Sociology 549,Lecture 3

Graphs

by Paul von Hippel

Common graphs for frequency distributions

• Pie chart

• Line chart (frequency polygon)

• Bar chart

• Histogram

Other common graphs

• Time series

• Statistical map

Common distortions

• False perspective– e.g., tilting a pie chart

• Shortening an axis; e.g.,– not starting the vertical at 0– breaking the vertical– squishing the horizontal

• Reasons– Add visual interest– Make small differences look big,– Or make big differences look small

Shapes of distributions

• Symmetric• Skewed

– Positively skewed– Negatively skewed

• Modal– unimodal– bimodal– multimodal

Pie chart

• Rare in research• Common in media• Hard to compare

wedges (different orientations)

• Can’t show order– Restrict to nominal

variables

Majors in Soc 549

Criminology35%

Psychology4%

Sociology61%

Perspective distortion

• Add a meaningless 3rd dimension

• Tilt pie away– Edge adds to front

– Perspective shrinks back

– Comparisons even harder

Sociology61%

Criminology35%

Psychology4%

Pie Charts in politics• Federal budget, from the website of the War Resisters’ League

Human Resources

32%

General Government

16%

Physical Resources

6%

Past Military20%

Current Military

26%

• Redrawn

Bar chart(column chart)

• In research,more common than pie

• Can show order– Appropriate for ordinal

and interval

– (as well as nominal)

• Easy to compare vertical distances

Majors in Soc 549

0

2

4

6

8

10

12

14

16

Sociology Criminology Psychology

Axis distortion

• Start vertical above zero– Exaggerates all

differences

• Similar distortion:– Break vertical axis

Majors in Soc 549

7

8

9

10

11

12

13

14

15

Sociology Criminology Psychology

Perspective distortion

• Add meaningless 3rd dimension– Reduces differences

(caps same size)

0

2

4

6

8

10

12

14

Psychology Criminology Sociology

Perspective distortion (continued)

• Add 3rd dimension and overlap

• Exaggerates differences– Hides side of smaller

bars

– Also hides part of top

• Rotation would make it worse

0

2

4

6

8

10

12

14

Psychology Criminology Sociology

Line chart(frequency polygon)

• Common in research• Can show order

– Appropriate for ordinal and interval variables

0

2

4

6

8

10

12

14

16

Sociology Criminology Psychology

Axis distortions

• Start vertical above zero– Or break vertical

7

8

9

10

11

12

13

14

15

Sociology Criminology Psychology

Perspective distortion

• Add meaningless 3rd dimension

• Tilt horizontal– Exaggerates trend

SociologyCriminology

Psychology

S1

0

2

4

6

8

10

12

14

Bar vs. line: similarities

• Bar and line charts almost equivalent– Start with a bar chart

• Connect tops

• remove bottoms

• You get a line chart!

Majors in Soc 549

0

2

4

6

8

10

12

14

16

Sociology Criminology Psychology

0

2

4

6

8

10

12

14

16

Sociology Criminology Psychology

Bar vs. line: Differences

• Line suggests trend more strongly– Helpful with ordinal or

interval variables

– Misleading with nominal

0

2

4

6

8

10

12

14

16

Sociology Criminology Psychology

0

2

4

6

8

10

12

14

16

Senior Junior Sophomore

Bar vs. line: Differences

0

2

4

6

8

10

12

14

16

Sociol

ogy

Crimino

logy

Psych

ology

Physic

al th

erapy

Comm

unica

tions

Biolog

y

Social statistics

Sociology of Sport

• Line eases comparison of groups

0

2

4

6

8

10

12

14

16

Socio

logy

Crimin

ology

Psych

ology

Physic

al ther

apy

Communic

ation

s

Biolog

y

Social statistics

Sociology of Sport

Histograms• Like bar chart,

except– Variable typically

continuous– Bars touch

• usually

– Horizontal can represent equal class intervals (“bins”)

• Bin shown by center value (e.g. 35.0)

• Or by ends of class interval (e.g. 33.75-36.25)

Starting salary in thousands

47.545.042.540.037.535.032.530.027.525.022.5

Starting salaries for BAs in sociology, 2000-2001

National Association of Colleges and Employers

survey of college placement off ices

30

20

10

0

Std. Dev = 4.31

Mean = 28.7

N = 96.00

Summary: Graphical display of distributions

Nominal Ordinal IntervalPie √ Book approvesBar √ √Line Book disapproves √ √Histogram if continuous

Shape of distributions: Positive or right skew

• Positive or right skew• Characteristics:

– Peak on left

– Long right tail• Stretched (Skewed)

to the right

– A few large values

• Common cause– Floor but no ceiling

Starting salary in thousands

47.545.042.540.037.535.032.530.027.525.022.5

Starting salaries for BAs in sociology, 2000-2001

National Association of Colleges and Employers

survey of college placement off ices

30

20

10

0

Std. Dev = 4.31

Mean = 28.7

N = 96.00

Negative or left skew

• Negative or left skew• Characteristics

mirror positive skew:– Peak on right– Long left tail

• Stretched (Skewed) to the left

– A few small values

• Common cause– Ceiling but no floor Assignment 1 scores

100.0

95.0

90.0

85.0

80.0

75.0

70.0

65.0

60.0

55.0

50.0

45.0

40.0

35.0

Assignment 1 scores, sociology 549, winter 200114

12

10

8

6

4

2

0

Std. Dev = 15.79

Mean = 75.4

N = 101.00

Symmetry

• Symmetry, no skew– Two tails,

or no tails

• Important example:– The normal curve

Height of adult males (inches)

82.00

80.40

78.80

77.20

75.60

74.00

72.40

70.80

69.20

67.60

66.00

64.40

62.80

61.20

59.60

58.00

Fre

qu

en

cy

200

100

0

Dummy variables

• Describe the shape of this distribution.

Sex distribution, Soc 549, winter 2003

0

5

10

15

20

25

30

0 1

Sex dummy (1=female)

Number of

students

Unimodal distributions

• Mode– peak

– most common value

• Unimodal– one peak

– e.g., starting salaries• mode around $27K

• Interpretation– the most common salaries

– are in the high $20sStarting salary in thousands

47.545.042.540.037.535.032.530.027.525.022.5

Starting salaries for BAs in sociology, 2000-2001

National Association of Colleges and Employers

survey of college placement off ices

30

20

10

0

Std. Dev = 4.31

Mean = 28.7

N = 96.00

Bimodal distributions

• Bimodal– two modes– e.g., # children

• modes at 0 and 2• Interpretation?

NUMBER OF CHILDREN

EIGHT OR MORE

7

6

5

4

3

2

1

0

Co

un

t

500

400

300

200

100

0

Multimodal distributions

• Multimodal– more than 2 modes

– e.g., hours worked by OSU sociology students

• modes at 0, 20, 40

(primary)mode

secondarymodes

Review of shape

• Shapes– Symmetric– Skewed

• Positive (right)

• Negative (left)

– Unimodal, bimodal, multimodal

Time series:don’t show distributions,show change over time

BAs in social science and history(National Center for Educational Statistics)

0%5%

10%15%20%25%30%35%40%45%50%

1970 1975 1980 1985 1990 1995

% women

Axis distortion:start (or break) vertical

above zeroBAs in social science and history

30%

32%

34%

36%

38%

40%

42%

44%

46%

1970 1975 1980 1985 1990 1995

% women

Axis distortion:Squeeze vertical

or stretch horizontal

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

1970 1975 1980 1985 1990 1995

% women

Axis distortion:Squeeze horizontal or stretch vertical

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

1970 1980 1990

% women

Axis distortion in business• NASDAQ stock index, reported by Yahoo!

•Redrawn

0

500

1000

1500

2000

2500

6-Jan-02 6-Jan-03

NASDAQ stock index

Graphical distortion: Summary

• Axis distortion– Squeeze one axis

• Honest aspect ratio is 3:2 (Tufte)

– Start or break vertical axis above zero

• Perspective distortion– Add disproportionate areas in a meaningless 3rd

dimension– Use blocking & tilting

Graphics: Good advice

• Keep it simple– Don’t stretch axes– Don’t start or break axes above zero– Don’t use 3-D

• If you have to use 3D, avoid abuses

– With just a few numbers,consider a table instead of a graph

Graphics: Evil advice

• Use every trick (3D, distorted axes)– Maximize differences that serve your purpose– Minimize differences that work against you