Stat 1010: Shapes of distributions
4.2 Shapes of Distributions
  Symmetry
 Symmetrical
 If
or asymmetrical
symmetrical, mounded or flat?
  Skew
 Right,
  Peaks
left
or Modes
 Unimodal,
bimodal, multiple peaks
  Spread
 Narrow
spread or wide spread
1
Histogram of Octane
Histogram of Octane Rating
10
9
8
Symmetrical
Frequency
One peak
7
6
5
4
3
2
1
0
86
87
88
89
90
91
92
93
94
95
96
Octane
A distribution is symmetric if its left half is a mirror
image of its right half.
2
Flat or Uniform
Figure 4.4
Perfectly flat
3
Stat 1010: Shapes of distributions
Flat or Uniform
Not perfectly flat, but
close. We want to
describe the general
shape of the distribution.
4
Not symmetrical
  A
distribution that is not symmetric must have
values that tend to be more spread out on one
side than on the other. In this case, we say that
the distribution is skewed.
5
Figure 4.7 (a) Skewed to the left (left-skewed): The mean
and median are less than the mode. (b) Skewed to the right
(right-skewed): The mean and median are greater than the
mode. (c) Symmetric distribution: The mean, median, and
mode are the same.
6
Stat 1010: Shapes of distributions
Right-skewed
pH of Pork Loins
80
70
Frequency
60
50
40
30
20
10
0
5.0
5.5
6.0
6.5
7.0
pH
7
Right-skewed
‘ski slope’ to
the right
40
0
20
Frequency
60
Salary
50000
100000
150000
sarlar(dollars)
8
Left-skewed
‘ski slope’ to
the left
20
Flexibility Index of Young Adult Men
Frequency
15
10
5
0
1
2
3
4
5
6
7
8
9
10
Flexibility Index
9
Stat 1010: Shapes of distributions
Right-skewed or Left-skewed
  A distribution
is left-skewed if its values
are more spread out on the left side.
  A distribution
is right-skewed if its values
are more spread out on the right side.
10
Number of Modes
  If
there are numerous obvious peaks, we
say there are multiple modes.
 One
peak  unimodal
 Two
peaks  bimodal
 More
than two peaks  multiple modes
  Some
peaks can be ‘major’ peaks and
some can be ‘minor’ peaks
11
Multiple Peaks
Major peak
Size of Diamonds (carats)
15
Frequency
Minor peak
10
5
0
0.1
0.2
0.3
0.4
Size (carats)
12
Stat 1010: Shapes of distributions
Time-Series Diagrams – example
Homes sold in Iowa City by
zip code and month
Multiple peaks
13
Year (data by the month)
Measures of Center
  These
  A
help describe a distribution, too.
typical or representative value.
 Mean,
Median, Mode
  Summary
of the whole batch of numbers.
  For symmetric distributions – easy.
14
Histogram of Octane
Histogram of Octane Rating
10
9
8
Frequency
7
6
5
4
3
2
1
0
86
87
88
89
90
91
92
Octane
93
94
95
96
Center
15
Stat 1010: Shapes of distributions
Spread
  Variation
matters.
 Tightly
clustered?
 Spread out?
 Low and high values?
Variation describes how widely data are
spread out about the center of a data set.
16
Spread
  Variation
0
5
matters.
10
0
5
10
0
5
10
17
Comparing Distributions
  How
do the distributions compare in terms
of…
 Shape?
 Center?
 Spread?
18
Stat 1010: Shapes of distributions
Workout Times: Men
5
3
Count
4
2
1
30
40
50
60
70
80
90
100
19
Workout Times: Women
5
3
Count
4
2
1
30
40
50
60
70
80
90
100
20
5
3
Count
4
Men
2
1
30
40
50
60
70
80
90
100
5
3
Count
4
2
1
30
40
50
60
70
80
90
Women
100
21
Stat 1010: Shapes of distributions
To describe a distribution, use…
  Shape
  Center
  Spread
22