psc200 3. descriptive statistics. level of measurement 1.nominal 2.ordinal 3.interval

PSC2003. Descriptive Statistics

Level of Measurement

1. Nominal

2. Ordinal

3. Interval

Lecture OverviewDescriptive Statistics

• Frequency Distribution• Data= Information –but too much information.

How do we summarize data?

• Central Measure of Tendency– Mode Nominal, Ordinal, Interval– Median Ordinal, Interval– Mean Interval

• Measures of Dispersion– Variance Interval

Frequency Distribution of Age in NES 2000

1820 22 24 26 28 30 32 34 36 3840 42 44 46 48 50 52 54 56 5860 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 95 97. 97 and older

Respondent age

0

10

20

30

40

50

Co

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t

Understanding Distributions

• What is “typical”?? Mode? Median? Mean?– Example: 2, 2, 2, 4, 6, 8, 8– Mode: 2– Median:4– Mean: 4.57

• Where does each measure of central tendency apply?

Nominal DataMeasure of Central Tendency: Mode

• What is typical?

1. Northeast 2. North Central 3. South 4. West

Census region of interview

0

100

200

300

400

500

600

700

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Nominal Data

• SPSS: =>Analyze =>Descriptive Stats=>Frequencies…

• What measures of central tendency & disperson can you identify?

• What’s the difference between percent and valid percent?

Statistics

career59

2

Valid

Missing

N

career

22 36.1 37.3 37.3

17 27.9 28.8 66.1

14 23.0 23.7 89.8

6 9.8 10.2 100.0

59 96.7 100.0

2 3.3

61 100.0

law

politics

business

academic/edu

Total

Valid

.Missing

Total

Frequency Percent Valid PercentCumulative

Percent

Nominal DataDisplay matters!

law politics business academic/edu

career

0

10

20

30

40

Pe

rc

en

t

career

Charts

Bar Charts

Percentage

Nominal Data

Would vote for ___?

9 14.8 15.3 15.3

11 18.0 18.6 33.9

3 4.9 5.1 39.0

20 32.8 33.9 72.9

2 3.3 3.4 76.3

13 21.3 22.0 98.3

1 1.6 1.7 100.0

59 96.7 100.0

2 3.3

61 100.0

Clark

Kerry

Sharpton

Bush

Kucinich

Dean

0

Total

Valid

.Missing

Total

Frequency Percent Valid PercentCumulative

Percent

Clark Kerry Sharpton Bush Kucinich Dean Edwards

Would vote for ___?

0

10

20

30

40

Pe

rce

nt

Would vote for ___?

Your Presidential Preference

“If the presidential election were today, for whom would you vote?”

Nominal Data

Primary Reason for Attending UR

9 14.8 15.0 15.0

19 31.1 31.7 46.7

3 4.9 5.0 51.7

29 47.5 48.3 100.0

60 98.4 100.0

1 1.6

61 100.0

Reputation

Financial Package

Safety School

Quality of Education

Total

Valid

.Missing

Total

Frequency Percent Valid PercentCumulative

Percent

Reputation Financial Package Safety School Quality of Education

Primary Reason for Attending UR

0

10

20

30

40

50

Pe

rce

nt

Primary Reason for Attending UR

Why UR?

“What was your primary reason for coming to UR?”

Ordinal Data

• Sequence matters, e.g. rankings• Median now has meaning• Example:

A12. Approve/disapprove Clinton job

Do you approve or disapprove of the way Bill Clinton is handling his job as president?

1. APPROVE5. DISAPPROVE 8. DON'T KNOW --> SKIP TO B1 9. RF 0. NA

1 5 8 9Count 1177 565 55 10

A12a. Strength of approval/disapproval of Clinton

IF R APPROVES CLINTON HANDLING JOB AS PRESIDENT/ IF R DISAPPROVES CLINTON HANDLING JOB AS PRESIDENT:

Strongly or not strongly? 1. STRONGLY 5. NOT STRONGLY 8. DK 9. RF 0. NA; INAP, 8,9,0 in A12

0 1 5 8 9 Count 65 1145 587 8 2

Summary: Approval/Disapproval of Clinton Job as President

Do you approve or disapprove of the way Bill Clinton is handling his job as president?

Strongly or not strongly? SUMMARY: APPROVAL/ DISAPPROVAL OF CLINTON JOB AS Built

from A12 and A12a.

1. Approve strongly 2. Approve not strongly 4. Disapprove not strongly 5. Disapprove strongly 8. DK (in A12 or A12a) 9. RF (in A12 or A12b) 0. NA

Summary Approval/Disapproval Clinton Job

1. Approve strongly 2. Approve not strongly 4. Disapprove not strongly

5. Disapprove strongly

Summary app/disapp Clinton job

0

200

400

600

800C

ou

nt

Ordinal DataMeasure of Central Tendency: Median

jobclint Summary app/disapp Clinton job

745 41.2 43.0 43.0

425 23.5 24.5 67.6

162 9.0 9.4 76.9

400 22.1 23.1 100.0

1732 95.8 100.0

75 4.2

1807 100.0

1 1. Approve strongly

2 2. Approve not strongly

4 4. Disapprove notstrongly

5 5. Disapprove strongly

Total

Valid

SystemMissing

Total

Frequency Percent Valid PercentCumulative

Percent

Ordinal DataMeasure of Central Tendency: Median

1. Never permit 2. Rape, incest, health only

3. Clear need 4. Always permit

Abortion scale

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400

600

800

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Ordinal DataMeasure of Central Tendency: Median

Abortion scale

215 11.9 12.2 12.2

525 29.1 29.9 42.1

265 14.7 15.1 57.2

753 41.7 42.8 100.0

1758 97.3 100.0

49 2.7

1807 100.0

1 1. Never permit

2 2. Rape, incest,health only

3 3. Clear need

4 4. Always permit

Total

Valid

SystemMissing

Total

Frequency Percent Valid PercentCumulative

Percent

Nominal Data

Party ID

14 23.0 23.0 23.0

24 39.3 39.3 62.3

15 24.6 24.6 86.9

4 6.6 6.6 93.4

4 6.6 6.6 100.0

61 100.0 100.0

Republican

Democrat

Independent

Other

Don't Know

Total

ValidFrequency Percent Valid Percent

CumulativePercent

Republican Democrat Independent Other Don't Know

Party ID

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10

20

30

40

Perc

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Party ID

Party Identification

“Generally speaking, do you usually consider yourself as a Republican, a Democrat, an Independent, or what?”

Interval Data

• Continuous: numbers on the real line• Mean (arithmetic):

• Example: 2, 3, 3, 5, 5, 6, 7, 7, 10, 201, 987– Mean =(2+3+3+5+5+6+7+7+10+201+987)/11 = 112.36

– Median?

– Modes?

Dichotomous or Dummy Variables• Nominal Data: Two Values• Can be treated as interval data

1. Male 2. Female

Gender

0

200

400

600

800

1,000

1,200

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Interval DataSkewed Distributions

Skewed Distributions

• Skewness:– For data Y1, Y2,…YN

Skewness =

Where is the mean, s is the standard deviation, and N is the number of data points

31

3

)1(

)(

sN

YYN

ii

Y

• Median = 45• Mean = 47.2• Modes = 37, 42

1820 22 24 26 28 30 32 34 36 3840 42 44 46 48 50 52 54 56 5860 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 95 97. 97 and older

Respondent age

0

10

20

30

40

50

Co

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Interval Data Grouped Into Categories for Visual Presentation

Variance

How dispersed or spread out the data is

Variance is the average squared deviation from the mean

1

)(1

2

2

n

XXs

n

ii

Standard Deviation = square root of variance = s

Use and Abuse of Descriptive Stats

Grofman, Koetzle, McGann. LSQ 2002. Congressional Leadership, 1965-96

• Are congressional leaders more extreme than their followers?

• Discern between theories that claim that

– leaders are more extreme

– leaders are more centrist

Measures of House Partisanship

House Party Members and Leaders

Conclusion: leaders not necessarily centrist but drawn from party mode.

Gary Jacobson. 1987. The Marginals Never Vanished. AJPS.

• “Marginal” – competitive elections

• Do incumbents have a growing advantage in elections?

• Do they win elections more easily than in the past?

• Has electoral competition declined? Incumbent behavior changed?

• Implications for democracy…

Incumbent Vote Share in House, 1952-82

Incumbents seem to be winning more votes in later years…

… but are incumbents winning more often?

All House Incumbents

Are incumbents winning more often?

Freshman Incumbents

Do Incumbents Win More Often?

Senior Incumbents

Jacobson’s Conclusion

• No net change in overall security for incumbents (same proportion, ca. 6-7%, lose)

• Marginals do increase but so does vote swing.

• First-term incumbents safer, senior incumbents not

• Explains absence of change in incumbent behavior