Univariate Statistics

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Univariate Statistics

• Analysis of a single variable
• Two general varieties
• Descriptive Statistics Describe Variables
  (where data are any collection of observations,
  sample/population)
• Inferential Statistics Make inferences about the
  population based on characteristics of sample data

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List of Variable Values
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Frequency Distribution

• A summary of the observations for a variable
• Includes a list of the values of the variable and
  the frequency of observations for each value

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Example Interval/Ratio

• Freq. distribution of midterm grades

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Example Interval/Ratio
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Example Interval/Ratio
Freq. / Total
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Example Interval/Ratio
Freq. / Total100
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Example - Nominal

• Freq. distribution of active hate group
  organizations in 1999

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Example - Nominal
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Summarizing Data in Graphs

• Pie charts, Bar charts appropriate for nominal
  variables and ordinal variables (small number of
  categories)

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Example Bar Chart
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Summarizing Data in Graphs

• Histograms appropriate for all interval/ratio
  variables with a large number of possible values
  data are collapsed into intervals, and axis
  labels represent interval boundaries or interval
  midpoints

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Histogram of County Unemployment Rates in Fla
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Measures of Central Tendency

• Mean
• _
• Y ? Yi / N
• Appropriate for interval/ratio variables ONLY

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Measures of Central Tendency

• Median Defined as the value of the variable in
  the middle of the distribution.
• Odd of obs 2 2 5 9 11 median5
• Even of obs 2 2 5 9 11 15
• median(59)/2 7
• Appropriate for ordinal, interval and ratio

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Measures of Central Tendency

• Mode Defined as the value that occurs most
  often.
• 2 2 5 9 11 15
• Mode2
• Appropriate for all levels of measurement

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Measures of Dispersion

• 1. Range Ymax - Ymin
• Weakness?
• 2. Percentiles - For variable Y, the pth
  percentile represents the value of Y below which
  p of the observations fall.
• 50th percentile median
• IQR Y75pct - Y25pct

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Measures of Dispersion (contd)

• More complex measures Based on mean
  deviations _ Yi Y
  
• 
  _
• Average Mean Deviation(?) S (Yi Y) / N
• 
  _
• Mean Absolute Deviation S Yi Y / N
• could use as measure of variation
• 
  _
• Mean Squared Deviation S(Yi Y)2 / N

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Variance (sample)

• _
• s2Y S (Yi - Y)2 / (N-1)
• Standard Deviation
• sY vs2Y
• Numerator Sum of Squares
• Denominator degrees of freedom

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The Normal Distribution

• Symmetric
• Bell-shaped
• MeanMedianMode

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The Normal Distribution
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Deviations from the normal distribution

• Bimodal distributions
• Skewed distributions
• Left skew vs. right skew
• Mean is pulled in direction of skew

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Histogram of County Unemployment Rates in Fla
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Descriptive Statistics for County Unemployment
Rates in Fla

• . sum unemp, detail
• unemp
• --------------------------------------------------
  -----------
• Percentiles Smallest
• 1 2 1.7
• 5 2.4 1.7
• 10 2.7 1.7 Obs
  3149
• 25 3.4 1.7 Sum of Wgt.
  3149
• 50 4.4 Mean
  4.809908
• Largest Std. Dev.
  2.129031
• 75 5.5 19.5
• 90 7.2 19.5 Variance
  4.532774
• 95 8.6 19.6 Skewness
  2.30285
• 99 13 19.7 Kurtosis
  12.11621

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Sampling Distribution (sample means)

• Population
• Draw Random Sample of Size N
• Calculate sample mean
• Repeat until all possible random samples are
  exhausted
• The resulting collecting of sample means is the
  sampling distribution of sample means

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Sampling Distribution of Sample Means

• A frequency distribution of all possible sample
  means for a given sample size (N)
• The mean of the sampling distribution will be
  equal to the population mean.
  

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Sampling Distribution of Sample Means

• When N is reasonably large (gt30), the sampling
  distribution will be normally distributed
• The standard error of the sampling distribution
  can be reliably estimated as (where sY sample
  standard deviation for Y and N sample size).
• sY /vN
• 
  

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Standard Error

• How the sample means vary from sample to sample
  (i.e. within the sampling distribution) is
  expressed statistically by the value of the
  standard deviation (i.e. standard error) of the
  sampling distribution.
• (Standard deviation the average distance of
  each observation from the mean)

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Using the Standard Error to Calculate a 95
Confidence Interval

• Calculate the mean of Y
• Calculate the standard deviation of Y
• Calculate the standard error of Y
• Calculate a 95 confidence interval for the
  population mean of Y
• _
• 95 CI Y 1.96(standard error)

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Example

• Hillary Clinton Feeling Thermometer (NES 2004)

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Example

• Hillary Clinton Feeling Thermometer (NES 2004)
• Mean 64.137, s.d. 88.408, N 1212

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Example

• Hillary Clinton Feeling Thermometer (NES 2004)
• Mean 64.137, s.d. 88.408, N 1212
• Standard Error 88.408 / v1212 2.539

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Example

• Hillary Clinton Feeling Thermometer (NES 2004)
• Mean 64.137, s.d. 88.408, N 1212
• Standard Error 88.408 / v1212 2.539
• 95 CI 64.137 1.96 2.539
• 59.158, 69.116