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Measures of Central Tendency © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION Chapter 2 © Jones & BartlettMeasures Learning, LLC of Central© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION Tendency © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION Learning Objectives ■■©Understand Jones & how Bartlett frequency Learning, tables are used LLC in statistical analysis. © Jones & Bartlett Learning, LLC ■■NOTExplain FOR the conventionsSALE OR for DISTRIBUTION building distributions. NOT FOR SALE OR DISTRIBUTION ■■ Understand the mode, median, and mean as measures of central tendency. ■■ Identify the proper measure of central tendency to use for each level of measure- ment. © Jones & Bartlett■■ Explain Learning, how to calculate LLC the mode, median, and mean.© Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION Key Terms bivariate median central tendency mode © Jones & Bartlett Learning,frequency LLC distribution © Jonesmultivariate & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTIONmean NOT univariateFOR SALE OR DISTRIBUTION Now we begin statistical analysis. Statistical analysis may be broken down into three broad categories: © Jones■■ Univariate & Bartlett analyses Learning, LLC © Jones & Bartlett Learning, LLC NOT■■ Bivariate FOR SALE analyses OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION ■■ Multivariate analyses These divisions are fairly straightforward. Univariate analyses deal with one variable at a time. Bivariate analyses compare two variables to each other to see how they dif- © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 17 © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC.© NOTJones FOR SALE & OR Bartlett DISTRIBUTION. Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 34032_CH02_Walker.indd 17 11/5/11 8:01:18 AM © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 18 Chapter 2 n Measures of Central Tendency fer or ©if theyJones are the & same. Bartlett In statistical Learning, analysis LLC this could take the form of association© Jones & Bartlett Learning, LLC or correlation.NOT FOR Multivariate SALE analysesOR DISTRIBUTION compare three or more variables to one NOTanother, FOR SALE OR DISTRIBUTION or compare more than two variables (called independent variables) in how they influ- ence a variable we are interested in knowing something about (called a dependent variable). The focus of this chapter and of Chapter 3 is on univariate analysis. Univariate © Jones descriptive& Bartlett statistics Learning, are used LLC to describe and interpret the meaning© Jones of a& distribution. Bartlett Learning, LLC NOT FORThese SALE statistics OR tellDISTRIBUTION a lot about a data set by revealing a fewNOT critical FOR characteristics. SALE OR DISTRIBUTION You should always undertake this process before doing any bivariate or multivari- ate analyses; but you can do sound research just using the analyses we will address in these chapters. Univariate descriptive statistics make compact characterizations of © Jones & Bartlett Learning,distributions LLC in terms of three properties of© the Jones data: central & Bartlett tendency ,Learning, which translates LLC NOT FOR SALE OR DISTRIBUTIONto the average, middle point, or most commonNOT valueFOR of SALE the distribution; OR DISTRIBUTION dispersion, which relates to how spread out the values are around the central measure; and the form of the distribution, which relates to what the distribution would look like if it were displayed graphically. © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC 2-1 NOTFrequency FOR SALE Distributions OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION Most statistical analyses are displayed through tables (and sometimes charts). Through- out these chapters, you will see different kinds of tables and graphs displaying various types of data. At the most basic level, data is displayed through a frequency distribu- © Jones tion& Bartlett (which often Learning, contains univariateLLC statistics displayed with© Jones the data). & FrequencyBartlett Learning, LLC NOT FORdistributions SALE OR are DISTRIBUTION helpful in summarizing data (scores) becauseNOT they FOR provide SALE a visual- OR DISTRIBUTION ization of how scores are spread over categories. The most common type of distribu- tion for current statistical packages is a combination distribution. This is the type of distribution shown in Table 2-1. In SPSS a combination distribution includes percent- age, valid percentage, and cumulative percentage. A lot can be learned from these © Jones & Bartlett Learning,frequency distributions,LLC but we want to focus© Jones on the &univariate Bartlett descriptive Learning, statistics LLC NOT FOR SALE OR DISTRIBUTIONthat usually are a part of these printouts. NOT FOR SALE OR DISTRIBUTION Conventions for Building Distributions Within statistical analysis, there are conventions and standards for presenting tables and distributions.© Jones You& Bartlett will soon realizeLearning, that many LLC people do not follow these conven-© Jones & Bartlett Learning, LLC tions; NOThowever, FOR if you SALE adhere OR to these DISTRIBUTION guidelines when building your own tablesNOT and FOR SALE OR DISTRIBUTION distributions, they will be clearer and the results will be easier to interpret. First, the title of the table or distribution should clearly explain the contents. It should contain the variables included in the table and any variable label or explanation © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC.© NOTJones FOR SALE & OR Bartlett DISTRIBUTION. Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 34032_CH02_Walker.indd 18 11/5/11 8:01:18 AM © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 2-1 Frequency Distributions 19 ©EDUCATION Jones & What Bartlett is your highestLearning, level of education?LLC © Jones & Bartlett Learning, LLC NOTValue LabelFOR SALE ORValue DISTRIBUTION Frequency Percent Valid CumulativeNOT FOR SALE OR DISTRIBUTION Percent Percent Less than High School 1 16 4.6 4.8 4.8 GED 2 59 17.0 17.6 22.3 © Jones & BartlettHigh School Learning, Graduate LLC 3 8 2.3© Jones 2.4 & Bartlett 24.7 Learning, LLC NOT FOR SALESome OR College DISTRIBUTION4 117 33.7NOT FOR 34.8 SALE 59.5 OR DISTRIBUTION College Graduate 5 72 20.7 21.4 81.0 Postgraduate 6 64 18.4 19.0 100.0 Total 336 96.8 100.0 © Jones & Bartlett Learning,System LLC Missing © Jones11 & Bartlett3.2 Learning, LLC NOT FOR SALE OR DISTRIBUTION TotalNOT 347 FOR 100.0SALE OR DISTRIBUTION Table 2-1 Combination Table for Education from 1993 Little Rock Community Policing Survey © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION EDUCATION What is your highest level of education? Value Label Value Frequency Percent Valid % Cum % Less than High School 1 16 4.6 4.8 4.8 © Jones & BartlettGED Learning, LLC2 59 17.0© Jones 17.6 & Bartlett 22.3 Learning, LLC NOT FOR SALEHigh OR School DISTRIBUTION Graduate 3 8 2.3NOT FOR 2.4 SALE 24.7 OR DISTRIBUTION Some College 4 117 33.7 34.8 59.5 College Graduate 5 72 20.7 21.4 81.0 Postgraduate 6 64 18.4 19.0 100.0 © Jones & Bartlett Learning,System LLC Missing © 11Jones &3.2 Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT347 FOR 100.0 SALE OR DISTRIBUTION Table 2-2 Constructed in Excel of© the Jones variable & that Bartlett makes it Learning, clear what data LLC the distribution contains. In Table© Jones 2-1, the & Bartlett Learning, LLC variableNOT FOR is education SALE, and OR the DISTRIBUTION explanation is the wording from the surveyNOT from whichFOR SALE OR DISTRIBUTION the variable was drawn: “What is your highest level of education?” If the origin of the data is not expressed clearly in the discussion, the title should also contain this © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC.© NOTJones FOR SALE & OR Bartlett DISTRIBUTION. Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 34032_CH02_Walker.indd 19 11/5/11 8:01:18 AM © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION 20 Chapter 2 n Measures of Central Tendency information.© Jones This &may Bartlett include the Learning, year the data LLC was collected, the geographic© region Jones & Bartlett Learning, LLC of collection,NOT FOR and any SALE specific OR information DISTRIBUTION about how the data was gathered. NOT FOR SALE OR DISTRIBUTION Possibly more important than the title is proper labeling of the columns and cat- egories. The categories of the data should be clearly identified in the left column, called the stub of the table or distribution. These categories should be labeled even if the values are included. In Table 2-1, simply including the values 1 to 6 does not
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