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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
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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., 6070 inches, 22
28 pounds, 9—14 points, (POiii) 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 lyeanolds, 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 hit280 M1 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 Theaverage 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 (is6.14)»? + (543.14)2 + (Tr6.14)?! + (56.14)2 + (swamp + (enemy? + (96.14)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 [23 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 nonmouse 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! .. 995.“  —> \ 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_._BoThis.' . . . . . . . ..
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. tinStress. 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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