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The Scientific Study of Politics (POL 51) . Professor B. Jones University of California, Davis. Fun With Numbers. Some Univariate Statistics Learning to Describe Data. Useful to Visualize Data. Main Features. Exhibits “Right Skew” Some “Outlying” Data Points?

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the scientific study of politics pol 51

The Scientific Study of Politics (POL 51)

Professor B. Jones

University of California, Davis

fun with numbers
Fun With Numbers
  • Some Univariate Statistics
  • Learning to Describe Data
main features
Main Features
  • Exhibits “Right Skew”
  • Some “Outlying” Data Points?
  • Question: Are the outlying data points also “influential” data points (on measures of central tendency)?
  • Let’s check…
the mean
The Mean
  • Formally, the mean is given by:
  • Or more compactly:
our data
Our Data
  • Mean of Y is 260.67
  • Mechanically…
    • (263 + 73 + … + 88)/67=260.67
  • Problems with the mean?
  • No indication of dispersion or variability.
variance
Variance
  • The variance is a statistic that describes (squared) deviations around the mean:
  • Why “N-1”?
  • Interpretation: “Average squared deviations from the mean.”
our data1
Our Data
  • Variance= 202,431.8
  • Mechanically:
    • [(263-260.67)2 + (73-260.67)2 + ••• + (88-260.67)2 ]/66
  • Interpretation:
    • “The average squared deviation around Y is 202,431.
  • Rrrrright. (Who thinks in terms of squared deviations??)
  • Answer: no one.
  • That’s why we have a standard deviation.
standard deviation
Standard Deviation
  • Take the square root of the variance and you get the standard deviation.
  • Why we like this:
    • Metric is now in original units of Y.
  • Interpretation
    • S.D. gives “average deviation” around the mean.
    • It’s a measure of dispersion that is in a metric that makes sense to us.
our data2
Our Data
  • The standard deviation is: 449.92
  • Mechanically:

{[(263-260.67)2 + (73-260.67)2 + ••• + (88-260.67)2 ]/66}½

  • Interpretation: “The average deviation around the mean of 260.67 is 449.92.
  • Now, suppose Y=Votes…
  • The average number of votes is “about 261 and the average deviation around this number is about 450 votes.”
  • The dispersion is very large.
  • (Imagine the opposite case: mean test score is 85 percent; average deviation is 5 percent.)
skewness and the mean
Skewness and The Mean
  • Data often exhibit skew.
  • This is often true with political variables.
  • We have a measure of central tendency and deviation about this measure (Mean, s.d)
  • However, are there other indicators of central tendency?
  • How about the median?
median
Median
  • “50th” Percentile: Location at which 50 percent of the cases lie above; 50 percent lie below.
  • Since it’s a locational measure, you need to “locate it.”
  • Example Data: 32, 5, 23, 99, 54
  • As is, not informative.
median1
Median
  • Rank it: 5, 23, 32, 54, 99
  • Median Location=(N+1)/2 (when n is odd)
  • =6/2=3
  • Location of the median is data point 3
  • This is 32.
  • Hence, M=32, not 3!!
  • Interpretation: “50 percent of the data lie above 32; 50 percent of the data lie below 32.”
  • What would the mean be?
  • (42.6…data are __________ skewed)
median2
Median
  • When n is even: -67, 5, 23, 32, 54, 99
  • M is usually taken to be the average of the two middle scores:
    • (N+1)/2=7/2=3.5
    • The median location is 3.5 which is between 23 and 32
    • M=(23+32)/2=27.5
  • All pretty straightforward stuff.
median voter theorem a sidetrip
Median Voter Theorem (a sidetrip)
  • One of the most fundamental results in social sciences is Duncan Black’s Median Voter Theorem (1948)
  • Theorem predicts convergence to median position.
  • Why do parties tend to drift toward the center?
  • Why do firms locate in close proximity to one another?
  • The theorem: “given single-peaked preferences, majority voting, an odd number of decision makers, and a unidimensional issue space, the position taken by the median voter has an empty winset.”
  • That is, under these general conditions, all we need to know is the preference of the median chooser to determine what the outcome will be. No position can beat the median.
dispersion around the median
Dispersion around the Median
  • The mean has its standard deviation…
  • What about the median?
    • No such thing as “standard deviation” per se, around the median.
    • But, there is the IQR
  • Interquartile Range
    • The median is the 50th percentile.
    • Suppose we compute the 25th and the 75th percentiles and then take the difference.
    • 25th Percentile is the “median” of the lower half of the data; the 75th Percentile is the “median” of the upper half.
iqr and the 5 number summary
IQR and the 5 Number Summary
  • Data: -67, 5, 23, 32, 54, 99
  • 25th Percentile=5
  • 50th Percentile=54
  • IQR is difference between 75th and 25th percentiles: 54-5=49
  • Hence, M=27.5; IQR=49
  • “Five Number Summary” Max, Min, 25th, 50th, 75th Percentiles:
  • -67, 5, 27.5, 54, 99
finding percentiles
Finding Percentiles
  • General Formula
  • p is desired percentile
  • n is sample size
  • If L is a whole number:
    • The value of the pth percentile is between the Lth value and the next value. Find the mean of those values
  • If L is not a whole number:
    • Round L up. The value of the pth percentile is the Lth value
example
Example
  • -67, 5, 23, 32, 54, 99
  • 25th Percentile: L=(25*6)/100=1.5
    • Round to 2. The 25th Percentile is 5.
  • 75th Percentile: L=(75*6)/100=4.5
    • Round to 5. The 75th Percentile is 54.
    • 50th Percentile: L=(50*6)/100=3
    • Take average of locations 3 and 4
    • This is (23+32)/2=27.5.
our data3
Our Data
  • Median=120 Votes (i.e. [50*67]/100)
  • 25th Percentile=46 Votes
  • 75th Percentile=289 Votes
  • IQR: 243 Votes
  • 5 number summary:
    • Min=9, 25th P=46, Median=120, 75th P=289, Max=3407
  • (massive dispersion!)
  • Mean was 260.67. Median=120.
  • The Mean is much closer to the 75th percentile.
  • That’s SKEW in action.
influential observations
“Influential Observations”
  • Two data points:
    • Y=(1013, 3407)
  • Suppose we omit them (not recommended in applied research)
  • Mean plummets to 200.69 (drop of 60 votes)
  • s.d. is cut by more than half: 203.92
  • Med=114 (note, it hardly changed)
  • Let’s look at a scatterplot
main features1
Main Features?
  • Y and X are positively related.
  • There are clearly visible “outliers.”
  • With respect to Y, which “outlier” worries you most?
  • Influence!
simple description
Simple Description
  • You can learn a lot from just these simple indicators.
  • Suppose that our Y was a real variable?
descriptive statistics help to clarify some issues
Descriptive Statistics Help to Clarify Some Issues.
  • Palm Beach County
    • Largely a Jewish community
    • Heavily Democratic
    • Yet an overwhelming number of Buchanan Votes
  • The Ballot created massive confusion.
  • Margin of Victory in Florida: 537 votes.
  • Number of Buchanan Votes in PBC: 3407
univariate statistics
Univariate Statistics
  • We can clearly learn a lot from very simple statistics
  • Some quick illustrations in R using data from last year’s election (on Prop. 8)
univariate quantities in r
Univariate Quantities in R
  • Our Data
    • Yes on Proposition 8 by County
  • Graphical Displays of data
    • Histogram
    • Dot Chart
    • Box Plots
    • Stem and Leaf
    • Strip Plot
first the basic statistics in r
First, the basic statistics in R
  • Mean (by county):
    • > mean(proportionforprop8)
    • [1] 56.7202
  • Standard deviation:
    • > sd(proportionforprop8)
    • [1] 13.39508
  • Five-number summary:
    • > fivenum(proportionforprop8)
    • [1] 23.50787 46.93203 59.25364 68.03883 75.37070
stem and leaf
Stem and Leaf

The decimal point is 1 digit(s) to the right of the |

2 | 459

3 | 4888

4 | 024445578

5 | 0113344566779

6 | 0000023344457889

7 | 0011123334455

r code for previous
R Code for Previous

row.names<- cbind(county)

hist(proportionforprop8, xlab="Percentage Yes on Prop. 8", ylab="Frequency", main="Histogram of Yes on 8 by County", col="yellow")

dotchart(proportionforprop8, labels=row.names, cex=.7, xlim=c(0, 100), main="Yes on 8 by County", xlab="Percent Yes")

abline(v=50)

abline(h=16)

boxplot(proportionforprop8, col="light blue", names=c("Proposition 8"), xlab="California Counties",

ylab="Percent Yes on 8", main="Box Plots for Prop. 8 by County", sub="Source: Los Angeles Times")

abline(h=50)

stem(proportionforprop8)

stripchart(proportionforprop8, method="stack", xlab="Percentage Yes on Prop. 8", main="Vote on Prop. 8 by County: Strip Chart", pch=1)

plots of two variables
Plots of Two Variables

plot(proportionforprop8, proportionforprop4, xlab="Prop. 8 Vote", ylab="Prop. 4 Vote", main="Prop. 8 Vote by Prop. 4 Vote“, col=“red”)

> abline(h=50)

> abline(v=50)