1. January 17 2006Lecture 1bSlide 1 Statistics Refresher: Topics Central tendency –Expected value and means Dispersion –Population variance, sample variance, standard deviations Measures of relations Covariation –covariance matrices Correlations Sampling distributions Characteristics of sampling distributions Class Data –2005 National Security Survey (phone and web) –Stata application Means, Variance, Standard Deviations The Normal Distribution Medians and IQRs Box Plots and Symmetry Plots

2. January 17 2006Lecture 1bSlide 2 Measures of Central Tendency In general: E[Y] = µ Y For discrete functions: For continuous functions: An unbiased estimator of the expected value:

3. January 17 2006Lecture 1bSlide 3 Rules for Expected Value E[a] = a -- the expected value of a constant is always a constant E[bX] = bE[X] E[X+W] = E[X] + E[W] E[a + bX] = E[a] + E[bX] = a + bE[X]

4. January 17 2006Lecture 1bSlide 4 Measures of Dispersion Var[X] = Cov[X,X] = E[X-E[X]] 2 Sample variance: Standard deviation: Sample Std. Dev:

5. January 17 2006Lecture 1bSlide 5 Rules for Variance Manipulation Var[a] = 0 Var[bX] = b 2 Var[X] From which we can deduce: Var[a+bX] = Var[a] + Var[bX] = b 2 Var[X] Var[X + W] = Var[X] + Var[W] + 2Cov[X,W]

6. January 17 2006Lecture 1bSlide 6 Measures of Association Cov[X,Y] = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y] Sample Covariance: Correlation: Correlation restricts range to -1/+1

7. January 17 2006Lecture 1bSlide 7 Rules of Covariance Manipulation Cov[a,Y] = 0 (why?) Cov[bX,Y] = bCov[X,Y] (why?) Cov[X + W,Y] = Cov[X,Y] + Cov[W,Y]

8. January 17 2006Lecture 1bSlide 8 Covariance Matrices Correlation Matrices (Example). correlate p2_age p1_edu p100d_in (obs=2500) | p2_age p1_edu p100d_in -------------+--------------------------- p2_age | 1.0000 p1_edu | 0.0322 1.0000 p100d_in | -0.0456 0.3234 1.0000

9. January 17 2006Lecture 1bSlide 9 In-Class Dataset: National Security Survey Review the Frequency Report –Public perspectives on national security, domestic and international –Telephone and Internet survey –Dates: April 2005-June 2005 –Knowledge, beliefs, policy preferences Class data: n=3006 –Variable types Nominal Ordinal scales, Likert-type scales Ratio scales Stata format

10. January 17 2006Lecture 1bSlide 10 Characterizing Data Rolling in the data -- before modeling –A Cautionary Tale Sample versus population statistics ConceptSample StatisticPopulation Parameter Mean Variance Standard Deviation

11. January 17 2006Lecture 1bSlide 11 Properties of Standard Normal (Gaussian) Distributions Can be dramatically different than sample frequencies (especially small ones) Stata Tails go to plus/minus infinity The density of the distribution is key: +/- 1.96 std.s covers 95% of the distribution +/- 2.58 std.s covers 99% of the distribution Student’s t tables converge on Gaussian

12. January 17 2006Lecture 1bSlide 12 Standard Normal (Gaussian) Distributions So what? –Only mean and standard deviation needed to characterize data, test simple hypotheses –Large sample characteristics: honing in on normal n i =300 n i =100 n i =20

13. January 17 2006Lecture 1bSlide 13 Order Statistics Medians –Order statistic for central tendency –The value positioned at the middle or (n+1)/2 rank –Robustness compared to mean Basis for “robust estimators” Quartiles –Q1: 0-25%; Q2: 25-50%; Q3: 50-75% Q4: 75-100% Percentiles –List of hundredths (say that fast 20 times)

14. January 17 2006Lecture 1bSlide 14 Distributional Shapes Positive Skew Negative Skew Approximate Symmetry Md Y

15. January 17 2006Lecture 1bSlide 15 Using the Interquartile Range (IQR) IQR = Q 3 - Q 1 Spans the middle 50% of the data A measure of dispersion (or spread) Robustness of IQR (relative to variance) If Y is normally distributed, then: –S Y ≈IQR/1.35. So: if Md Y ≈ and S Y ≈IQR/1.35, then –Y is approximately normally distributed

16. January 17 2006Lecture 1bSlide 16 Example: The Observed Distribution of Age (p2_age) (Distribution of Age)

17. January 17 2006Lecture 1bSlide 17 Interpreting Box Plots Median Age = ~49; IQR = ~25 years

18. January 17 2006Lecture 1bSlide 18 Quantile Normal Plots Allow comparison between an empirical distribution and the Gaussian distribution Plots percentiles against expected normal Most intuitive: – Normal QQ plots Evaluate

19. January 17 2006Lecture 1bSlide 19 Data Exploration in Stata Access National Security dataset (new) Using Age: univariate analysis Stata Using Age: split by survey mode Stata Exercises: –Univariate analysis of age By mode, gender –Graphing: Produce Histograms Box plots Q-Normal plots

20. January 17 2006Lecture 1bSlide 20 For Next Week Read Hamilton –Appendix 1 (review carefully) –Pages 1-23; 29-37 Review Herron and Jenkins-Smith –Homework #1 Bivariate Regression Analysis –Theoretical model –Model formulation –Model assumptions –Residual analysis