Introduction to Quantitative Methods - Harvard Law School

1 Introduction to Quantitative MethodsParina PatelOctober 15, 2009 Contents1 Definition of Key Terms22 Descriptive Frequency Tables .. Measures of Central Tendencies .. Measures of Variability .. Summary of Central Tendencies and Variability ..63 Inferential More Definitions and Terms .. Comparing Two or More Groups .. Association and Correlation .. Explaining a Dependent Variable .. 14 List of Tables1 Frequency Table Socioeconomic Class ..42 Crosstab of Music Preference and Age ..53 Summary of Univariate Statistics ..64 Comparing Group Means .. 105 Association and Correlation .. 126 Explaining a Dependent Variable .. 131 Empirical Law SeminarParina Patel1 Definition of Key Terms1. Unit of Analysis (also referred to as cases): The most elementary part ofwhat is being studied or observed.

2 Some examples include individuals,households, court cases, countries, states, firms, industries, Variables: Concepts, characteristics, or properties that canvary, orchange, from one unit of analysis to another. Please note thatall variables must vary, if there is no variation among the differentcases then it is not a variable. Some examples of variables includegender, social class, education, age, level of public enforcement, typeof bankruptcy, etc.(a) Dependent Variable DV: Variables whose change the researcherwishes to explain(b) Independent Variable IV: Variables that help explain the changein the dependent variable3. Hypothesis: An empirical statement which seeks to test the relation-ship between at least two variables. For instance, As levels of publicenforcement increases, levels of stock development also increases.

3 Thishypothesis has two variables: (1) public enforcement independent vari-able, and (2) stock development dependent Levels of Measuring Variables(a) Nomial: A nominal variable has qualitative categories that can-not be ranked in a meaningful way in terms of degree or mag-nitude. Examples of nominal variables include RACE, TYPEOF BANKRUPTCY, TYPE OF CORPORATION, NAME. Allof these variables have qualitative categories that cannot be or-dered in terms of magnitude or degree. This is the least powerfultype of (b) Ordinal: An ordinal variable has qualitative categories that areordered in terms of degree or magnitude. Examples of a nomi-nal variable include CLASS or DEGREE OBTAINED. The vari-able DEGREE OBTAINED may include the following categories:1 Alphabetizing the categories does not count as ordering the variable, because theordering has to be in terms of degree or Law SeminarParina PatelNone, High School Diploma, College/University Degree, Masters,Advanced Degree (JD/PHD/MD).

4 All of these categories are qual-itative and are ordered in terms of the amount of education eachindividual has completed.(c) Interval/Ratio: An interval variable has Quantitative values (ornumbers). Some examples of interval variables include AGE (inyears), NUMBER OF SHARES OUTSTANDING, and AMOUNTIN DEBT (in dollars). For all of these variables the response isgoing to be a number or value. This is most powerful type ofvariable because you can do the most with it that if a variable has qualitative categories that ARE orderedand there are numerical values assigned to each category which arealso ordered, we can treat this variable like an interval level vari-able. An example would be questionnaire that asks respondentsabout their feelings towards President Obama s handling of theeconomy on a scale of 1 to 5 where (1=very bad job, 2=bad job,3=neither bad nor good, 4=good job, and 5=very good job).

5 Therespondents are asked to choose a category that is ordered, butsince it has ordered numbers attached to the categories, we cantreat it as an interval level variable with some (d) Dichotomous/Dummy: A dichotomous variable is a variable withtwo (and only two) categories. These categories can be qualitativeor Quantitative Descriptive StatisticsDescriptive statistics are often used to describe variables. Descriptive statis-tics are performed by analyzing one variable at a time (univariate analysis).All researchers perform these descriptive statistics before beginning any typeof data such restriction being the dependent variable in regression analysis. In order toperform regression (see section ) your dependent variable must be a proper is possible to convert nominal variables into numerous dichotomous/dummy Law SeminarParina Frequency TablesFrequency tables are a detailed description of the categories/values for onevariable.

6 A frequency table most often includes all of the following:41. Absolute frequency (or just frequency): This tells you how many timesa particular category in your variable occurs. This is a tally, count, orfrequency of occurrence of each individual category/value in the Relative frequency (or percent): This tells you the percentage of eachcategory/value relative to the total number of Cumulative frequency: This is simply a cumulation of the relative fre-quency for each 1 provides an example of a frequency table for an ordinal variable(note it is ordinal because the categories are qualitative and ordered)named Socioeconomic Class. If there were numbers assigned to eachcategory that were also ordered, we could treat this as an interval 1: Frequency Table Socioeconomic ClassSocioeconomic ClassFrequencyPercentCumm.

7 Crosstabulations: This is also referred to as a grouped frequency tablefor two variables. A crosstab simply presents the absolute frequencybroken down by categories of two or more variables. It is also possibleto find percentages in these types of tables. For instance, using the4 The stata command for frequency isfreortab. Before you use thefrecommand youneed to install it onto your computer, so you need to type the following command: sscinstall fre, which will install thefrecommand onto your computer. For a frequencytable of a variable named class, type either fre class or tab class 4 Empirical Law SeminarParina Patelexample below, we can find the percentage of young people that listento 2: Crosstab of Music Preference and AgeAGEP referenceYoungMiddle Measures of Central TendenciesMeasures of central tendencies provide the most occurring or middle value/categoryfor each variable.

8 There are three measures of central tendencies mode, me-dian, and mean. See Table 3 for a summary of measures of central Measures of VariabilityMeasures of variability is defined as the dispersion (or deviation) away fromthe mean for each variable. Measures of variability only exist for interval levelvariables. There are three measures of variability range, standard deviation,and variance. A discussion of each can be found below followed by a summarytable (Table 3).1. Range: The range is found by taking the highest value of a variableminus the lowest value of that Standard deviation: The standard deviation exists for all interval vari-ables. It is the average distance of each value away from the samplemean. The larger the standard deviation, the farther away the val-ues are from the mean; the smaller the standard deviation the closer,the values are to the mean.

9 Suppose you passed out a questionnaire5 The stata command for a crosstab is eithertabortab2. For a crosstab of twovariables named age and preference, type the following command into stata: tabage preference 5 Empirical Law SeminarParina Patelasking randomly selected individuals to rate President Obama s jobperformance on a scale from 1 to 10. You find that on average theseindividuals give the President a rating of , and this variable has astandard deviation of This means that on average, each rating ofthe President is approximately points away from (the samplemean).3. Variance: The variance is always going to the the standard deviationsquared. The variance cannot be interpreted as meaning anything otherthan the standard deviation Summary of Central Tendencies and VariabilityTable 3: Summary of Univariate StatisticsUnivariateVariablesDescription StatisticModeNominal, Ordinal, and Intervalmost frequent category/valueMedianOrdinal and Intervalcategory/value that lies in the middleMeanIntervalvalue that represents the averageRangeIntervalhighest value minus lowest valueStandardIntervalon average how much each individualDeviationvalue is dispersed around the meanVarianceIntervalstandard deviation squared3 Inferential More Definitions and Terms1.

10 Normal CurveAn interval variable is said to be normally distributed if it has all ofthe following characteristics:(a) A bell shape stata, the easiest way to find the mode is by looking at a frequency table and findingthe value/category that occurs most frequently. The median, mean, standard deviation,and variance can be found by using the following command:sum var1 var2 .., detail6 Empirical Law SeminarParina Patel(b) It is perfectly symmetrical.(c) All measures of central tendencies (mode, median, and mean) liein the middle middle of the curve. These measures of centraltendencies divide the curve in half (where 50% of the values lie tothe left of the mean, and 50% lie to the right).(d) Approximately 95% of the values are found two standard devia-tions away from the mean (in both directions).