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Chapter 16

Statistical Analysis

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Chapter OutlineIntroductionDescriptive StatisticsInferential StatisticsOther Multivariate Techniques

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Descriptive StatisticsStatistical computations that describe thecharacteristics of a sample or therelationship among variables in a sample.Inferential statistics make inferencesabout the larger population from which the

sample observations were drawn.

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P artial Raw Data Matrix

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H ypothetical Raw Data on

Education and P rejudice

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Measures of AssociationDescriptive statistics summarizing therelationships between variables.Many measures of association arebased on a proportionate reduction of error ( P RE) model.

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P roportionate Reduction of

Error ( P RE) A logical model for assessing the strength of arelationship by asking how much knowingvalues on one variable would reduce our errorsin guessing values on the other.

Example:If we know how much education people

have, we can improve our ability toestimate how much they earn, thusindicating there is a relationship betweenthe two variables.

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P roportionate Reduction of

Error ( P RE)Based on a comparison of:

1. The number of errors we would make in

attempting to guess the attributes of a givenvariable for each of the cases under study -if we knew nothing but the distribution of attributes on that variable.

2.The number of errors we would make if weknew the joint distribution overall and weretold for each case the attribute of onevariable each time we were asked to guessthe attribute of the other.

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Q uestion ______________ is a logical model for assessing the strength of a relationship.

A. MRIB. DVUC. OMGD. P RE

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Answer: DPRE is a logical model for assessing thestrength of a relationship.

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N ominal VariablesIf two variables consist of nominal data(gender, religious affiliation, race),lambda (l) would be one appropriatemeasure.Lambda is based on your ability to guess

values on one of the variables: the P REachieved through knowledge of values onthe other variable.

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Ordinal VariablesIf the variables being related are ordinal(social class, religiosity, alienation),g amma ( g ) is one appropriate measure of association.

Lambda is based on guessing exact

values, g amma is based on guessingthe ordinal arrangement of values.

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GammaComputed from two quantities:

1. The number of pairs having the same

ranking on the two variables.2. The number of pairs having the opposite

ranking on the two variables.For pairs having the same ranking:

The frequency of each cell in the table ismultiplied by the sum of all cells below andto the right of it, with all products summed.

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Q uestion A. If the variables being related are ordinal,

_________ is an appropriate measure of

association. A. GammaB. Lambda

C. RhoD. chi square

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Answer: AIf the variables being related are ordinal,g amma is an appropriate measure of association.

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Interval or Ratio VariablesIf interval or ratio variables (age, income, gradepoint average, and so forth) are beingassociated, one appropriate measure of association is P earsons product-moment correlation (r) .r reflects how closely you can guess the valueof one variable through your knowledge of thevalue of another.Errors are measured in terms of sum of thesquared differences between the actual valeand mean. This is called total variation.

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Regression Analysis A method of data analysis in which therelationships among variables are represented

in the form of an equation, called a regressionequation.Linear regression analysis

A form of statistical analysis that seeks theequation for the straight line that bestdescribes the relationship between two ratiovariables.

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Q uestionIf interval or ratio variables are beingassociated, one appropriate measure is

____________. A. gammaB. lambdaC. P earsons productD. none of these choices

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Answer: CIf interval or ratio variables are beingassociated, one appropriate measure isPe arsons product .

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Simple Scattergram of

Values of X and Y

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A Scattergram of the Values of Two

Variables with Regression Line Added

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Multiple Regression Analysis A form of statistical analysis that seeksthe equation representing the impact of two or more independent variables on asingle dependent variable.

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P artial Regression Analysis A form of regression analysis in which theeffects of one or more variables are heldconstant, similar to the logic of theelaboration model.

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Curvilinear Regression

Analysis A form of regression analysis that allowsrelationships among variables to beexpressed with curved geometric linesinstead of straight ones.

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Inferential StatisticsThe body of statistical computationsrelevant to making inferences fromfindings based on sample observations tosome larger population.

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N onsampling Error Those imperfections of data quality thatare a result of factors other than samplingerror.

Examples: misunderstandings of questions by respondents, erroneous

recordings by interviewers and coders,and keypunch errors.

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Statistical Significance A general term referring to the likelihood thatrelationships observed in a sample could be

attributed to sampling error alone.Te sts of Statistical Si g nificanc e

A class of statistical computations thatindicate the likelihood that the relationshipobserved between variables in a sample canbe attributed to sampling error only.

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The Logic of Statistical

Significance1. Assumptions regarding the independence of

two variables in the population study.

2. Assumptions regarding the representativenessof samples selected through conventionalprobability-sampling procedures.

3. The observed joint distribution of sampleelements in terms of the two variables.

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Q uestion _________________ indicate thelikelihood that the relationship observed

between variables in a sample can beattributed to sampling error only. A. ex post facto hypothesizingB. tests of statistical significanceC. disconfirmationD. all of these choices

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Answer: BTe sts of statistical si g nificanc e indicatethe likelihood that the relationshipobserved between variables in a samplecan be attributed to sampling error only.

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H ypothetical P opulation of Men and Women

Who Favor or Oppose Sexual Equality

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A Representative Sample

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Level of SignificanceIn the context of tests of statisticalsignificance, the degree of likelihood that

an observed, empirical relationship couldbe attributable to sampling error.

A relationship is significant at the .05 level

if the likelihood of its being only a functionof sampling error is no greater than 5 outof 100.

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Representative Sample from a P opulation in

Which Variables Are Related

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Chi SquareComputed as follows.For each cell in the tables, the researcher:

Subtracts the expected frequency for that cellfrom the observed frequency.Squares this quantity.Divides the squared difference by the

expected frequency.This procedure is carried out for each cell in thetables.

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Path Analysis

A form of multivariate analysis in whichthe causal relationships among variables

are presented in a graphical format.

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Diagramming the Religious

Sources of Anti-Semitism

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Factor Analysis An analytical method of discovering thegeneral dimensions represented by a

collection of actual variables.These factors are calculated hypotheticaldimensions that are not perfectlyrepresented by any of the empiricalvariables under study but are highlyassociated with groups of empiricalvariables.

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Analysis of Variance

(AN

OVA)Based on comparing variations betweenand within groups and determining

whether between-group differences couldreasonably have occurred in simplerandom sampling or whether they likely

represent a genuine relationship betweenthe variables involved.

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Discriminant AnalysisSeeks to account for variation in somedependent variable by finding a

hypothetical, composite dimension thatseparates categories of the dependentvariable.Results in an equation that scores peopleon the basis of that hypotheticaldimension and allows us to predict their values on the dependent variable.

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Question _________________ is a causal modelfor understanding relationships between

variables. A. ex post facto hypothesizingB. tests of statistical significanceC. path analysisD. all of these choices

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Two Distribution P atterns of the Incomes of

Republicans and Democrats

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Six Writers: Three Who Write by H and

and Three Who Use Computers

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P lotting the Six Writers in

Terms of Age and Income

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Income Alone Is Sufficient toP

redict Writing Method

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A Slightly More ComplicatedP

attern

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Separating the P ens from the

Computers

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Geographic Information

Systems (GIS)Map quantitative data that describegeographic units for a graphical display.

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1. _____________ is the applied branch of

mathematics especially appropriate to avariety of research analyses. A. calculus

B. probabilityC. statisticsD. none of these choices

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Answer: CStatistics is the applied branch of mathematics especially appropriate to a

variety of research analyses.

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2. Gamma is composed of: A. the number of pairs having the same

ranking on two variablesB. the number of pairs having the

opposite ranking on the two variablesC. both a and bD. none of these choices

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Answer: C2. Gamma is composed of th e numb e r of

pairs havin g th e sam e rankin g on two

variabl e s , and th e numb e r of pairshavin g th e opposit e rankin g on th e two variabl e s .

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3. A __________permits the estimation of

values on one variable from values onthe other.

A. multivariate analysis

B. indirect analysisC. regression lineD. exploratory study

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4 . ______________ are statistical

measures used for making inferencesfrom findings based on sampleobservations to a larger population.

A. descriptive statisticsB. inferential statisticsC. ex post facto statistics

D. none of these choices

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Answer: BInf e r e ntial statistics are statisticalmeasures used for making inferences

from findings based on sampleobservations to a larger population.

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5. A____________ analysis represents

changes in a variable over time. A. regressionB. bivariate

C. time-series analysisD. all of these choices

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Answer: C A tim e- s e rie s analysis representschanges in a variable over time.

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6. Which type of statistics assists

researchers in drawing conclusions fromtheir observations.

A. descriptive statistics

B. inferential statisticsC. ordinal statisticsD. interval statistics

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Answer: BInf e r e ntial statistics assistsresearchers in drawing conclusions from

their observations.