Statistics Economics 91 - Pitzer Collegepzacad.pitzer.edu/~lyamane/ec91wk2.pdf · Stem –all but...
Transcript of Statistics Economics 91 - Pitzer Collegepzacad.pitzer.edu/~lyamane/ec91wk2.pdf · Stem –all but...
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Data
• Measurement– What is being measured?
– How is it being measured?
– Why is it being measured?
• Mean and Variation– What is the central tendency of the data?
– Why are all the observations in the dataset not the same?
• Relationships with other variables
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Examining Data
• Begin with graphs
– Then go on to numerical summaries
• First look at each variable by itself
– Then look at relationships between variables
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Graphing Data
• Graphs for qualitative data– Bar graphs (count or percent)
– Pie charts (percents)
– * Be careful with 3D figures
• Graphs for quantitative data– Stem plots (stem and leaf plots)
– Histogram
– Time series graphs
– * Pay attention to the scale
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0
5
10
15
20
25
30
Female Male
Gender Distribution
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0%
10%
20%
30%
40%
50%
60%
70%
Female Male
15
26
Pe
rce
nt
Gender Distribution
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63.41%
36.59%
Gender Distribution
men women
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15
26
Gender Distribution
Female Male
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0
50
100
150
200
250
300
350
400
White Latino Asian Black
Nu
mb
er
Ethnic Group
Diversity
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0
0.1
0.2
0.3
0.4
0.5
0.6
White Latino Asian Black
Percent
Ethnic Group
Diversity
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336
120
80
42
Diversity
White Latino
Asian Black
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58%21%
14%
7%
Diversity
White
Latino
Asian
Black
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0
5000
10000
15000
20000
25000
30000
35000
40000
45000
UnitedStates
HongKong
Japan Taiwan SouthKorea
Mexico Brasil China Peru ElSalvador
Bolivia
Country
GDP Per Capita
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0
20000
40000
60000
80000
100000
120000
Professor President
Do
llars
Income Inequality
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0
20000
40000
60000
80000
100000
120000
Professor President
Do
llarsIncome Inequality
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0
20000
40000
60000
80000
100000
120000
Professor President
Do
llars
Income Inequality
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Graphing Data
• Graphs for qualitative data– Bar graphs (count or percent)
– Pie charts (percents)
– * Be careful with 3D figures
• Graphs for quantitative data– Stem plots (stem and leaf plots)
– Histogram
– Time series graphs
– * Pay attention to the scale
![Page 17: Statistics Economics 91 - Pitzer Collegepzacad.pitzer.edu/~lyamane/ec91wk2.pdf · Stem –all but the right most digit 2. Leaf –final digit 3. Write stems in a vertical column,](https://reader034.fdocuments.us/reader034/viewer/2022042107/5e865da726ba6053c21dc928/html5/thumbnails/17.jpg)
Stem Plots
1. Stem – all but the right most digit
2. Leaf – final digit
3. Write stems in a vertical column, largest to smallest
4. Write each leaf in a row next to the stem
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Stem Plots
Examine the overall distribution of the data
- overall pattern and striking deviations
- shape, center, and spread
- outliers: fall outside the overall pattern
Does the distribution have one peak (unimodal) or several peaks?
Is the distribution symmetric, skewed to the right, or skewed to the left?
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Symmetric or Skewed
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Histograms
• Stem plots are of limited usefulness
– Hard with large datasets
– Can’t choose your own interval sizes
• Histograms work better
• You can choose the size of your intervals
• You can display counts or percentages
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Time Series Plots
• When data are collected over time
• Plot observations in time order
• Observe any seasonal variation
– Repetition at regular known intervals
• Observe trends over time
– Persistent long term rise or fall
• * Notice the scale of the axes!
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Numerical summaries
• Use numbers to describe the center and spread of any dataset
• Measures of Center– Mean: average value– Median: middle value– Mode: most common value
• Measures of spread– Range– Interquartile Range– Five number summary (box plots)– Variance and Standard Deviation
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Choosing measures of center and spread
• Use the sample mean and sample standard deviation if you have a symmetric distribution
• Use the five number summary if you have a skewed distribution
• A plot or graph gives you the best overall picture of a distribution
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Changing units of measure
• When you change the units of measure– Feet to inches– Pounds to kilograms
• The mean, variance and standard deviation will change
• Example: Let Y be a linear transformation of XY = aX + b where a and b are constants
ത𝑌 = 𝑎 ത𝑋 + 𝑏𝑠𝑌 = 𝑎 𝑠𝑋𝑠𝑦2 = 𝑎2(𝑠𝑋
2)
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How to explore your data
• Plot your data with a stem plot or histogram
• Look at the overall pattern and any striking deviations
• Calculate some numerical summaries to describe the center and the spread of the distribution
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Relationships between variables
• Is there a relationship between two variables?
• Is it a positive relationship or negative relationship?
• How strong is this relationship?
• Is it an explanatory relationship?– Dependent variable: response variable measuring
the outcome of a study
– Independent variable: explanatory variable which explains the change in the dependent variable
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Quantitative Data
• Start with a graphical display, then add numerical summaries
• Look for overall patterns and deviations from those patterns
• If the pattern is regular, we can try to model the relationship and use regression analysis
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Scatterplot
• Displays the relationship between to quantitative variables on the same entity
• Put the dependent (response) Y variable on the vertical axis
• Put the independent (explanatory) X variable on the horizontal axis
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Do taller people weight more?
100
120
140
160
180
200
220
240
60 62 64 66 68 70 72 74 76
Wei
ght
Height
Height and Weight
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0
2
4
6
8
10
12
2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9 4.1
Ho
urs
of
Stu
dy
GPA
Hours of Study and GPA
Is it worth it?
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64
66
68
70
72
74
76
78
80
60 61 62 63 64 65 66 67 68 69 70 71
Fath
er H
eigh
t
Mother Height
Heights of Mothers and Fathers
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0
1
2
3
4
5
6
7
8
0 5 10 15 20 25 30 35 40 45 50
Day
s w
ith
Bre
akfa
st
Hours of Exercise
Exercise and Breakfast
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2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8
4
0 1 2 3 4 5 6 7 8
GP
A
Days with Breakfast
GPA and Breakfast
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1000
1100
1200
1300
1400
1500
1600
3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4
Tota
l SA
T
GPA
SAT and GPA
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Measuring Relationships
• Covariance (sample)
𝑐𝑜𝑣𝑥𝑦 =σ𝑁 𝑋𝑖 − ത𝑋 𝑌𝑖 − ത𝑌
𝑁 − 1
• Correlation Coefficient
𝑟𝑥𝑦 =𝑐𝑜𝑣𝑥𝑦
𝑆𝐷𝑥𝑆𝐷𝑦
−1 ≤ 𝑟𝑥𝑦 ≤ 1
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Qualitative Data
• Examine relationships by looking at tables
• You can present counts or percent
• You can have percent of row totals or column totals or both
• * Be careful of inappropriate aggregation (Simpson’s paradox)
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Region and Major
US Europe Asia Total
Engineering 61,941 158,931 280,772 501,644
Natural Science 111,158 140,126 242,879 494,163
Social Science 182,166 116,353 236,018 534,537
Total 355,265 415,410 759,669 1,530,344
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Marginal Distribution
US Europe Asia
Engineering 4.05% 10.39% 18.35% 32.78%
Natural Science 7.26% 9.16% 15.87% 32.29%
Social Science 11.90% 7.60% 15.42% 34.93%
23.21% 27.14% 49.64% 100.00%
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Conditional on Region
US Europe Asia
Engineering 17.44% 38.26% 36.96%
Natural Science 31.29% 33.73% 31.97%
Social Science 51.28% 28.01% 31.07%
100.00% 100.00% 100.00%
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Conditional on Major
US Europe Asia
Engineering 12.35% 31.68% 55.97% 100.00%
Natural Science 22.49% 28.36% 49.15% 100.00%
Social Science 34.08% 21.77% 44.15% 100.00%
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Death Penalty and Race
• Examination of 326 death penalty cases
• About half involve a white defendant
• About half involve a black defendant
• Every defendant was convicted of killing someone.
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Death Penalty
Defendant Yes No Total
White 19 141 160
Black 17 149 166
Total 36 290 326
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Death Penalty
Defendant Yes No Total
White 11.87 88.13 100%
Black 10.24 89.76 100%
Total 11.04 88.95 100%
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Death Penalty
• The probability of getting the death penalty appears to be about the same for whites and blacks.
• But what if we look at the data more carefully?
• Is killing a black person the same as killing a white person?
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White Defendant Death Penalty
Yes No Total
White Victim 19 132 151
Black Victim 0 9 9
Total 19 141 160
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Black Defendant Death Penalty
Yes No Total
White Victim 11 52 63
Black Victim 6 97 103
Total 17 149 166
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White Defendant Death Penalty
Yes No Total
White Victim 12.5 87.5 100%
Black Victim 0 100 100%
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Black Defendant Death Penalty
Yes No Total
White Victim 17.4 82.5 100%
Black Victim 5.8 94.2 100%
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Simpson’s Paradox
• Blacks are more likely to get the death penalty
• But the overall probability of getting the death penalty looked to be the same
• You are more likely to get the death penalty for killing a white person
• White are more likely to kill other whites
• Blacks are more likely to kill other blacks
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Entering
Smallville, Kansas
Established 1793
Population 7943
Elevation 710
Average 3,482
Simpson’s Paradox
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Batting Average
MLB Batting
Averages
1995 1996 Combined
Derek
Jeter
12/
48 0.25
183/
582 0.314
195/
630 0.31
David
Justice
104/
411 0.253
45/
140 0.321
149/
551 0.27
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Batting Averages
1995 1996 1997 Combined
Derek
Jeter
12/
48 0.25
183/
582 0.314
190/
654 0.291
385/
1284 0.3
David
Justice
104/
411 0.253
45/
140 0.321
163/
495 0.329
312/
1046 0.298
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College Admissions
Admission to UC Berkeley
Applicants % admitted
Men 8442 44%
Women 4321 35%
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College Admissions
Major Men Women
Applicants % admitted Applicants % admitted
A 825 62% 108 82%
B 560 63% 25 68%
C 325 37% 593 34%
D 417 33% 375 35%
E 191 28% 393 24%
F 272 6% 341 7%
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Kidney Stones
Kidney Stone Treatment
Treatment A Treatment B
78% (273/350) 83% (289/350)
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Kidney Stones
Treatment A Treatment B
Small StonesGroup 1 Group 2
93% (81/87) 87% (234/270)
Large Stones
Group 3 Group 4
73% (192/263) 69% (55/80)
Both78% (273/350) 83% (289/350)
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Statistics
• Statistics– The collection, organization, presentation, analysis and
interpretation of numerical facts and data
• Descriptive Statistics– The collection, organization and presentation of data
(summarizing and describing a given data set)
• Inferential Statistics– The way we draw general conclusions about the
phenomena under consideration, beyond the facts of the observed data• Deriving rational decisions from incomplete data• Wise decision making in the face of uncertainty
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Inferential Statistics
• Population– Total set of observations on measurements or
outcomes.
– Size of the population can be finite or infinite.
• Sample– Set of measurements or outcomes selected from a
population
– Some samples are created by the experimenter, but most samples in economics are created by “nature.”
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Statistical Inference
• How we generalize the sample characteristics to the entire population?
– How certain are we that the implications are true?
– How certain are we that a given theory is true?
• Since samples are drawn randomly, we need to understand some things about randomness and thus probability.
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Probability
• Gerolamo Cardano
– 1501-1576
• Book of Games of Chance
– 1526 (published 1663)
• Gambler, effective cheating methods