AS we were teaching a multivariate statistics

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1 RESEARCH NOTE What Are Degrees f Freedm? Shanta Pandey and Charltte Lyn Bright AS we were teahing a multivariate statistis urse fr dtral students, ne f the students in the lass asked,"what are degrees f freedm? I knw it is nt gd t lse degrees f freedm, but what are they?" Other students in the lass waited fr a lear-ut respnse. As we tried t give a textbk answer, we were nt satisfied and we did nt get the sense that ur students understd. We lked thrugh ur statistis bks t determine whether we uld find a mre lear way t explain this term t sial wrk students.the wide variety f language used t define degrees jfredm is enugh t nfuse any sial wrker! Definitins range frm the brad, "Degrees f freedm are the number f values in a distributin that are free t vary fr any partiular statisti" (Healey, 1990, p. 214), t the tehnial; Statistiians start with the number f terms in the sum [f suares], then subtrat the number f mean values that were alulated alng the way. The result is alled the degrees f freedm, fr reasns that reside, believe it r nt. in the thery f thermdynamis. (Nrman & Streiiier, 23, p. 43) Authrs wh have tried t be mre speifi have defined degrees f freedm in relatin t sample size (Trhim,25;Weinbah & Grinne]],24), ell size (Salkind, 24), the mmiber f relatinships in the data (Walker, 1940),and the differene in dimensinahties f the parameter spaes (Gd, 1973).The mst mmn definitin inludes the number r piees f infrmatin that are free t vary (Healey, 1990; Jaard & Beker, 1990; Pagan, 24; Warner, 28; Wnnatt & Wnnatt, 1990). These speifiatins d nt seem t augment students' understanding f this term. Hene, degrees f freedm are neptually diffiult but are imprtant t reprt t understand statistial analysis. Fr example, withut degrees f freedm, we are unable t alulate r t understand any underlying ppulatin variability. Als, in a bivariate and multivariate analysis, degrees f freedm are a funtin f sample size, number f variables, and number f parameters t be estimated; therefre, degrees f freedm are als assiated with statistial pwer. This researh nte is intended t mprehensively define degrees f freedm, t explain hw they are alulated, and t give examples f the different types f degrees f freedm in sme mmnly used analyses. DEGREES OF FREEDOM DEFINED In any statistial analysis the gal is t understand hw the variables (r parameters t be estimated) and bservatins are linked. Hene, degrees f freedm are a funtin f bth sample size (N) (Trhim, 25) and the number f independent variables (k) in ne's mdel (Tthaker & Miller, 1996; Walker, 1940; Yu, 1997).The degrees f fi^edm are eual t the number f independent bservatins {N),r the number f subjets in the data, minus the number f parameters (k) estimated (Tthaker & Miller, 1996; Walker, 1940). A parameter (fr example, slpe) t be estimated is related t the value f an independent variable and inluded in a statistial euatin (an additinal parameter is estimated fr an interept iu a general linear mdel). A researher may estimate parameters using different amunts r piees f infrmatin,and the number f independent piees f infrmatin he r she uses t estimate a statisti r a parameter are alled the degrees f freedm {dfi (HyperStat Online, n.d.). Fr example,a researher rerds inme f N number f individuals frm a mmunity. Here he r she has Nindependent piees f infrmatin (that is, N pints f inmes) and ne variable alled inme (t); in subseuent analysis f this data set, degrees f freedm are asiated with bth Nand k. Fr instane, if this researher wants t alulate sample variane t understand the extent t whih inmes vary in this mmunity, the degrees f freedm eual N- f.the relatinship between sample size and degrees f freedm is CCCCde: 107O-S3O9/O8 (3. O28 Natinal Assiatin f Stial Wrkers 119

2 psitive; as sample size inreases s d the degrees f freedm. On the ther hand, the relatinship between the degrees f freedm and number f parameters t be estimated is negative. In ther wrds, the degrees f freedm derease as the number f parameters t be estimated inreases. That is why sme statistiians define degrees f freedm as the number f independent values that are left after the researher has applied all the restritins (Rsenthal, 21; Runyn & Haber, 1991); therefre, degrees f freedm vary frm ne statistial test t anther (Salkind, 24). Fr the purpse f larifiatin, let us lk at sme examples. A Single Observatin with One Parameter t Be Estimated If a researher has measured inme (k = 1) fr ne bservatin {N = 1) frm a mmunity, the mean sample inme is the same as the value f this bservatin. With this value, tbe researher has sme idea t the mean inme f this mmunity but des nt knw anything abut the ppulatin spread r variability (Wnnatt & Wnnatt, 1990). Als, the researher has nly ne independent bservatin (inme) with a parameter that he r she needs t estimate. The degrees f freedm here are eual t N - f.thus, there is n degree f freedm in this example (1-1 = 0). In ther wrds, the data pint has n freedm t vary, and the analysis is limited t the presentatin f the value f this data pint (Wnnatt & Wnnatt, 1990; Yu, 1997). Fr us t understand data variability, N must be larger than 1. Multiple Observatins (N) with One Parameter t Be Estimated Suppse there are N bservatins fr inme. T examine the variability in inme, we need t estimate nly ne parameter (that is, sample variane) fr inme (k), leaving the degrees f freedm f N k. Beause we knw that we have nly ne parameter t estimate, we may say that we have a ttal f N 1 degrees f freedm. Therefre, all univariate sample harateristis that are mputed with the sum f suares inluding the standard deviatin and variane have N 1 degrees f freedm (Warner, 28). Degrees f freedm vary frm ne statistial test t anther as we mve frm univariate t bivariate and mtiltivariate statistial analysis, depending n the nature f restritins applied even when sample size remains unhanged. In the examples that fllw, we explain hw degrees f freedm are alulated in sme f the mmnly used bivariate and mujtivariate analyses. 1W Samples with One Parameter (r t Test) Suppse that the researher has tw samples, men and wmen, r n, + n^ bservatins. Here, ne an use an independent samples t test t analyze whether the mean inmes f these tw grups are different. In the mparisn f inme variability between these tw independent means (r k number f means), the researher will have n^ + n.,-2 degrees f freedm. The ttal degrees f freedm are the sum f the number f ases in grup 1 and grup 2 minus the number f grups. As a ase in pint, see the SAS and SPSS utputs f a t test mparing the literay rate (LITERACY, dependent variable) f pr and rih untries (GNPSPLIT, independent variable) in Table l.au in all, SAS utput has fur different values f degrees ffreedm (tw f whih are als given by SPSS).We review eah f them in the fllwing paragraphs. The first value fr degrees f freedm under t tests is 1 (reprted by bth SAS and SPSS).The tw grups f untries (rih and pr) are assumed t have eual varianes in their literay rate, the dependent variable. This first value f degrees f freedm is alulated asm^ + n^-2 (the sum f the sample size f eah grup mpared in the f test minus the number f grups being mpared), that is = 1. Fr the test f euality f variane, bth SAS and SPSS use the F test. SAS uses tw different values f degrees f freedm and reprts flded F statistis. The numeratr degrees f freedm are alulated as n 1, that is 64 1 = 63. The denminatr degrees f freedm are alulated as n^ - 1 r 38-1 = 37.These degrees f freedm are used in testing the assumptin that the varianes in the tw grups (rih and pr untries, in ur example) are nt signifiantly different.these tw values are inluded in the alulatins mputed within the statistial prgram and are reprted n SAS utput as shwn in Table 1. SPSS, hwever, mputes Levene's weighted F statisti (seetable 1) and uses k-\ and N- kdegrees ffreedm,where k stands fr the number f grups being mpared and N stands fr the ttal number f bservatins in the sample; therefre, the degrees f freedm assiated with the Levene's F statisti 120 Sial Wrk Researh VOLUME JI, NUMBER Z JUNE ZOO8

3 lu a. '^ # la 5" n-1 m O GNPSPLiT LITERACT 0 (pr) 1 [rih) Levene's Test fr Euality f Varianes LITERACY Eual varianes assumed 14,266.0 Eual varianes nt assumed 1 T UJ 1 E ren tt; u 'O <ri IT rj IN Z^ ^ -^ r-i fs t t 5 p 1 ^ 96.9 "i L I lij g S UJ > E S LITERACY t,j d s >^ la.3 1 a -TJ (N v r^ rs S fs -«p la ^ S C- w I. 2 ^3 I 13

4 are the same (that is,fe-1 =2-1 = \,N-k = = 1) as the degrees ffireedmassiated with "eual" variane test disussed earlier,and therefre SPSS des nt reprt it separately. If the assumptin f eual variane is vilated and the tw grups have different varianes as is the ase in this example, where the flded F test r Levene's F weighted statisti is signifiant, indiating that the tw grups have signifiantly different varianes, the value fr degrees f freedm (1) is n lnger aurate.therefre, we need t estimate the rret degrees f freedm (SAS Institute, 1985; als see Satterthwaite, 1946, fr the mputatins invlved in this estimatin). We an estimate the degrees f freedm arding t Satterthwaite's (1946) methd by using the fllwing frmula: (//"Satterthwaite = («-1)1- (H-])(»,-!) where n^ - sample size f grup \,n^= sample size f grup 2, and S^ and S^ are the standard deviatins f grups 1 and 2,respetiveIy.By inserting subgrup data frm Table 1, we arrive at the mre aurate degrees f freedm as fllws: (64-1) (38-1) 25.65= X X 38 (64-1) 1 - / 25.65^ X 38 + (38-1) / 25.65^ X 38 V 18.07^ X 64, U 18.07^x64, h ; [ ]^ _5_0(i(X J '''[ & , X X = Beause the assumptin f euality f varianes is vilated, in the previus analysis the Satterthwaite's value fr degrees f freedm, (SAS runds it t 97), is aurate, and ur earlier value, 1, is nt. Frtunately, it is n lnger neessary t hand alulate this as majr statistial pakages suh as SAS and SPSS prvide the rret value fr degrees f freedm when the assumptin f eual variane is vilated and eual varianes are nt assumed. This is the furth value fr degrees f freedm in ur example, whih appears in Table 1 as 97 in SAS and in SPSS. Again, this value is the rret number t reprt, as the assumptin f eual varianes is vilated in ur example. Cmparing the Means f g Grups with One Parameter (Analysis f Variane) What if we have mre than tw grups t mpare? Let us assume that we have «, +...+«grups f bservatins r untries gruped by phtial freedm (FP^EDOMX) and that we are interested in differenes in their literay rates (LITERACY, the dependent variable ).We an test the variability f^ means by using the analysis f variane (ANOVA).Th ANOVA predure prdues three different types t degrees f freedm, alulated as fllws: The first type f degrees f freedm is alled the between-grups degrees f freedm r mdel degrees f freedm and an be determined by using the number f grup means we want t mpare. The ANOVA predure tests the assumptin that the g grups have eual means and that the ppulatin mean is nt statistially different frm the individual grup means. This assumptin reflets the null hypthesis, whih is that there is n statistially signifiant differene between literay rates in g grups f untries (ji^= ^^ = fx^).the alternative hypthesis is that the g sample means are signifiantly different frm ne anther. There are g - 1 mdel degrees f freedm fr testing the null hypthesis and fr assessing variability amng the^ means.this value f mdel degrees f freedm is used in the numeratr fr alulating the F rati in ANOVA. The send type f degrees ffreedm,alled the within-grups degrees f freedm r errr degrees ffreedm, is derived frm subtrating the mdel degrees ffreedm frm the rreted ttal degrees f freedm. The within-grups 122 Sial Wrk Researh VOLUME 32, NUMBER 2 JUNE 28

5 degrees f freedm eual the ttal number f bservatins minus the number f grups t be mpared, n^ n^-g. This value als aunts fr the denminatr degrees f freedm fr alulating the F statisti in an ANOVA. Calulating the third type f degrees f freedm is straightfrward. We knw that the sum f deviatin frm the mean r 2(Y - F) = O.We als knw that the ttal sum f suares r 2(y - Y)^ is nthing but the sum f N- deviatins frm the mean. Therefre, t estimate the ttal sum f suares 2(y -?)-, we need nly the sum f N- 1 deviatins frm the mean.therefre.with the ttal sample size we an btain the ttal degrees f freedm, r rreted ttal degrees f freedm, by using the frmula N- 1. hitable 2, we shw the SAS and SPSS utput with these three different values f degrees f freedm using the ANOVA predure.the dependent variable.literay rate.is ntinuus,and the independent variable, plitial freedm r FREEDOMX, is nminal. Cuntries are lassified int three grups n the basis f the amunt f plitial freedm eah untry enjys: Cuntries that enjy high plitial freedm are ded as 1 («= 32), untries that enjy mderate plitial freedm are ded as 2 (n = 34), and untries that enjy n phtiai freedm are ded as 3 (tt = 36). The mean literay rates (dependent variable) f these grups f untries are examined.the null hypthesis tests the assumptin hat there is n signifiant differene in the literay r.ites f these untries arding t their level f plitial freedm. The first f the three degrees f freedm, the between-grups degrees f freedm, euals g - \. Beause there are three grups f untries in this analysis, we have 3-1 = 2 degrees f freedm.this aunts fr the numeratr degrees t freedm in estimating the Fsta tis ti. Send, the wi thin-grups degrees f freedm, whih aunts fr the denminatr degrees f freedm fr alulating the F statisti in ANOVA, euals ri^ n^-g. These degrees f freedm are alulated as = 99. Finally, the third degrees f freedm, the ttal degrees f freedm, are alulated as N - 1 (102-1 = 101).When reprting Fvalues and their respetive degrees f freedm, researhers shuld reprt them as fllws: The independent and the dependent variables are signifiantly related [F(2, 99) ^ 16.64,p<.01]. Degrees f Freedm in Multiple Regressin Analysis We skip t multiple regressin beause degrees f freedm are the same in ANOVA and in simple regressin. In multiple regressin analysis, there is mre than ne independent variable and ne dependent variable. Here, a parameter stands fr the relatinship between a dependent variable (Y) and eah independent variable (X). One must understand fur different types f degrees f freedm in multiple regressin. The first type is the mdel (regremti) degrees f freedm. Mdel degrees f freedm are assiated with the number f independent variables in the mdel and an be understd as fllws: A null mdel r a mdel withut independent variables will have zer parameters t be estimated. Therefre, predited V* is eual t the mean f Vand the degrees f freedm eual 0. A mde! with ne independent variable has ne preditr r ne piee f useful infrmatin (fe = 1) fr estimatin f variability in Y. This mdel must als estimate the pint where the regressin line riginates r an interept. Hene, in a mdel with ne preditr, there are (fe + t) parameters k regressin effiients plus an interept t be estimated, with k signifying the number f preditrs. Therefre,there are \{k + ])- l],rfe degrees f freedm fr testing this regressin mdel. Ardingly, a multiple regressin mdel with mre than ne independent variable has sme mre useful infrmatin in estimating the variability in the dependent variable, and the mdel degrees f freedm inrease as the number f independent variables inrease.the null hypthesis is that all f the preditn have the same regressin effiient f zer, thus there is tily ne mmn effiient t be estimated (Dallal,2(X)3).The alternative hypthesis is that the regressin effiients are nt zer and that eah variable explains a different amunt f variane in the dependent variable. Thus, the researher must estimate fe effiients plus the interept. Therefre, PANDEY AND BRtGHT / WhtAr Dtgrfs ffrefdm? 123

6 ! Sum f Suares CV O Subset (a =.05) ! Between Grups Within Grups Ttal 1 FREEDOMX " t: 'S 0 a y II II II f^ (N 1 <^ II i I Si a fn la frt >O (TV la la O rs Ml * rs vi rj r^ v I 3 mbei tial 3 2 J il

7 there are (fe + 1) - 1 r it degrees ffreedm tr testing the null hypthesis (Dallal, 23). In ther wrds, the mdel degrees ffreedm eual the number f useful piees finfrmaiii available fr estimatin f variability in the dependent variable. The send type is tbe residual, r errr, degrees ffreedm. Residual degrees f freedm in multiple regressin invlve infrmatin f bth sample size and preditr variables. In additin, we als need t aunt fr the interept. Fr example, if ur sample size euals N, we need t estimate k + l parameters, r ne regressin fiu-ient fr eah f the preditr variables {k) plus ne fr the interept. The residual degrees f freedm are alulated N - {k + l).this is the same as the frmula fr the errr, r within-grups, degrees ffreedm in the ANOVA. It is imprtant t nte that inreasing the number f preditr variables has impliatins fr the residual degrees f freedm. Eah additinal parameter t be estimated sts ne residual degree ffreedm {Dallal,2()03).The remaining residual degrees ffreedm are used t estimate variability in the dependent variable. Tbe third type f degrees ffreedm is the ttal, r rreted ttal, degrees ffreedm. As in ANOVA, this is alulated N- \. Finally, the furth rype f degrees ffreedm that SAS (and nt SPSS) reprts under the parameter estimate in multiple regressin is wrth mentining. Here, the null hypthesis is that there is n relatinship between eah independent variable and the dependent variable. The degree f freedm is always t fr eah relatinship and therefre, sme statistial sfrware.sub as SPSS,d nt bther t reprt it. In the example f multiple regressin analysis (see Table 3), there are fur different values f degrees f freedm. The first is the regressin degrees f freedm. This is estimated as (fe + 1) - 1 r (6 + 1) - 1 =6, where k is the number f independent variables in the mdel. Send, the residual degrees ffreedm are estimated as N- {k + 1). Its value here is 99 - (6 + 1) = 92.ThinJ, the ttal degrees ffreedm are alulated N- 1 (r 99-1 = 98). Finally, the degrees f freedm shwn under parameter estimates fr eah parameter always eual 1. as explained abve. F values and the respetive degrees f freedm frm the urrent regressin utput shuld be reprted as fllws: Tbe regressin mdel is statistially signifiant with F(6, 92) = 44.86,p<.01. Degrees f Freedm in a Nnparametri Test Pearsn's hi suare, r simply tbe hi-suare statisti, is an example fa nnparametri test that is widely used t examine the assiatin between tvi' nminal level variables. Arding t Weiss (1968) "the number f degrees ffreedm t be assiated with a hi-suare statisti is eual t the number f independent mpnents that entered int its alulatin" (p. 262). He further explained that eah ell in a hi-suare statisti represents a single mpnent and that an independent mpnent is ne where neither bserved nr expeted values are determined by the freuenies in ther ells. In ther vfrds, in a ntingeny table, ne rw and ne lumn are fated and tbe remaining ells are independent and are free t vary. Therefre, the hi-suare distributin has (r- 1) X (- 1) degrees f freedm, where r is the number f rws and f is tbe number f lumns in the analysis (Chen, 1988; Walker, 1940; Weiss, 1968). We subtrat ne frm bth the number f rws and lumns simply beause by knwing the values in ther ells we an tell the values in the last ells fr bth rws and lumns; therefre, these last ells are nt independent. \ As an example, we ran a hi-suare test t exatnine whether grss natinal prdut (GNP) per apita f a untry (GNPSPLIT) is related t its level f plitial freedm (FREEDOMX). Cuntries (GNPSPLIT) are divided int tw ategries rih untries r untries with higji GNP per apita (ded as 1) and pr untries r untries with lw GNP per apita (ded as O),and plitial freedm (FREEDOMX) has three levels free (ded as 1), partly free (ded as 2), nt free (ded as 3) (see Table 4). In this analysis, the degrees ffreedm ar (2-1) X (3-1) = 2. In ther wrds, by knwing the number f rih untries, we wuld autmatially knw the number f pr untries. But by knwing the number f untries that are free, we wuld nt knw the number f untries that are partly free and nt free. Here, we need t knw tw f the three mpnents fr instane, the number f untries that are free and partly free s that we will knw the number f untries I'ANDEV AND BRIGHT / What Are Degrees fiivedm? 125

8 s ^_ s 1/* "^ (N p s <? (N "T CN "" Estimate enrs p. 1 Standa: 1 Ceffi (N fs p rn Adjusted /^' Mde!,729,745 s UJ part] 2 r 1 t3 II u Vari - I.2 -a Sum f Suares Mdel v^ fn ON CN Regressin 24120,239 Residual Ttal GNP 2 0 U J2-9 tvari 1 V u S B - v> a n U S 0 D effi C del s CN (N 1 (Cn <1 fs 1 CO SO part p w Z EDFU CN CO PUPl CO 90 d SCHC 5. Z ON «(N -ii- "A ^» m V IN I/N t s O V V 92 -I s. <} O O JJ -.r ^*j i^ w^ P (N CN IN.g r^ (*) \.'I ON ^-^ O P S I C IN S s 1/5 (S CN 111 J4.S ^ S E E 3 3a Z Z 2 IT p -V t^ B -0 2 ^

9 n 1.0 Missing a < 192 :; Ttal FREEDOMX % % % 35 28,2% % % % % % % % % Asymptti, p (2-tailea) GNPSPLIT FREEDOMX Value fn (N GNPSPLIT 0 Cunt % f Ttal 1 Cunt % f Ttal Ttal Cunt % f Ttal Pearsn X" Likelihd Rati " Linear-by-Linear Assiatin Valid Cases (N) u

10 that are ntfree.therefre,in this analysis there are tw independent mpnents that are free t vary, and thus the degrees f freedm are 2. Readers may nte that there are three values under degrees f freedm in Table 4. The first tw values are alulated the same vi'ay as disussed earlier and have the same values and are reprted mst widely. These are the values assiated with the Pearsn hi-suare and likelihd rati hi-suare tests.the fmal test is rarely used. We explain this briefly.the degree f freedm fr the Mantel-Haenszel hi-^ suare statisti is alulated t test the hypthesis that the relatinship between tw variables (rw and lumn variables) is linear; it is alulated as (N " 1) X f^, where r is the Pearsn prdut-mment rrelatin between the rw variable and the lumn variable (SAS Institute. 1990).This degree f freedm is always 1 and is useful nly when bth rw and lumn variables are rdinal. CONCLUSION Yu (1997) nted that "degree f freedm is an intimate stranger t statistis students" (p. l).this researh nte has attempted t derease the strangeness f this relatinship with an intrdutin t the lgi f the use f degrees f freedm t rretly interpret statistial results. Mre advaned researhen, hwever, will nte that the infrmatin prvided in this artile is limited and fairly elementary. As degrees f freedm vary by statistial test (Salkind, 24), spae prhibits a mre mprehensive demnstratin. Anyne with a desire t learn mre abut degrees ffreedmin statistial alulatins is enuraged t nsult mre detailed resures,suh as Gd (1973),Walker (1940),andYu (1997). Finally, fr illustrative purpses we used Wrld Data that reprts infrmatin at untry level. In ur analysis, we have treated eah untry as an independent unit f analysis. Als, in the analysis, eah untry is given the same weight irrespetive f its ppulatin size r area. We have ignred limitatins that are inherent in the use f suh data. We warn readers t ignre the statistial findings f ur analysis and take away nly the disussin that pertains t degrees f feedm. Healey.J. F. (1990). Statistis: A tl fr sial researh (2nd ed,), Belmnt, CA: Wadswrth, HyperStat Onlitie. (n,d,). Degrees f freedm. Retrieved May 30, 26, frm A42408.htmt Jaard,J., & Beker, M. A. (1990). Statistiafr the behaviral sienes (2nd ed.), Belmnt, CA:Wadswrth, Nrman. G. R,, & Streiner, D. L, (20113), PD statistis (3rd ed,). Hamiltn, Ontari, Canada: BC Deker, Pagan. R, R, (24), Understanding statistis in the behaviral sienes (7th ed,). Belmnt, CA: Wadswrth, Rsenthal, j.a, (21), Statistis and data interpretaliu fr the helping prfessins. Belmnt. CA: Wadswrth. Runyn. R. P,,& Haber,A. (1991). Fundamntak f behaviral statistis (7th ed.). New Yrk: MGraw-Hill, Salkind, N.J, (21)04), Statistis fr peple wh (think they) hate statistis (2nd ed.).thusand Oaks, CA: Sage Publiatins, SAS Institute. (1985). SAS user's guide: Statistis, versin 5.1. Cary, NC: SAS Institute. SAS Institute, (1990), SAS predure ^uide, versin 6 (3nJ ed.). Cary, NC: SAS Institute, ' Satterthwaite, F. E. (1946), An apprximate distributin f estimates f variane mpnents. Bimetris Bulletin, 2, Tthaker, L, E., &: MiUer, L, (1996), Intrdutry statistis fr the behaviral sienes (2nd ed.). Paifi Grve, CA: Brks/Cle, Trhim,W.M.K. (25), Researh methds:the nise knwledge base. Cininnati: Atmi Dg, Walker, H,W, (1940), Degrees f freedm.^hrrti/ f Ediuatinal Psyhlgy, 31, Warner. R. M. (28). Applied smti.(fi«,thusand Oaks, CA: Sage Publiatins, Weinbah, R. W., & Griniiell, R. M.,Jr. (24). Statistis fr sial wrkers (6th ed.). Bstn: Pearsn. Weiss, R. S. (1968). Statistis in sial researh:an intrdudin. New Yrk: Jhn Wiley & Sns. Wnnatt,T, H., & Wiinatt, R.J, (1990). Intrdutry statistis (5th ed,). New Yrk: Jhn Wiley & Sns. Yu, C. H, (1997), Illustrating degrees f freedm in terms f sample size and dimensinality. Retrieved Nvember 1,27, frm m/mputer/sas/df.hnnl Shanta Pandey, PhD, is assiate prfes.<:r, and Charltte Lyn Bright, MSH^is a dtral student, Gerge Warren Brwn Shl f Sial Wrk, Washingtn University, St. Luis. The authrs are thankful t a student whse urius mind inspired them t wrk n this researh nte. Crrespndene nerning this artile shuld be sent t Shanta Pindey, Getge Warren Brwn Shl f Sial Wrk, Washingtn University, St. Luis, MO 63130; parideys@unistl.edu. Original manusript reeived August 29, 26 Final revisin reeived Nvember 15, 27 Aepted January 16, 28 REFERENCES Cheti,J, (1988), Statistial pwer analysis fr the behaviral sienes (2nd ed,), Hillsdale, NJ: Lawrene Eribauni. Dallal. G. E, (23). Derees f freedm. Retrieved May 30, 26. trtn http;// Gd. LJ. (1973), What are degrees f freedm? Amerkim Statistiian, 21, Sial Wrk Researh VOLUME 32, NUMBER 2 JUNE 18

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