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Chapter 11 Basic Data Analysis for Quantitative Research McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Statistical Analysis - Overview • Every set of data collected needs some summary information that describes the numbers it contains – Central tendency and dispersion – Relationships of the sample data – Hypothesis testing 11-2 Measures of Central Tendency Mean • The arithmetic average of the sample • All values of a distribution of responses are summed and divided by the number of valid responses Median • The middle value of a rank-ordered distribution • Exactly half of the responses are above and half are below the median value Mode • The most common value in the set of responses to a question • The response most often given to a question 11-3 Dialog Boxes for Calculating the Mean, Median, and Mode (in ‘Frequencies’ function) 11-4 Measures of Dispersion Range • The distance between the smallest and largest values in a set of responses Standard deviation • The average distance of the distribution values from the mean Variance • The average squared deviation about the mean of a distribution of values 11-5 SPSS Output for Measures of Dispersion 11-6 Type of Scale and Appropriate Statistic 11-7 Univariate Statistical Tests • Used when the researcher wishes to test a proposition about a sample characteristic against a known or given standard • Appropriate for interval or ratio data • Test: “Is a mean significantly different from some number?” – Example: “Is the ‘Reasonable Prices’ average different from 4.0?” 11-8 Univariate Hypothesis Test Using X-16 – Reasonable Prices 11-9 Bivariate Statistical Tests • Test hypotheses that compare the characteristics of two groups or two variables • Three types of bivariate hypothesis tests – Chi-square – t-test – Analysis of variance (ANOVA) 11-10 Cross-Tabulation (“Cross-tabs”) • Used to examine relationships and report findings for two categorical (i.e. ‘nominal’) variables • Purpose is to determine: – if differences exist between subgroups of the total sample on a key measure – whether there is an association between two categorical variables • A frequency distribution of responses on two or more sets of variables 11-11 Cross-Tabulation: Ad Recall vs. Gender 11-12 Chi-Square Analysis • Assesses how closely the observed frequencies fit the pattern of the expected frequencies – Referred to as a “goodness-of-fit” • Tests for statistical significance between the frequency distributions of two or more nominally scaled (i.e. “categorical”) variables in a crosstabulation table to determine if there is any kind of association between the variables 11-13 SPSS Chi-Square Crosstab Example Do males and females recall the ads differently? Comparing Means: Independent Versus Related Samples • Independent samples: Two or more groups of responses that supposedly come from different populations • Related samples: Two or more groups of responses that supposedly originated from the same population – Also called “Matched” or “Dependent” samples – SPSS calls them “Paired” samples • Practical tip: Ask yourself, “Were the subjects reused (“Paired”) or not re-used (“Independent”) in order to obtain the data? 11-15 Using the t -Test to Compare Two Means • t-test: A hypothesis test that utilizes the t distribution – Used when the sample size is smaller than 30 and the standard deviation is unknown • Where, X 1 m ean of sam ple 1 X 2 m ean of sam ple 2 S X 1 X 2 standard error of the difference betw ee n the tw o m eans 11-16 Comparing two means with Paired Samples t-Test Do average scores on variables X-18 and X-20 differ from each other? 11-17 Comparing Two Means with Independent Samples t-Test Do males and females differ with respect to their satisfaction? 11-18 Analysis of Variance (ANOVA) • ANOVA determines whether three or more means are statistically different from each other • The dependent variable must be either interval or ratio data • The independent variable(s) must be categorical (i.e. nominal or ordinal) • “One-way ANOVA” means that there is only one independent variable • “n-way ANOVA” means that there is more than one independent variable (i.e. ‘n’ IVs) 11-19 Analysis of Variance (ANOVA) • F-test: The test used to statistically evaluate the differences between the group means in ANOVA 11-20 Example of One-Way ANOVA Does distance driven affect customers’ likelihood of returning? 11-21 Analysis of Variance (ANOVA) • ANOVA does not tell us where the significant differences lie – just that a difference exists • Follow-up (Post-hoc) tests: Analysis that flags the specific means that are statistically different from each other – Performed after an ANOVA determines there is an “Omnibus” differenc between means • Some Pairwise Comparison Tests (there are others) – Tukey – Duncan – Scheffé 11-22 Results for Post-hoc Mean Comparisons 11-23 n-Way ANOVA • ANOVA that analyzes several independent variables at the same time – Also called “Factorial Design” • Multiple independent variables in an ANOVA can act in concert together to affect the dependent variable – this is called Interaction Effect 11-24 n-way ANOVA: Example • Men and women are shown humorous and non-humorous ads and then attitudes toward the brand are measured. • IVs (factors) = (1) gender (male vs. female), and (2) ad type (humorous vs. non-humorous) • DV = attitude toward brand • Need 2-way ANOVA design here (also called “factorial design”) because we have two factors – 2 x 2 design (2 levels of gender x 2 levels of ad type) 11-25 n-Way ANOVA Example Does distance driven and gender affect customers’ likelihood of recommending Santa Fe Grill? 11-26 n -Way ANOVA Post-hoc Comparisons 11-27