4.2 shapes of distributions - university of iowardecook/stat1010/notes/section_4.2... · 4.2 shapes...
TRANSCRIPT
1
4.2 Shapes of Distributions ! Symmetry
" Symmetrical or asymmetrical " If symmetrical, mounded or flat?
! Skew " Right, left
! Peaks or Modes " Unimodal, bimodal, multiple peaks
! Spread " Narrow spread or wide spread
2
Histogram of Octane
86 87 88 89 90 91 92 93 94 95 96
0
1
23
4
5
6
7
89
10
Octane
Freq
uenc
y
Histogram of Octane Rating
Symmetrical
One peak
A distribution is symmetric if its left half is a mirror image of its right half.
4
Flat or Uniform
Not perfectly flat, but close. We want to describe the general shape of the distribution.
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
6
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.
8
Right-skewed Salary
sarlar(dollars)
Frequency
50000 100000 150000
020
4060
‘ski slope’ to the right
9
Left-skewed
1 2 3 4 5 6 7 8 9 10
0
5
10
15
20
Flexibility Index
Freq
uenc
y
Flexibility Index of Young Adult Men‘ski slope’ to the left
10
! 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.
Right-skewed or Left-skewed
11
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
12
Multiple Peaks
0.1 0.2 0.3 0.4
0
5
10
15
Size (carats)
Freq
uenc
y
Size of Diamonds (carats)Major peak
Minor peak
13 13
Homes sold in Iowa City by zip code and month
Year (data by the month)
Time-Series Diagrams – example
Multiple peaks
14
Measures of Center
! These help describe a distribution, too. ! A typical or representative value.
" Mean, Median, Mode ! Summary of the whole batch of numbers. ! For symmetric distributions – easy.
15
Histogram of Octane
9695949392919089888786
10
98
7
6
5
4
32
1
0
Octane
Freq
uenc
yHistogram of Octane Rating
Center
16
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.
18
Comparing Distributions
! How do the distributions compare in terms of… " Shape? " Center? " Spread?