Download Notes on ppt

Numerical Methods for
Describing Data
From Graphical to Numerical
1. Traumatic knee dislocation often requires surgery to repair
ruptured ligaments. One measure of recovery is range of motion
(measured by the angle formed when, starting with the leg straight,
the knee is bent as far as possible). The article “Reconstruction of
the Anterior and Posterior Cruciate Ligaments after Knee
Dislocation” reported the following post surgical range of motion
for a sample of 13 patients.
154 142
137
133
122
126
135
135
108
120
127
134
154+142+137+…+122
13
122
1695
x 
13
x  130.38
108 120 122 122 126 127 133 134 135 135 137 142 154
Median = 133
2. The paper “The Pedaling Technique of Elite Endurance Cyclists”
reported the accompanying data on single-leg power at a high
workload.
160 174 176 180 180 183187 191 194 200 205 211 211 244
2696
x
14
x  192.57
Median = 189
160 174 176 180 180 183187 191 194 200 204 205 211 211
2656
x
14
x  189.71
Median = 189
The mean is nonresistant to outliers.
Trimmed mean = 191
2292
x
12
x  191
3. The ages (measured by last birthday) of the employees of Dewey,
Cheatum and Howe are listed below.
22
31
21
49
26
42
42
30
28
31
39
39
20
37
32
36
35
33
45
47
49
38
28
48
x  35.33 years old
4 9 2 2 5 7 9 8
3 1 0 1 9 9 7 2 6 5 3 8
2 2 1 6 8 0 8
Median = 35.5
4H 5 7 8 9 9
4L 2 2
3H 5 6 7 8 9 9
3L 0 1 1 2 3
2H 6 8 8
2L 0 1 2
42
35.5
29
20 21 22 26 28 28 30 31 31 32 33 35 36 37 38 39 39 42 42 45 47 48 49 49
Q1
median
Q3
4. In the Consumer’s Report April 2007 issue, the following gas
mileage was reported in mixed driving for the following five brands
of Subaru:
Subaru B9 Tribeca
Subaru Forester
Subaru Impreza
Subaru Legacy
Subaru Outback
1
𝑛−1
16 mpg
22 mpg
23 mpg
18 mpg
19 mpg
𝑥𝑖 − 𝑥
x  19.6 mpg
Median = 19
2
Variance: the average of the squares of the
deviations of the observations from their mean
Observations:
Deviations:
xi
xi  x
16
16 – 19.6
22
22 – 19.6
Squared deviations
𝑥𝑖 − 𝑥
2
 3.6   12.96
2
 2.4   5.76
2
 3.4 
2
 11.56
23
23 – 19.6
18
 1.6   2.56
2
 .6   .36
19 – 19.6
2
  xi  x   33.2
19
18 – 19.6
2
S2 =
1
𝑛−1
2
𝑥𝑖 − 𝑥
Standard deviation 
1
  33.2 
4
s 2  8.3
variance
s  8.3  2.88 mpg
Roughly speaking, the standard deviation of 2.88 mpg measures a
typical or average distance between the individual gas mileage ratings
and the mean gas mileage rating for a Subaru.
With Toyota Prius
x  23.667
 x  x 
i
2
 529.333 s  10.29 mpg
2. The percentage of juice lost after thawing for 19 different strawberry
varieties appeared in the article “Evaluation of Strawberry Cultivars with
Different Degrees of Resistance to Red Scale”:
6L
5H
5L
4H
4L
3H
3L
0
5
0 0 1 3 3 4
6 6 6 6 8
1 1 2 4 4
3
Key: 3 | 3 means 33
Q1 = 44
Median = 46
IQR = 9
Q3 = 53
lowest = 33
highest = 60
Five number summary: lowest value, Q1, median,
Q3, highest value
Interquartile range (IQR) = Q3 – Q1
Outlier (below): smaller than Q1 – 1.5IQR
Outlier (above): larger than Q3 + 1.5IQR
44 – 1.5(9) = 30.5
No outliers!
53 + 1.5(9) = 66.5
Extreme outliers use 3IQR for calculations.
Boxplot
33
25
30
44
35
40
46
53
IQR
1.5 IQR
45
50
55
Percent of Juice Lost
60
60
65
Modified Boxplot
Suppose the data set replaces 33 with 28
41
44 46
28
53
60
*
25
30
35
40
1.5 IQR
45
50
55
Percent of Juice Lost
60
65
Recap of Numerical Descriptors
• Measures of center: mean, median
• Measures of variability: spread, range, IQR,
variance, standard deviation
• Mean, range, spread, variance and standard
deviation are nonresistant to outliers.
• How does shape affect the measures of
center and variability?
How the tail pulls the mean
MEDIAN = 9 = MEAN
AVERAGES
I=
9
II =
9
III =
9
IX
IX
VIII VIII
IV =
9
V=
9
IV VII VII IV
VI =
9
III
VI
VI
III
I
II
V
V
II
I
4
6
8
10
12
14
VII = 9
VIII = 9
IX =
9
2
16
18
20
How the tail pulls the mean
MEDIAN = 9
AVERAGES
I=
10
II =
10
III =
9
MEAN = 9.33
IX
VIII IX
IV =
9
V=
10
IV VII VIII V
VI =
9
III
VI VII IV
I
II
V
VI
III
II
I
4
6
8
10
12
14
16
VII = 9
VIII = 9
IX =
9
2
18
20
How the tail pulls the mean
MEDIAN = 9
AVERAGES
I=
12
II =
11
III =
10
MEAN = 10
IX
IV VIII
IV =
10
V=
10
III VII IX
VI =
10
II
VI VIII VI
IV
I
V
VII
V
III
II
I
6
8
10
12
14
16
18
VII = 9
VIII = 9
IX =
9
2
4
20
How the tail pulls the mean
MEDIAN = 9
AVERAGES
I=
13
II =
12
III =
11
IX
MEAN = 10.44
VIII
VII
IV =
11
V=
10
III
VI
VI =
10
II
V VIII VI
I
IV VII
VII = 9
IX
V
IV
III
II
I
12
14
16
18
20
VIII = 9
IX =
9
2
4
6
8
10