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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 Sec
2、urity Survey (phone and web) Stata application Means, Variance, Standard Deviations The Normal Distribution Medians and IQRs Box Plots and Symmetry Plots,Measures of Central Tendency,In general: EY = Y For discrete functions: For continuous functions: An unbiased estimator of the expected value:,Rul
3、es for Expected Value,Ea = a - the expected value of a constant is always a constant EbX = bEX EX+W = EX + EW Ea + bX = Ea + EbX = a + bEX,Measures of Dispersion,VarX = CovX,X = EX-EX2 Sample variance: Standard deviation: Sample Std. Dev:,Rules for Variance Manipulation,Vara = 0 VarbX = b2 VarX From
4、 which we can deduce: Vara+bX = Vara + VarbX = b2 VarX VarX + W = VarX + VarW + 2CovX,W,Measures of Association,CovX,Y = E(X - EX)(Y - EY) = EXY - EXEY Sample Covariance: Correlation: Correlation restricts range to -1/+1,Rules of Covariance Manipulation,Cova,Y = 0 (why?) CovbX,Y = bCovX,Y (why?) Cov
5、X + W,Y = CovX,Y + CovW,Y,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,In-Class Dataset: National Security Survey,Review the Frequency Report Publi
6、c 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,Characterizing Data,Rolling in the d
7、ata - before modeling A Cautionary Tale Sample versus population statistics,Concept Sample Statistic Population Parameter Mean Variance Standard Deviation,Properties of Standard Normal (Gaussian) Distributions,Can be dramatically different than sample frequencies (especially small ones) Stata Tails
8、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 Students t tables converge on Gaussian,Standard Normal (Gaussian) Distributions,So what? Only mean and standard deviation needed to characteri
9、ze data, test simple hypotheses Large sample characteristics: honing in on normal,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:
10、 75-100% Percentiles List of hundredths (say that fast 20 times),Distributional Shapes,Positive Skew Negative Skew Approximate Symmetry,MdY,MdY,MdY,Using the Interquartile Range (IQR),IQR = Q3 - Q1 Spans the middle 50% of the data A measure of dispersion (or spread) Robustness of IQR (relative to va
11、riance) If Y is normally distributed, then: SYIQR/1.35. So: if MdY and SY IQR/1.35, then Y is approximately normally distributed,Example: The Observed Distribution of Age (p2_age),(Distribution of Age),Interpreting Box Plots,Median Age = 49; IQR = 25 years,Quantile Normal Plots,Allow comparison betw
12、een an empirical distribution and the Gaussian distribution Plots percentiles against expected normal Most intuitive: Normal QQ plots Evaluate,Data Exploration in Stata,Access National Security dataset (new) Using Age: univariate analysis Stata Using Age: split by survey mode Stata Exercises: Univar
13、iate analysis of age By mode, gender Graphing: Produce Histograms Box plots Q-Normal plots,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,
