4.2 Shapes of Distributions - University of Iowardecook/stat1010/notes/Section_4.2... · 4.2 Shapes of Distributions ! ... values that tend to be more spread out on one ... and median

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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

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Histogram of Octane

86 87 88 89 90 91 92 93 94 95 96

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23

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89

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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.

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Flat or Uniform

Perfectly flat

Figure 4.4

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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.

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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.

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Right-skewed

7.06.56.05.55.0

80

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pH

Freq

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ypH of Pork Loins

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Right-skewed Salary

sarlar(dollars)

Frequency

50000 100000 150000

020

4060

‘ski slope’ to the right

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Left-skewed

1 2 3 4 5 6 7 8 9 10

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Flexibility Index

Freq

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Flexibility Index of Young Adult Men‘ski slope’ to the left

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! 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

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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

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Multiple Peaks

0.1 0.2 0.3 0.4

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Size (carats)

Freq

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y

Size of Diamonds (carats)Major peak

Minor peak

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Homes sold in Iowa City by zip code and month

Year (data by the month)

Time-Series Diagrams – example

Multiple peaks

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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.

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Histogram of Octane

9695949392919089888786

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98

7

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32

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Octane

Freq

uenc

yHistogram of Octane Rating

Center

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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.

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Spread ! Variation matters.

0 5 10 0 5 10 0 5 10

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Comparing Distributions

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

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Workout Times: Men

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Count

30 40 50 60 70 80 90 100

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Workout Times: Women

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Count

30 40 50 60 70 80 90 100

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Count

30 40 50 60 70 80 90 100

Women

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Count

30 40 50 60 70 80 90 100

Men

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To describe a distribution, use…

! Shape ! Center ! Spread