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Statistical%20Significance

Statistical%20Significance - 96 Chapter 7 Descriptive...

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Unformatted text preview: 96 Chapter 7 / Descriptive Statistics Qesoripfives is anoTher Treauenle used SPSS procedure. DescripTive sTaTisTics are designed To give you inTorrnaTion abouT The disTribuTions 0T your variables. WiThin This broad caTegory are measures oi cenTral Tendency (Mean, Median, Mode), measures oi variabiliTy around The mean (83d deviation and yariancel, measures of deviarion from normaliTy (Skeflness and {garton sis), iniormaTion concerning The spread oi The disTribuTion (Magirnum, Migimum, and flange}, and inTormaTion abouT The sTabiliTy or sampling error 01‘ cerTain measures including sTandard error (S.E.) oi The mean (3.5,. mean), SE. 01‘ The kuriosis, and SE. oT The skewness (included by deTaulT when skewness and kurTosis are reauesTed). Using The Qescriptives command, i“? is Eff possible To access all OT These sTaTisTics or any subseT oT Them. in This inTroducTory secTion of The ill chapTer, we begin wiTh a brief descripiion oT sTaTisTical significance (included in all Terms oi daTa analysis) and The normal disTribuTion (because mosT sTaTisTical procedures require normally iii. disTribuTed daTa). Then each oi The sTaTisTics idenTiTied above is briefly described and illusTraTed. STATISTSCAL SIGNIFICANCE All procedures in The chapTers ThaT Tallow involve TesTing The significance OT The resulTs 0T each analysis. AlThough sTaTisTical significance is noT employed in The presenT chapTer if was Thoughi desirable To cover The concepT oi sTaTisTical significance (and normal disTribuTions in The secTion "if ThaT Tollows) early in The book. Significance is Typically designated wiTh words such as ”significance", ”sTaTisTical significance”, or ”probabiliTy”. "The laTTer word is The source 0T The leTTer ThaT represen’rs significance, The leTTer ”p”. [he p value idenTiTies The likelihood ThaT a pariicalar ouTcome may have occurred by -. chance. For insTance, group A may score an average oi 3.7 on a scale oi depression while. group 8 scores 4i on The same scale. IT a fTesT deTermines ThaT group A diliers from group B _ aT a pf: .01 level oi significance, iT may be concluded ThaT There is a l in TOO probabiliTy Thai The resuiTing dih‘erence happened by chance, and a 99 in iOO probabiliTy ThaT The discrepancy j in scores is a reliable finding. Regardless oT The Type oi analysis The ,0 value idenTiTies The likelihood ThaT a pariicular ouicome occurred by chance. A Chi—square analysis idenTiTies wheTher observed values diTTer signifi- canTiy Tram expecTed values; a TTesT onANOVA idenTilies whe’rher The mean OT one group dil- Ters significanily Tram The mean 0T anoTher group or groups; correla’rions and regressions iden— TiTy wheTher Two or more variables are signihcan’rly relaTed To each oTher. in all'insTances a sig- nificance vaiue will be calculaTed idenTiTying The likelihood ThaT a pariicular ouTcome is or is noi reliable. WiThin The conTexT of research in The social sciences, noThing is ever ”proved”. TT is " demonsTraTed or supporTed aT a cerTain level oT likelihood or significance. The smaller The p ' value, The greaTer The iikelihood Thai The Tindings are valid. ' Social scienTisTs have generally accepTed ThaT ii The p value is less Than .05 Than The resulT is considered statistically significant. Thus, when There is less Than a i in 20 probabiliTy ThaT a ‘ cerTain oaTcome occurred by chance, Then ThaT resulT is considered sTaTisTically signiTicanT. An- oTher Trequenle observed convenTion is ThaT when a significance level Tails beTween .05 and .i0, The resui’r is considered marginaily significant. When The significance level Tails Tar below .05 (e.g., .OOT , .OOOi, eTc.) The smaller The value The greaTer confidence The researcher has ThaT his or her Tindings are valid. When one wriTes up The Tindings oi a parTiculor sTudy, ceriain sTaTisTical iniormaTion and p Val‘ ; aes are always included. WheTher or noT a signiTicanT resulT has occurred is The key locus 0i .1. d to are eon {120* set, lard _ l by it is the s of oily sch ion tir- an- ig- iOl' Chapter 7 1 Descriptive Statistics 97 most studies that invoive statistics. Notice that when we use the letter ”p” in this section, it is italicized. In most manuscripts of scientific writing, letters associated with test statistics or levels of significance are underlined. When such material is published, the underlined material is then changed to italics. THE NORMAL DISTRIBUTION Many naturally occurring phenomena produce distributions at data that approximate a normal distribution. Some examples include the height at adult humans in the worid, the weight at coliie dogs, the scoring averages oi players in the NBA, and the le of residents at the United States. in oil at these distributions, there are many mid—range vaiues (e.g., 60-70 inches, 22 28 pounds, 9—14 points, (PO-iii) IQ points) and tew extreme values (e.g., 30 inches, 80 pounds, 60 points, T? %Q points). There are other distributions that approximate normaiéty but deviate in predictabie ways. For instance, times of runners in a iO—kilometer race will have few values iess than 30 minutes (none iess than 26:22), but many values greater than 40 minutes. The maiority at values will tie above the mean (average) value. This is called a negafr've/y skewed distribution. Then there is the distribution at ages at persons living in the United States. While there are individuais who are i year old and others who are TOO years old, there are tar more l-yeanolds, and in general the popuiation has more values below the mean than above the mean. This is calied a positive/y skewed dism'buffon. it is possible tor distributions to devi~ ate from normaiity in other ways, some of which are described in this chapter. A normal distribution is symmetric about the mean or average value. in a normai distribution, 68% at vaiues wiil lie between plus—ormminus (i) 1 standard deviation (described below) at the mean, 95r5% of values wili lie between i 2 standard deviations at the mean, and 99.7% of values will lie between i 3 standard deviations at the mean. A normal distribution is illustrated in the tigurp below. / l i i _,// | g , i T M” r ‘‘‘‘‘ M—SSD hit-280 M-1 SD MEAN (M) M+1SD {Vi-+230 M+GSD |<-—W-- 68% W>i i<- 95.5% ————————————————————————— >3 i< -------------------------------------------- 99.7% ------------------------------------------ >1 A tinal example will complete this section. The average (or mean) height at an American aduit male is 69 inches (5 it 9 in.) with a standard deviation at 4 inches. Thus, 68% at American 98 Chapter 7 / Descriptive Statistics men are beTween 5 TT 5 in. and 6 Ti i in. (69 i 4), 95.5% 01‘ American men are beTween 5 TTT in. and 6 TT 5 in. (69 ir 8], and 99.7% 0T American men are beTween 4 TT 9 in. and 6 TT 9 in. (69 i l2) in heighi (don‘T leT The NBA Tool you]. MEASURES OF CENTRAL TENDENCY The Mean is The-average value of The disTribuTion, or, The sum oT all values divided by The num» ber of values. The mean CT The disTribuTion [3 5 7 5 6 8 9] is (3+s+7+5+6+8+9v7=§_.1_4_. The Median is The middle value of The disTribuTion. The median 0T The disTribuTion [3 5 7 5 6 8 9], is 6, The middle value (when reordered Tram small To large, 3 5 5 6 7 8 9). The Mode is The mosT Trequenle occurring value. The mode 01‘ The disTribuTion [3 5 7 5 6 8 9] is 5, because The 5 occurs mosT Trequeniiy (Twice, all oTher values occur only once}. MEASURES OF VARIABILITY AROUND THE MEAN The Variance is The sum 01‘ squared deviaTions from The mean divided by N— i. The variance Tor The disTribuTion [3 5 7 5 6 8 9] is (is-6.14)»? + (543.14)2 + (Tr-6.14)?! + (5-6.14)2 + (swamp + (enemy? + (9-6.14)2)16 = 4.1429 Variance is used mainly Tor compuTaTional purposes. STandard deviaTion is The more commonly used measure 0T variabiliiy. The Starfdard deviation is The posiTive square rooT 01‘ The variance. For The disTribuTion [3 5 7 5 6 8 9], The siandard deviaTion is The square rooT 0T 4.1429, or 2.0354. 5 MEASURES BF DEVIATION FROM NORMALITY Kurtosis is a measure 0’? The ”peakedness“ or The “TlaTnessH aT a disTribuTioe. A l<uriosis value near zero (0) indicaies a shape close To normal. A posiTive value Tor The l<urTosis indicaTes a disTribuTion more peaked Than normal. A negaTive kurTosis indicaTes a shape TlaTTer Than nor— mal. An exTreme negaTive |<urTosis (e.g., < ~50) indicaTes a disTribuTion where more 0“? The values are in The Tails oT The disTribuTion Than around The mean. A l<urTosis value beTween ii .0 is considered excellenT Tor mos? psychomeiric purposes, buT a value beTween :20 is in many cases also accepTable, depending on The parTicular applicaTion. Example oi posiTive l<urTosis Example 0T aegaTive l<urTosis in. ‘qh UH) Chapter 7 / Descriptive Statistics 99 Skewness measures to what extent a distribution at values deviates from symmetry around the mean. A value at zero (0) represents a symmetric or evenly baianced distribution. A positive skewness indicates a greater number at smallervalues (sounds backward, but this is correct). A negative skewness indicates a greater number at iarger values. As with kartosis, a skewness value between it .0 is considered excellent tor most psychometric purposes, but a value be— tween i2.0 is in many cases aiso acceptable, depending on your particular application. "M-..\ F _. mm. /_Example at positive skewness /Emeple at negative skewnessk MEASURES FOR SIZE OF THE DISTRIBUTION For the distribution [3 5 7 5 6 8 9], the Maximum vaiue is 2, the Minimum vaiue is 3, and the Range IS 9— 3—— ‘55; TheSum otthe scores is 3+5+7+5+6+8+ 9"” M £13 MEASURES OF STABILITY: STANDARID ERROR SPSS computes the Standard errors tor the mean, the kurtosis, and the skewness. As indicated above, standard error is designed to be a measure at stabiiity or at sampiing error. The logic behind standard error isthis: It you take a random sampie tram a population, you can comw pute the mean, a single number it you take another sample 0! the same size lrom the same population you can again compute the mean—a number likely to be slightly ditterent from the tirst number. ityou collect many such samples, the standard error at the mean is the standard deviation at this sampling distribution at means. A similar logic is behind the computation at standard error tor kuriosis or skewness. A smail value (what is ”small" depends on the nature at your distribution) indicates greaterstability or smal/ersampling error. The tile we use to iiiastrate the Qescriptives command is our example described in the tirst chapter. The data tile is calied grades.sav and has an N = i035. This analysis computes de- scriptive statistics for variables gpa, totai, final, and percent. STEP BY STEP Bescriptives l Create and name a data file or edit {i'tnecessaij/j an already existing fi/e {see Chapter 3) To enter SIDES, a click on Start in the taskbar (bottom ofscreen) activates the start menu: ' . ,t _) «g fisrssriereerrooom Fronri t 100 Chapter 7 / Descriptive Statistics Affer clicking The 5/955 program icon, Screen 7 appears on The moniior. Screens 7 and 2 (dis- played on The inside froni cover] allow you To access ihe dafa Tile used in concluding The analy— sis ofinieresi‘. The following sequence accesses The grades.sav file for fun‘her analyses: . @Eiie —>_Qpen —-> _. type gradee.eav —-> gradessav] Whether Tirsi em‘ering 5/955 or rerurning from earlier operafions The srandard menu of com- mands across The Top is required {shown below). /-ls long as if is visible you may perform any analyses. li‘ is nor‘ necessary for ihe dafa window io be visible. J 2:4 dwEIr " ' _. .. J __ _ ' " J Eile idit Eiew 'Qate- iransiarm finefyze graphs fltiliiies window fielp This menu 0T commands disappears or modifies when using pivo’r Tables or ediTing graphs. To_'_ uncover The sTandard menu of commands simply click on The [2-3 or The icon. " ' Ah‘er compleiion of Siep 3 a screen wiih ihe desired menu bar appears. When you click a command {from The menu bar), a series of opfions will appear (usually) below The selecied if command. Wiih each new sef of opz‘ions, click The desired ifem. The sequence To access De? :5 scripfive Siaiisfics begins of any screen wifh The menu of commands visible: " ' ._ In. “ream . J Q analyze -> Descriptive Statistics —~> Qescriptives A new screen now appears (below) ThaT allows you To selecT variabies Tor which you wish To: 53;: compuie descripiives. The procedure involves clicking The desired variable name in The box so The leTT and Then pasiing ii inTo The yariablds) (or ”acTive”) box To The righT by ciicking The righ’r arrow (“ in The middle OT The screen. if The desired variable is noi visible, use The scroli ba arrows (a I) To bring H To view. To deseleci a variable (Thai is, To move iT Tram The Mari able(s) box back To The originai lisT), ciici< on she variable is The acTive box and The in The : cenTer will become a -. Ciick on The leTT arrow To move The variable back. To clear ail varil— ables from The acTive box, click The geset buTTon. The Descripfives Window '55 4%» year __:: @iawup ' seamen e» gee :2 (e extrcred vie) review Chapter 7 / Descriptive Statistics 101 2 {Cl/3- The only check box on The ini’rial screen, Save standardiged vaiues as variables, will converT GNU/y- all designaTed variables (Those in The Mariablds) box) To Z scores. The original variahies will remain, buT riew variables wiTh a ”z” aTTached To The TronT will be included in The lisT ol vari- ables. For insTance, iT you click The Save standardiged values as variables opTion, and The variable final was in The _\[_ariable(s) box, iT would be iisTed in Two ways: final la The original scale, and zfinai Tor The same variable convened To 2 scores. You may Then do analyses wiTh eiTher The original variable or The variable canver’red To zscores. Recall ThaT zscores are values Thai have been maThemaTically Transposed To creaie a disTribaTion wiTh a mean 01‘ zero and a sTandard deviaTion 01‘ one. See The glossary for a more compleTe delieiTion. Also noTe Thair Tor non-mouse users, The SPSS peopie have cleverly underlined The 2 in The word ”sTandardized” as a genTle reminder ThaT sTandardized scores and zscores are The same Thing. To creafe a Table of The defau/f descriph'ves (mean, sfana’ara’ devfaffon, maxmwm, minimum) for The variables gpa and total, perform The Ira/lowing sequence ofsfeps: ihijc'r-‘eea 9.9 This! .. 99-5.“ - -—> \ total fl)“;- -> “1.3 -JT is: ll you wish To calculaTe more Than The Tour deTauiT sTaTisTics, aTTer selecTing The desired vari— De- ables, belore clicking The OK, H is necessary To click The thions baTTon (aT The boTTom oT screen 7.1). Here every descripTive sTaTisTic presenTed earlier in This chapTer is included wiTh a couple oi excepTioris: Median and mode are accessed Through The Frequencies command oniy. See ChapTer 6 To deTermine how To access These values. Also, The sTandard errors j (”SE”) oTEThe kurTosis and skewness are noT included. This is because when you click eiTher i<urTosis or skewness, The sTandard errors oT Those values are auTornaTically included. To seiecT l0 The desired; descripiive sTaTisTics, The procedure is simplyio click. (so as To ieave an in The box l0 To The leTT of The desired value) The descripTive sTaTisTics you wish. This is Tollowed by a click of l” Continue and OK. The Display order opTions include (0) Variable list (The delaulein The [3” same order as displayed in The daTa ediTor), (b) Alphabetic (names 03‘ variables ordered alpha— : beTically), (c) Asgending means (ordered Tram smallesT mean value To largesi mean value in The oquuT), and (d) Qescending means (Tram iargesT To smallesT). “'2 The Descrrpi‘fves.’ Oph‘ons Who/ow El '3'" .fiiirtasis 102 Chapter 7 / Descriptive Statistics To se/ecf The variables final, percent, gpa, and totai, and Then se/eci a// desired descripi‘ive sioiisr‘ics, perform The fo//owing sequence of sreps: Press The Beset bun‘on if There are unde— sired variob/es in The acfive box. Step" 5.3.5 w) £39m ->® -> Q .iu-‘s.¢.iéan_._Bo-This.' . . . . . . . .. figfinai “9%- --) “QT percent we £$ total *9 £3— -> % thions Q aii desired descriptive statistics (so 3 appears in each box) w“) Continue ‘4 OK -' Upon compleTion of eiTher sTep 5 or sTep 50, Screen 7.3 will appear (below). The resulTs of The iusT—compieTed analysis are included in The Top window labeled Output1 w SPSS Output Navi- gator. Click on The .To The righT of This Title ii you wish The output To iiii The enTire screen, Then make use oi the arrows on The scroli bar (I. iii“: ii To view The resulTs Even when viewing oquUT, The sTandard menu of commands is sTiIl lisTed across The Top of The window. - Furiher analyses may be conducted wiThouT returning To The dais screen. ParTiai oquuT from This analysis is included in The OUTpUT section The Oufp UH — 5P55 l Elm... {resumes 01,prer i” We Nawgafor i'fi ”we? - : I nun-[3 Descriptive senses WHO/63W : fleecript’m Stamina N i Mean i 531 I Skewnsss ‘ Kunssis Statistic i Statistic Statistic Statistic Std‘Eiror Statistic Tee Error FTNAL ‘ Si ‘48 7.64 x1335 ,235 T PERCENT 38.331 12.177 77844 236 FREVGF‘A 2.1ng 1533 _ ms: .235 TOTAL i .- 1335? i i530 -.937 i 238 \iaiitiN (“six/rise) l ‘isisseiimsmeiime PRINTING RESULTS Results UT The analysis (or analyses) ThaT have jusi been cenducied requires a window ThaT dis— piays The SlCli‘lCiCIT'Ci commends (Eiie gdit Qata Iransform Anaiyze . . .) across the Top. A Typical prinT procedure is shown below beginning with The sTandord oquuT screen (Screen it inside back cover). ' ‘ fiJ Chapter 7 / Descripttve Statistics 103 To print resu/ts, from the Output screen perform the I’d/owing sequence ofst‘eps: theatres. DoThis _. '. . .. . ' . . i Tiers-.96, Se/ect desired output or edit {see pages it 9 - 25) —> % Eiie —-9 «:3 Brim: l - Consider and se/ect desired print options on‘ered #179” ‘9‘ :2" To exit you may begin from any screen that shows the File command at the top. tin-Stress. Dons _ __ . . .. _ .. . __ ;- {stifle e‘tiEEsit w _...2:1 Note: Atter clicking Eggit, there will trequently be smalt windows that appear asking it you wish to save or change anything. Simpiy ciick each appropriate response. GUTPUT flescriptive Statistics What follows is output tram sequence step 50, page 102. Notice that the statistics requested included the N, the Mean, the Standard Deviation, the Vartonce, the Skewness, and the Korto- sis. The Standard Errors of the Skewness and Kurtosis are included by detault. SPSS for Windows: Descriptive Statistics ' N Mean Sld'. Variance Skewness E Kurtosis - Devratlon , Z Statistic Statistic Statistic ‘ Statistic Statistic Std. Error Statistic Std Error _ FINAL ’ ' _' ' 63.098 PERCENT I l t48.272 GPA : .583 TOTAL - 234.074 _ Vaiid N First observe that in this display the entire output tits neatly onto a single page or is entirely visi- late on the screen. This is rarety the case. When more extensive output ts produced, make use at the up, down, lett, and right scroll bar arrows to move to the desired place. You may also use the index in the tett window to move to particular output more quickly. Notice that all tour variables tall within the ”excellent” range as acceptable variables tor’ turther analyses; the skewness and i<urtosis values all lie between 3: H]. All terms are identified and described in the introductory portion at the chapter. The only undefined word is Itstwise. This means that any subiect that has a misséng value tor any variable has been deleted tram the onaiysts. Since in the gradessav hie there are no missing values, at! i05 subjects are included. EXERC‘ISES Answers to selected exercises are avattabie tor downtoad at httg:llwww.abacon.comlgeorge. l x u! s 7’? . f.)- . ‘h‘r‘ “wittih s tease“! ...
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