Introduction to Descriptive Statistics2/21/02Population vs. Sample Notations, b:, F, $RomansGreeksSampleVsPopulationTypes of VariablesNominal(Qualitative)~Nominal(Quantitative)OrdinalInterval orratioDescribing data--KurtosisPeaked--SkewnessSkewRange,Interquartile rangeVariance (standard deviation)SpreadMode, medianMeanCenterNon-mean based measureMomentMeanXnxnii≡≡∑=µ1Variance, Standard Deviationσµσµ≡−≡−∑∑==nxinxinxnx12212)(,)(Variance, S.D. of a Samplesnxsnxnxinxi≡−−≡−−∑∑==122121)(,1)(µµCoefficient of variation100.. ×=µσvcSkewnessSymmetrical distribution• IQ• SATValueFrequencySkewnessAsymmetrical distribution• GPA of MIT studentsValueFrequencySkewness(Asymmetrical distribution)• Income• Contribution to candidates• Populations of countries• “Residual vote” ratesValueFrequencySkewness( )( )smedianmeansemeannnxxxxniinii/)(3/)mod(2)1(2/312/313−×≈−≈−−×−−∑∑==SkewnessValueFrequencyKurtosisValueFrequencyk > 3k = 3k < 3A few words about the normal curve• Skewness = 0• Kurtosis = 3ValueFrequency22/)(21)(σµπσ−−=xexfMore words about the normal curveValueFrequency34%34%x47.7% 47.7%CEG example4.1-1.10.795.8Overall rating3.8-1.00.716.1Is friendly and supportive2.8-0.550.805.5Is available outside of class6.4-1.10.596.2Speaks clearly3.0-0.610.935.5Uses the blackboard well4.1-1.10.785.8Explains clearly and answers questions well5.8-1.50.755.9Gives well-prepared, relevant presentationsGraphKurtSkews.d.MeanThe instructor and/or section leader:Binary data)1()1(1 timeof proportion1)(2xxsxxsxXprobXxx−=⇒−=====Commands in STAT for getting univariate statistics• summarize• summarize, detail• graph, bin() normal• graph, box• tabulate [NB: compare to
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