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.

3

Flat or Uniform

Perfectly flat

Figure 4.4

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.

7

Right-skewed

7.06.56.05.55.0

80

70

60

50

40

30

20

10

0

pH

Freq

uenc

ypH of Pork Loins

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.

17

Spread ! Variation matters.

0 5 10 0 5 10 0 5 10

18

Comparing Distributions

! How do the distributions compare in terms of… " Shape? " Center? " Spread?

19

Workout Times: Men

1

2

3

4

5

Count

30 40 50 60 70 80 90 100

20

Workout Times: Women

1

2

3

4

5

Count

30 40 50 60 70 80 90 100

21

1

2

3

4

5

Count

30 40 50 60 70 80 90 100

Women

1

2

3

4

5

Count

30 40 50 60 70 80 90 100

Men

22

To describe a distribution, use…

! Shape ! Center ! Spread