.59
9.56
9.56
8.98
8.95
8.95
8.79
7.99
7.99
7.99
7.99
7.80
7.50
7.00
6.99
6.99
6.95
6.95
5.99
5.99
5.95
5.95
5.39
4.74
3.99
3.95
3.77
3.00
2.95
2.50
2.35
1.99
1.99
1.45
1.00
0.49
Copy the data columns into Minitab and for the data set, construct (1) a dot plot, (2) a stem-and-leaf display, (3) a frequency histogram with 10 intervals and with x-axis values at the midpoints of the intervals, (4) four relative frequency histograms with 5, 10, 13, and 26 intervals and with x-axis values at the end points of the intervals, and (5) a cumulative relative frequency histogram with 26 intervals and with x-axis values at the end points of the intervals.
To view some explanation of the mechanics for solving the problem and the final results, click here.
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