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Revisiting a 90-Year-Old Debate: The Advantages of the Mean Deviation Author(s): Stephen Gorard Reviewed work(s): Source: British Journal of Educational Studies, Vol. 53, No. 4 (Dec., 2005), pp. 417-430 Published by: Taylor & Francis, Ltd. on behalf of the Society for Educational Studies Stable URL: http://www.jstor.org/stable/3699276 . Accessed: 08/01/2013 12:35 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Taylor & Francis, Ltd. and Society for Educational Studies are collaborating with JSTOR to digitize, preserve and extend access to British Journal of Educational Studies. http://www.jstor.org This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions BRITISH JOURNAL OF EDUCATIONAL STUDIES, ISSN 0007-1005 VOL. 53, No. 4, DECEMBER2005, PP 417-430 REVISITING A 90-YEAR-OLDDEBATE: THE ADVANTAGESOF THE MEAN DEVIATION bySTEPHENGoRARD,University ofYork ABSTRACT: Thispaperdiscusses thereliance analysison ofnumerical relative the close theconcept the its standard and variance. deviation, of thattheoriginalreasonswhythestandarddeviation It suggests concept has permeated traditional statistics areno longerclearly valid,ifthey it is arguedhere,has many everwere.The absolutemeandeviation, as an It is moreefficient over standard the deviation. advantages the where situation a estimate in the of population parameter real-life normal or do notforma completely data containtinyerrors, perfect It is easiertouse,and moretolerant values,in distribution. ofextreme arenot themajority situations where parameters ofreal-life population and about It is learn easier new researchers to understand, required. for and also closely linkedto a number already techniques ofarithmetic to Wecouldcontinue usedin thesociology education and elsewhere. of much so because usethestandarddeviation do as we instead, presently, statistics is baseduponit (effect sizes,and the oftherestoftraditional convenience the we should F-test, ofthis forexample).However, weigh and a much solution some the simpler for ofcreating against possibility numeric morewidespread formany. formof analysis mean variation, variance,measuring Keywords: politicalarithmetic, statistics standarddeviation, socialconstruction deviation, of 1. INTRODUCTION Thispaper discussestherelianceofnumericalanalysison theconcept of the standarddeviation,and its close relativethe variance. Such a considerationsuggestsseveralpotentially importantpoints.First,it as a such basic acts reminder that even a concept as 'standard mathematical an deviation',with apparentlyimpeccable pedigree,is of sociallyconstructedand a product history(Porter,1986). Second, therefore,there can be equally plausible alternativesof which this paper outlinesone - the mean absolutedeviation.Third,we maybe able to create from this a simpler introductorykind of statisticsthat 417 Ltd.and SES 2005 Publishing ? Blackwell This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGESOF THE MEAN DEVIATION is perfectly useable formanyresearchpurposes,and thatwillbe far less intimidating fornew researchersto learn (Gorard,2003a). We could reassure these new researchers that, although traditional statisticaltheoryis oftenuseful,the mere act of using numbersin researchanalysesdoes not mean that theyhave to accept or even knowabout thatparticulartheory. 2. WHAT IS A STANDARD DEVIATION? The 'standarddeviation'is a measure of 'dispersion'or 'spread'. It is used as a commonsummaryof the rangeof scoresassociatedwith a measureof centraltendency- the mean-average.It is obtainedby summingthe squared values of the deviationof each observation fromthe mean, dividingby the totalnumberof observations,'and then takingthe positivesquare rootof the result.2For example,take the separatemeasurements: 13, 6, 12, 10, 11, 9, 10, 8, 12, 9. Their sum is 100, and theirmean is therefore10. Their deviations fromthe mean are: 3, -4, 2, 0, 1, -1, 0, -2, 2, -1. To obtain the standarddeviationwe firstsquare these deviationsto eliminatethe negativevalues,leading to: 9, 16, 4, 0, 1, 1, 0, 4, 4, 1. The sum of thesesquared deviationsis 40, and the averageof these (dividingby the numberof measurements)is 4. This is definedas the 'variance'of the originalnumbers,and the 'standarddeviation' is itspositivesquare root,or 2. Takingthe square root returnsus to a value of the same orderof magnitudeas our originalreadings.So a traditionalanalysiswouldshowthatthesetennumbershavea mean of 10 and a standarddeviationof 2. The lattergivesus an indication of how dispersedthe originalfiguresare, and so how representative the mean is. The main reason thatthe standarddeviation(SD) was created like this was because the squaring eliminatesall negative deviations,in a waythatmakesthe resulteasierto workwithalgebraicallythansimplyignoringthe signof negativedeviations(see below). 3. WHAT IS A MEAN DEVIATION? There are severalalternativesto the standarddeviation (SD) as a summary of dispersion. These include the range, the quartiles, and ? Blackwell Ltd.and SES 2005 Publishing 418 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGESOF THE MEAN DEVIATION theinter-quartile forSD as a measrange.The mostdirectalternative ure of dispersion,however,is the absolute mean deviation (MD). This is simplythe averageof the absolute differences betweeneach score and the overallmean. Take the separatemeasurements: 13, 6, 12, 10, 11, 9, 10, 8, 12, 9. Their sum is 100, and theirmean is therefore10. Their deviations fromthe mean are: 3, -4, 2, 0, 1, -1, 0, -2, 2, -1. To obtainthemean deviationwe firstignoretheminussignsin these deviationsto eliminatethe negativevalues,leading to: 3, 4, 2, 0, 1, 1, 0,2, 2, 1. These figuresnow representthe distancebetweeneach observation and the mean, regardlessof the directionof the difference.Their sum is 16, and the averageof these (dividingbythe numberofmeasurements)is 1.6. This is the mean deviation,and it is easierfornew researchersto understandthan SD, being simplythe averageof the deviations- the amount by which,on average,any figurediffers fromthe overallmean.3It has a clear meaning,whichthe standard deviationof 2 does not.4Why,then,is thestandarddeviationin common use and the mean deviationlargelyignored? 4. WHY DO WE USE THE STANDARD DEVIATION? As early as 1914, Eddington pointed out that 'in calculating the mean errorof a seriesof observationsit is preferableto use the simple mean residualirrespectiveof sign [i.e. MD] ratherthan the mean square residual [i.e. SD]' (Eddington,1914, p. 147). He had found, in practice,that the 'mean deviation' worked betterwith empiricaldata than SD, even though 'thisis contraryto the advice of mosttext-books; but it can be shownto be true' (p. 147). He also claimed thatthe majorityof astronomershad found subsequently the same. Fisher (1920) countered Eddington'sempiricalevidence witha mathematicalargumentthatSD was more efficientthan MD under idealcircumstances, and manycommentatorsnow accept thatFisher a provided complete defence of the use of SD (e.g. Aldrich,1997; MacKenzie,1981). Fisherhad proposedthatthequalityofanystatistic could be judged in termsof threecharacteristics. and The statistic, the population parameterthatit represents,should be 'consistent' (i.e. calculated in the same wayforboth sample and population). 419 ? BlackwellPublishingLtd. and SES 2005 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGESOF THE MEAN DEVIATION all of The statistic in the sense of summarising should be 'sufficient' the relevantinformationto be gleaned fromthe sample about the populationparameter.In addition,the statisticshould be 'efficient' in the sense of havingthe smallestprobable erroras an estimateof the populationparameter.Both SD and MD meet the firsttwo criteria (to the same extent). According to Fisher,it was in meeting the last criterionthatSD provessuperior.When drawingrepeated large samplesfroma normallydistributedpopulation,the standard deviation of theirindividualmean deviationsis 14 per cent higher than the standarddeviationof theirindividualstandarddeviations (Stigler,1973). Thus, the SD of such a sample is a more consistent estimateof the SD fora population,and is consideredbetterthanits as a wayofestimatingthe standarddeviationin plausiblealternatives a population using measurementsfroma sample (Hinton, 1995, p. 50). That is the main reason whySD has subsequentlybeen preferred,and whymuch of subsequentstatistical theoryis based on it. One furtherconcern has been that the absolute value symbols necessaryto create a formulafor the absolute mean deviationare quite difficultto manipulatealgebraically(http://infinity.sequoias. This cc.ca.us/faculty/woodbury/Stats/Tutorial/DispVarPop.htm). makesthe developmentof sophisticatedformsof analysismore complicatedthanwhen usingthe standarddeviation(http://mathworld. wolfram.com/MeanDeviation. html). So we now have a complex formofstatistics based on SD (and itssquare - thevariance)because SD is more efficientthan MD under ideal circumstances,and because it is easier to manipulatealgebraically.Of course, SD has now become a tradition,and much of the restof the theoryof stathe calcutisticalanalysisrestson it (the definitionof distributions, lation of effectsizes, analysesof variance,least squares regression, and so on). For example,SD is both based on and partof the defiThis has nitionof thewidespreadGaussianor 'normal' distribution. the benefitthatit enables commentatorsto statequite preciselythe proportionof the distribution lyingwithineach standarddeviation from the mean. Therefore,much of the expertiseof statisticians restson the basis of usingthe standarddeviation,and thisexpertise is whattheypass on to novices. 5. WHY MIGHT WE USE THE MEAN DEVIATION? On the otherhand, it is possibleto argue thatthe mean deviationis preferableand that,since Fisher,we have takena wrongturnin our analytichistory.The mean deviationis actuallymore efficientthan the standard deviation in the realistic situation where some of the 420 ? BlackwellPublishingLtd. and SES 2005 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGESOF THE MEAN DEVIATION otherthan measurements are in error,more efficient fordistributions perfectnormal,closelyrelatedto a numberofotherusefulanalytical techniques,and easierto understand.I discusseach of thesein turn. ErrorPropagation The standarddeviation,by squaringthe values concerned,givesus a distortedviewof the amount of dispersionin our figures.The act of squaringmakes each unit of distancefromthe mean exponentially(ratherthan additively)greater,and the act of square-rooting the sum of squares does not completelyeliminatethisbias. That is why,in the exampleabove,the standarddeviation(2) is greaterthan the mean deviation (1.6), as SD emphasisesthe largerdeviations. Figure 1 shows a scatterplotof the matchingmean and standard deviationsfor 255 sets of random numbers.Two thingsare noteSD is alwaysgreaterthanMD, but thereis morethanone possworthy. ible SD foranyMD value,and viceversa.Therefore,the twostatistics are not measuringpreciselythe same thing.Their Pearson correlation overanylarge numberof trials(such as the 255 picturedhere) isjust under 0.95, traditionally meaningthataround 90 per cent of theirvariationis common. If thisis sufficient to claim thattheyare measuringthe same thing,then eithercould be used, but in the absence of otherevidence the mean deviationshould be preferred because it is simpler,and easier fornewcomersto understand(see above). If,on the otherhand,theyare not measuringthesame thing then the mostimportantquestionis not whichis the more reliable but whichis measuringwhatwe actuallywantto measure? The apparentsuperiority of SD is not as clearlysettledas is usually in texts (see above). For example,the subsequentworkof portrayed and otherssuggeststhatEddingtonhad been right, Tukey (1960) and Fisherunrealisticin at least one respect.Fisher's calculations of the relativeefficiencyof SD and MD depend on there being no errors at all in the observations.But for normal distributions withsmallcontaminationsin the data, 'the relativeadvantageof the sample standarddeviationover the mean deviationwhichholds in the uncontaminatedsituationis dramatically reversed'(Barnettand An as 0.2 per cent (i.e. error element as small Lewis,1978, p. 159). two error points in 1,000 observations)completelyreversesthe advantageofSD overMD (Huber,1981). So MD is actuallymoreefficientin all life-likesituationswheresmall errorswilloccur in observation and measurement(being over twiceas efficientas SD when the errorelementis 5 per cent,forexample). 'In practicewe should certainlyprefer d, [i.e. MD] to s, [i.e. SD]' (Huber, 1981, p. 3). 421 ? BlackwellPublishingLtd. and SES 2005 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGESOF THE MEAN DEVIATION 45 40 :. 35- " 30 S15 10 5- 0 0 5 10 15 20 25 MeanDeviation 30 35 40 Figure1. Comparisonof mean and standarddeviationforsetsof randomnumbers Notethisexamplewasgenerated over255 trialsusingsetsof10 randomnumbers between0 and 100.The scatter and theoverallcurvilinear effect relationship, commontoall suchexamples, in are due to thesumsofsquaresinvolved SD. computing The assumptions underlying statistical inference are only mathematicalconveniences,and are usuallydefendedin practiceby a furtherassumption,presentedwithoutan explicitargument,that minorinitialerrorsshould lead to onlyminorerrorsin conclusions. This is clearlynot the case (see Gorard,2003b). 'Some of the most common statisticalprocedures (in particularthose optimized for an underlyingnormaldistribution)are excessively sensitiveto seeminglyminor deviationsfromthe assumptions'(Huber, 1981, p. 1). The differencebetweenFisherand Eddingtonis relatedto the differencebetween mathematicsand science. The firstis concerned withthe Platonicworldof perfectdistributions and ideal measurements.Perhapsagriculture, whereFisherworkedand wherevegetativereproductionof cases is possible,is one of the fieldsthatmost closelyapproximatesthisworld.The second is concernedwiththe Aristotelianworldof empiricalresearch.Astronomy, whereEddington workedand where the potentialerrorsin calculated distances are substantial,highlightsthe importanceof trackingthe propagation of measurementerrors.The imperfectmeasurementsthatwe use in social research are more like those fromthe largelynonexperimentalastronomythan thosefromagriculture. Anotherimportant,but too oftenoverlooked,assumptionunderof SD is thatit involvesworkingwith lyingthe purportedsuperiority samplesselectedrandomlyfroma fixedpopulation (because thisis 422 ? BlackwellPublishingLtd. and SES 2005 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGESOF THE MEAN DEVIATION how itsefficiency is calculated). However,thereis a range of analytical situationswherethisis not so, such as whenworkingwithpopulationfigures,or witha non-probability sample,or even a probability samplewithconsiderablenon-response.In all of thesesituationsit is perfectly properto calculatethevariationin thefiguresinvolved,but withoutattemptingto estimatea populationSD. Therefore,in what are perhaps the majorityof situationsfaced by practisingsocial is scientists,the supposed advantage of SD in termsof efficiency irrelevant. Some statisticians mayattemptto evade thisconclusionbyarguing thatthe data theywork,even where these are forpopulations,are actuallytakenfroman infinitely large super-population.For example, Camilli(1996) reportsa debate betweenseveralprominentcommentatorsin which it is argued that statisticiansare not really interestedin generalisingfroma sample to a specifiedpopulation but to an idealised super-populationspanningspace and time.It is claimed that 'social statisticiansare prettymuch forced to adopt the notion of a "superpopulation"when attemptingto generalise the resultsof an analysis'(p. 7). But it is clear thatif such a superpopulation is involvedthen itsvarianceis infinite(Fama, 1963), in whichcase thepurportedgreaterefficiency ofSD, based on estimating the populationvariance,is impossibleto establish.An analystcannot, of the therefore,use a super-populationand argue the efficiency standarddeviationat the same time. Distribution-free In addition to an unrealisticassumptionabout error-free measurements,Fisher'slogic also depends upon an ideal normaldistribution for the data. What happens if the data are not perfectlynormally or not normallydistributedat all? distributed, Fisherhimselfpointedout thatMD is betterforuse withdistributions other than the normal/Gaussiandistribution(Stigler,1973). This can be illustratedforuniformdistributionsthroughthe use of repeated simulations.However,we firsthave to consider what appears to be a tautologyin claims thatthe standard deviation of a sample is a more stable estimate of the standard deviation of the population than the mean deviationis (e.g. Hinton, 1995). We should notbe comparingSD fora sampleversusSD fora population withMD fora sampleversusSD fora population.MD fora sample should be comparedto the MD fora population,and Figure1 shows whythis is necessary- each value for MD can be associatedwith more than one SD and vice versa, giving what is merely an illusion 423 ? BlackwellPublishingLtd. and SES 2005 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGESOF THE MEAN DEVIATION of unreliability forMD when compared withSD. The exact inverse conclusion to thatdrawnby Fishercould be argued by comparing the SD fora sample and the MD of a population,withthe MD fora sample and the MD of a population.In thiscase, MD would be the betterestimateof the population parameter,but the comparison would be as unfairas the one Fishermade. Repeated simulationsshow thatthe efficiencyof MD is at least as good as SD fornon-normaldistributions. For example,I created 1,000 samples (with replacement) of 10 random numbers each between0 and 19, fromthepopulationof 20 integersbetween0 and 19. The mean of the populationis 9.5, the mean deviationis 5, and the standarddeviationis 5.77. The 1,000sampleSDs variedfrom2.72 to 7.07, and the sample MDs variedfrom2.30 to 6.48. The standard deviationof the 1,000 estimatedstandarddeviationsaround their truemean of 5.77 wasjust over 1.025.6The standarddeviationof the 1,000 estimatedmean deviationsaround theirtrue mean of 5 was just under 1.020. These values and theirdirectionof differenceare stableover repeated simulationswithfurthersetsof 1,000 relatively samples.Readers maylike to trythisforthemselves.This is an illustrationthat,foruniformdistributions of the kind involvingrandom of the mean deviationis at least as good as numbers,the efficiency thatof the standarddeviation. The normaldistribution, like the notionofmeasurementwithout willbe dealing error,is a mathematicalartifice.In practice,scientists with observationsthat merelyresemble or approximatesuch an ideal. But strictnormality was a basic assumptionofFisher'sproofof the efficiency of SD. What Eddington had realised was that small deviationsfromnormality, such as alwaysoccur in practice,have a considerableimpacton idealstatistical procedures (Hampel, 1997). In general,our observeddistributions havtend to be longer-tailed, ing more extreme scores, than would be expected under ideal assumptions.Because we thensquare the deviationsfromaverageto tend to produce SD, but not MD, such longer-taileddistributions 'explode' the variationin SD (Huber, 1981). The act of squaring makes each unit of distancefromthe mean exponentially(rather than additively)greater,and the act of square-rootingthe sum of squares does not completelyeliminate this bias. In practice, of course, this fact is often obscured by the widespread deletion of 'outliers'(Barnettand Lewis,1978). In fact,our use of SD rather than MD formspart of the pressure on analyststo ignore any extremevalues. The distortioncaused by squaring deviations is part of a culture in which advice is routinelygiven to students to remove or ignore valid 424 ? BlackwellPublishingLtd. and SES 2005 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGESOF THE MEAN DEVIATION measurementswithlarge deviationsbecause theseundulyinfluence the finalresults.This is done regardlessof theirimportanceas data, and it means thatwe no longer allow our priorassumptionsabout distributionsto be disturbedmerelyby the fact that theyare not matchedbythe evidence.Good science should treasureresultsthat show an interestinggulf between theoreticalanalysisand actual observations,but we have a long and ignoble historyof too often ignoringresultsthatthreatenour fundamentaltenets(Moss, 2001). Extremescoresare importantoccurrencesin a varietyofnaturaland social phenomena,includingcitygrowth,incomedistribution, earthquakes, traffic jams, solar flares,and avalanches.We cannot simply dismissthemas exogenous to our models. Ifwe take themseriously, as a fewcommentatorshave,thenwe findthatmanyapproximately normaldistributions show consistentdeparturesfromnormality. In particular,theyhave more of the extreme scores than expected. Statisticaltechniquesbased on the standarddeviationgivemisleadsuch as ... ing answersin these cases,and so 'conceptsofvariability, the absolute mean deviation ... are more appropriate measures of forthese distributions'(Fama, 1963, p. 491). variability RelatedTechniques Anotheradvantage in using MD lies in its linksand similaritiesto a range of other simple analyticaltechniques,a few of which are describedhere. In 1997, Gorard proposed the use of the 'segregation index' (S) forsummarisingthe unevennessin the distribution of individualsbetweenorganisationalunits,such as the clusteringof childrenfromfamiliesin povertyin specificschools (Gorard and Fitz,1997).7 The index is relatedto severalof the more established indices such as the dissimilarity index and the isolationindex. However,S has two major advantagesover both of these. It is strongly composition-invariant, meaningthatitis affectedonlybyunevenness of distribution and not at all byscaled changesin the magnitudeof the figuresinvolved(Gorard and Taylor,2002). Perhapseven more S has an easyto comprehendmeaning.It representsthe importantly, of proportion the disadvantagedgroup (childrenin poverty)who would have to exchange organisationalunits (schools) forthereto be a completelyeven distributionof disadvantagedindividuals. Other indices,especiallythose like the Gini coefficientthatinvolve the squaringof deviations,lead to no such easilyinterpretedvalue. The similarity betweenS and MD is striking.MD is whatyou would devise in adapting S to work with real numbers rather than the frequencies of categories. MD is also, like S, more tolerant of problems 425 ? BlackwellPublishingLtd. and SES 2005 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGESOF THE MEAN DEVIATION withinthe data, and has an easier to understandmeaning than its potentialrivals. In social research there is a need to ensure,when we examine thatthefigureswe use overtime,place or othercategory, differences are proportionate(Gorard,1999). Otherwisemisleadingconclusions can be drawn.One easywayof doing thisis to look at differences For example, betweenfiguresin proportionto thefiguresthemselves. when comparingthe numberof boysand girlswho obtain a particular examinationgrade,we can subtractthe scoreforboysfromthat of girlsand then divideby the overallscore (Gorard etal., 2001). If we call the score for boys b and forgirlsg, then the 'achievement gap' can be definedas (g- b)/(g+ b). This is verycloselyrelatedto a range of otherscores and indices,includingthe segregationindex (see above and Tayloret al., 2000). However,such approaches give resultsthatare difficult to interpretunless theyare used withratio values havingan absolute zero, such as examinationscores.When used witha value, such as the FinancialTimes (FT) index of share prices,whichdoes not have a clear zero, it is betterto considerdifferencesin termsof theirusual range of variation.If we divide the differencebetweenany two figuresby the past variationin the figures thenwe automatically deal withthe issueof proportionatescale as well. This approach, of creating'effect'sizes,is growingin popularity as a way of assessing the substantiveimportance of differences betweenscores,as opposed to assessingthe less useful'significance' of differences(Gorard,2006). The standardmethodis to dividethe differencebetween two means by theirstandarddeviation(s). Or, put anotherway,beforesubtractionthe twoscoresare each standardised throughdivisionby their standarddeviation(s).8We could, instead,use the mean deviation(s)as the denominator.Imagine,for example, that one of the means to be compared is based on two observations(x,y).Their sum is (x + y), theirmean is (x + y)/2, and their mean deviation is (I x- (x+y)/2 1+1y- (x+y)/2 1)/2. The wouldbe: standardisedmean,or mean dividedbythemean deviation, (x + y)/2 (I x- (x+y)/2 I+ Iy- (x+y)/2 I)/2 Since boththe numeratorand denominatorare dividedbytwothese can be cancelled,leading to : (x + y) (Ix- (x+y)/2 I+I y- (x+y)/2 I) 426 ? Blackwell Publishing Ltd. and SES 2005 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGESOF THE MEAN DEVIATION Ifbothx and yare thesame thenthereis no variation,and themean deviationwillbe zero. If x is greaterthan the mean then (x + y)/2 is subtractedfromx but y is subtractedfrom(x + y)/2 in the denominator.This leads to the result: (x+ y) (x- y) If x is smallerthan the mean then x is subtractedfrom(x + y)/2 but (x+ y)/2issubtracted fromyin thedenominator. Thisleadstotheresult: (x+ y) (y- x) For example,if the twovalues involvedare actually13 and 27, then theirstandardised scoreis 20/7.9 Therefore, forthetwovalueexample, an effectsize based on mean deviationsis the difference betweenthe reciprocalsof two 'achievementgaps' (see above). Similarities ofunifying such as thesemean thatthereis a possibility traditionalstatisticalapproaches,based on probabilitytheory,with simplerarithmeticapproaches,of the kinddescribedin thissection. This,in turn,wouldallownewresearchersto learnsimplertechniques forroutineuse withnumbersthatmayalso permitsimplecombinationwithotherformsof data (GorardwithTaylor,2004). Simplicity In an earlierera of computationit seemed easier to findthe square rootof one figureratherthantakethe absolutevaluesfora seriesof figures.This is no longer so, because the calculationsare done by computer.The standard deviation now has several potential disand thekeyproblem advantagescomparedto itsplausiblealternatives, it has fornewresearchersis thatit has no obviousintuitivemeaning. The act ofsquaringbeforesummingand thentakingthesquare root after dividing means that the resultingfigure appears strange. Indeed, it is strange,and its importanceforsubsequentnumerical analysisusuallyhas to be takenon trust.Studentsare simplytaught that theyshould use it, but in social science most opt not to use numeric analysisat all anyway(Murtonen and Lehtinen, 2003). Giventhatboth SD and MD do the samejob, MD's relativesimplicity of meaning is perhaps the most importantreason for henceforth using and teachingthe mean deviationbefore,or even insteadof, the more complex and less meaningful standard deviation. Most researchers wishing to provide a summary statisticof the dispersion 427 ? BlackwellPublishingLtd. and SES 2005 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGES OF THE MEANDEVIATION in their findingsgenerallydo not want to manipulate anything, whetheralgebraicallyor otherwise.For these,and formostconsumers of researchevidence,the algebraicadvantagesof SD are irrelevant,and using the mean deviationis more 'democratic'. 6. CONCLUSION Even one of the mostelementarythingstaughton a statistics course, the standarddeviation,is more complexthanit need be, and is considered here as an example of how conveniencefor mathematical manipulationoftenover-ridespragmatismin researchmethods.In thoseraresituationsin whichwe obtainfullresponsefroma random samplewithno measurementerror,and we wishto estimatethe dispersionin a perfectGaussianpopulationfromthe dispersionin our sample, then the standarddeviationhas been shown to be a more stable indicatorof its equivalentin the population than the mean deviationhas. Note thatwe can only calculate thisvia simulation, since in real-liferesearchwe would not knowthe actual population figure,else we would not be tryingto estimateit via a sample. In essence, the claim made for the standarddeviationis thatwe can computea number (SD) fromour observationsthathas a relatively consistentrelationshipwitha number computed in the same way fromthe populationfigures.This claim,in itself,is ofno greatvalue. Reliabilityalone does not make thatnumberof any valid use. For whateverfigureswere example,if the computationled to a constant used thentherewouldbe a perfectly consistentrelationshipbetween the parametersfor the sample and population. But to what end? Surelythe keyissue is not how stable the statisticis but whetherit we shouldnot use an inapencapsulateswhatwe wantit to. Similarly, propriatestatisticsimplybecause it makescomplex algebraeasier. Of course, much of the restof traditionalstatisticsis now based on the standarddeviation,but it is importantto realisethatit need not be. In fact,we seem to have placed our 'reliance in practiceon isolated pieces of mathematicaltheoryproved under unwarranted assumptions,[rather] than on empirical facts and observations' (Hampel, 1997, p. 9). One resulthas been the creationsince 1920 of methodsfordescriptivestatistics thatare more complex and less democraticthan theyneed be. The lack of quantitativeworkand skillin social science is usuallyportrayedvia a deficitmodel, and more researchersare exhortedto enhance theircapacityto conduct suchwork.One of thekeybarriers,however,could be deficitscaused by the unnecessarycomplexityof the methods themselvesrather than theirpotential users. The standard deviationis one such example. 428 ? BlackwellPublishingLtd. and SES 2005 This content downloaded on Tue, 8 Jan 2013 12:35:46 PM All use subject to JSTOR Terms and Conditions THE ADVANTAGES OF THE MEANDEVIATION 7. NOTES Sometimesthe sum of the squared deviationsis divided by one less than the numberofobservations. Thisarbitrary is intendedto makeallowancefor adjustment anysamplingerror. 2 SD = 4(X(IM - Oil)2/N), where M is the mean, Oi is observationi, and N is the numberof observations. MD = (IM - Oil)/N, where M is the mean, Oi is observationi, and N is the numberof observations. themean deviationhas sometimesbeen adjustedbytheuse ofBessel's Traditionally, formula,such that MD is multiplied by the square root of Pi/2 (MacKenzie, 1981). However,thisloses the advantageof its easyinterpretation. 5 SD is used here and subsequentlyfor comparison,because that is what Fisher used. 6 Of course,it is essentialat thispoint to calculate the variationin relationto the value we are tryingto estimate(the population figure)and not merelycalculate the internalvariability of the estimates. The segregationindex is calculated as 0.5 * I (IAi/A- Ti/TI), whereAi is the numberof disadvantagedindividualsin unit i, Ti is the numberof all individuals in unit i, A is the number of all disadvantagedindividualsin all units,and T is the numberof all individualsin all units. 8 There is a dispute over preciselywhichversionof the standarddeviationis used here. In theoryit is SD forthe population,but we willnot usuallyhave population figures,and ifwe did thenwe would not need thisanalysis.In practice,it is SD foreitheror both of the groups in the comparison. 9 Of course,it does not matterwhichvalue is x and whichis y.If we do the calculationwith27 and 13 insteadthen the resultis the same. 1 8. REFERENCES ALDRICH, J. (1997) R. A. Fisher and the makingof maximumlikelihood 191212 (3), 162-176. 1922, Statistical Science, Data (Chichester,John BARNETT, V. and LEWIS, T. (1978) Outliersin Statistical Wileyand Sons). CAMILLI, G. 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