Department of Statistics

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Department of Statistics
Statistical Methods
Quantitative Summary Graphical Display of
Information
Dr. Rick Edgeman, Professor Chair and Six Sigma
Black Belt Tel. [phone] Fax. 1
208-885-7959 Email redgeman_at_uidaho.edu
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Copier Abuses
Firm in multilevel building, each level has its
own photocopier Free usage the tenth level is
selected and each person is given a key to the
machine on their level and will have their usage
tracked. Third level employee usage is tracked
in aggregate but those employees are unaware of
monitoring. Data for 50 days follows.
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Day 10th 3rd Day 10th 3rd Day
10th 3rd 1 500 440 18 360
20 35 150 370 2 420
220 19 310 250 36 140 405 3
440 360 20 320 350 37
130 130 4 480 110 21 290
150 38 150 120 5 450
240 22 290 250 39 130 70 6
460 360 23 270 230 40
110 240 7 450 80 24 250
90 41 90 20 8 420
420 25 240 50 42 80 450 9
410 310 26 250 320 43
90 20 10 405 30 27 250
360 44 70 40 11 380
290 28 230 450 45 20 320 12
360 410 29 240 270 46
50 140 13 360 460 30 220
380 47 40 90 14 370
420 31 190 190 48 20 130 15
350 150 32 150 500 49
30 480 16 320 170 33 170
290 50 30 350 17 350 250
34 120 150
50 Days of Photocopier Usage
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Process Histograms

• Such plots generally show the number of data
  values in successive categories with categories
  being
• mutually exclusive,
• nonoverlapping, and
• exhaustive.
• Construction of a histogram is both art and
  science.

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Histogram Construction

• Determine the range, R max -min,
• for 10th Level data R 500 - 20 480.
• Determine the number of categories to be used, k.
  A useful rule of thumb is k log2(n) where n
  is the sample size. This gives k 5 (4 to 6)
• Determine W R/k, this is the minimum width that
  a category should be, and usually W will be
  rounded to a convenient value. That is W is
  between 480/4 120 and
  480/6 80
• Construct categories classify data.

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• Given these guidelines, the number of categories,
  k, for a histogram as
• n k
• 1-7 dont bother
• 8-15 3 /- 1 These are rule-of-
• 16-31 4 /- 1 thumb. We should
• 32-63 5 /- 1 not under- or over-
• 64-127 6 /- 2 resolve the data.
• 128-255 7 /- 2

Histogram Categories
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Process Shape Via the Histogram
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Box-and-Whisker Plots

• Most commonly called boxplots.
• Consist of a five-number summary and four
  outlier points
• min, Q1, Q2, Q3, max
• Inner LOP Q1-1.5(IQR) Outer LOP
  Q1-3(IQR)
• Inner UOP Q31.5(IQR) Outer UOP Q33(IQR)
• min 20 max 500
• Q1 127.5 Q2 250 Q3 362.5
• Inner LOP 127.5 - 1.5(235) -225.5 copies
  (N/A) Outer LOP 127.5 -
  3.0(235) -577.5 copies (N/A)
• Inner UOP 362.5 1.5(235) 615 days
  Outer UOP 362.5
  3.0(235) 1,067.5 copies
• NO OUTLIERS BY THESE MEASURES!

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Shape - The Rest of the Story?
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PATTERNS This graph shows
an indirect relation with one
trait
increasing in value as the other
trait decreases.
Correlation
sometimes indicates causality. Prediction
and possibly process guidance may be
possible.
Scatter Diagrams Correlation Causality
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Descriptive Statistics Copies10
Total
Sum of Variable Count Mean SE Mean StDev
Variance Sum Squares Minimum Copies10
50 248.1 20.3 143.3 20521.3 12405.0
4083225.0 20.0 Variable Q1 Median
Q3 Maximum Range IQR Copies10 127.5 250.0
362.5 500.0 480.0 235.0
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Time Series Plot
250
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Descriptive Statistics Rainfall
Sum
of Variable N N Mean SE Mean StDev
Variance Sum Squares Rainfall 28 0
20.00 1.53 8.11 65.70 560.00
12974.00 Variable Minimum Q1 Median Q3
Maximum IQR Rainfall 6.00 14.50 17.00
27.75 37.00 13.25
Computation of Outlier Points Inner Outlier
Points Lower Q1 1.5(IQR) 14.50
1.5(13.25) 14.50 19.875 ??? Upper Q3
1.5(IQR) 27.75 19.875 47.625 of
rain Outer Outlier Points Lower Q1 3(IQR)
14.50 39.75 ??? (Not applicable) Upper
Q3 3(IQR) 27.75 39.75 67.5 of rain
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Stem-and-leaf of Rainfall N 28 Leaf Unit
1.0 3 0 689 7 1 1224 (8) 1 66667777
13 2 013 10 2 55788 5 3 0011 1 3 7
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Full name Box-and-Whisker Plot
Max 37
Q3 27.75
Whisker

Mean 20
Q2 17
Box
Q1 14.50
Whisker
Min 6
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Scores Score2 75 5625 84 7056 68 4624 95 9025 87
7569 93 8649 56 3136 87 7569 83 6889 82 6724 80 64
00 62 3844 91 8281 84 7056 75 5625
Stem-and-Leaf Plot of Exam Scores 1 5 6 2
6 2 3 6 8 3 7 5 7 55 (5) 8 02344 5
8 77 3 9 13 1 9 5
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Descriptive Statistics Scores
Sum
of Variable N N Mean SE Mean StDev
Variance Sum Squares Scores 15 0
80.13 2.89 11.19 125.12 1202.00
98072.00 Variable Minimum Q1 Median Q3
Maximum IQR Scores 56.00 75.00 83.00
87.00 95.00 12.00
Computation of Outlier Points Inner Outlier
Points Lower Q1 1.5(IQR) 75 1.5(12)
75 18 57 Upper Q3 1.5(IQR) 87 18
105 (not applicable) Outer Outlier
Points Lower Q1 3(IQR) 75 36 39
Upper Q3 3(IQR) 87 36 123 (not
applicable)
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Max 95
Q3 87
Q2 83
Mean 80.13
Q1 75
Min 56
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Descriptive Statistics Survival

Sum of Variable N N Mean SE Mean StDev
Variance Sum Squares Minimum Survival 40
0 167.4 37.0 233.9 54727.6 6697.0
3255621.0 1.00 Variable Q1 Median
Q3 Maximum Range IQR Skewness
Kurtosis Survival 16.5 83.0 223.5 999.0
998.0 207.0 2.44 6.45
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Descriptive Statistics Survival

Sum of Variable Status N N Mean SE
Mean StDev Variance Sum Squares Survival
20 2 0 4.50 3.50 4.95 24.50
9.00 65.00 30 4 0 26.50
8.21 16.42 269.67 106.00 3618.00
40 7 0 21.9 10.3 27.3
742.8 153.0 7801.0 50 4 0
179 118 235 55226 716
293842 60 7 0 120.4 31.6
83.7 7003.0 843.0 143539.0 70
9 0 326.2 94.4 283.3 80232.2 2936.0
1599646.0 80 6 0 155.8
46.0 112.6 12681.0 935.0 209109.0
90 1 0 999.00
999.00 998001.00 Variable Status Minimum
Q1 Median Q3 Maximum Range
IQR Survival 20 1.00 4.50
8.00 7.00 30
16.00 16.75 19.50 43.25 51.00
35.00 26.50 40 2.00 8.00
12.0 21.0 82.0 80.0 13.0
50 12.0 12.8 96.0 428 512
500 416 60 43.0
44.0 100.0 153.0 287.0 244.0 109.0
70 11.0 141.0 250.0 415.0
991.0 980.0 274.0 80
54.0 55.5 142.0 235.8 340.0 286.0
180.3 90 999.00 999.00
999.00 0.000000000
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Descriptive Statistics Survival

Sum of Variable TRT-TYPE N N Mean
SE Mean StDev Variance Sum
Squares Survival 1 21 0 120.5
28.2 129.0 16647.7 2531.0 637999.0
2 19 0 219.3 70.6 307.7
94676.2 4166.0 2617622.0 Variable TRT-TYPE
Minimum Q1 Median Q3 Maximum Range
IQR Survival 1 8.00 12.0 82.0
188.5 419.0 411.0 176.5 2
1.00 19.0 84.0 340.0 999.0 998.0 321.0
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Descriptive Statistics Survival

Sum of Variable CANCER-T
N N Mean SE Mean StDev Variance Sum
Squares Survival 1 14 0 281.3
92.5 346.2 119867.5 3938.0 2665980.0
2 11 0 91.9 35.5 117.9
13899.3 1011.0 231913.0 3 5
0 42.4 18.3 40.9 1674.8 212.0
15688.0 4 10 0 153.6
34.3 108.6 11790.0 1536.0
342040.0 Variable CANCER-T Minimum Q1
Median Q3 Maximum Range IQR Survival 1
1.00 14.0 122.0 442.3 999.0
998.0 428.3 2 2.00 16.0
51.0 153.0 341.0 339.0 137.0 3
8.00 10.0 18.0 87.0 90.0 82.0
77.0 4 12.0 37.0 170.5
235.8 340.0 328.0 198.8