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Revisiting a 90-Year-Old Debate: The Advantages of the Mean Deviation
Author(s): Stephen Gorard
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Source: British Journal of Educational Studies, Vol. 53, No. 4 (Dec., 2005), pp. 417-430
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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
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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
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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).
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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
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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).
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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
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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
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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
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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
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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)
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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
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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.
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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.
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Correspondence
ProfessorStephen Gorard
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ofYork
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YO10O5DD
E-mail:[email protected]
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