Guinness, Gosset, Fisher, and Small Samples

Guinness, Gosset, Fisher, and Small Samples
Author(s): Joan Fisher Box
Source: Statistical Science, Vol. 2, No. 1 (Feb., 1987), pp. 45-52
Published by: Institute of Mathematical Statistics
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StatisticalScience
1987,Vol. 2, No. 1, 45-52
Guinness, Gosset, Fisher,and
Small Samples
Joan Fisher Box
Abstract.The environment
in whichW. S. Gosset(Student)workedas a
brewerat Guinness'Breweryat the turnof the centuryis describedfully
enoughto showhow it forcedhimn
to confront
problemsof smallsample
statistics,usingthe techniqueshe pickedup fromKarl Pearson. R. A.
Fisher'sinterestin humangeneticspromptedbiometrical
applicationsof
his mathematical
trainingevenas an undergraduate.
As soonas he consideredStudent'swork,he perceivedits importance
and beganto extendits
applications.Consequently,
whenhe startedworkat RothamstedExperimentalStation in 1919, he was readyto respondto the experimental
problemsby developingstatisticaltheoryalongwithappropriate
methods
ofexperimental
analysisand design.
Keywordsandphrases:Gosset,Student'st,Fisher,smallsamples,analysis
ofvariance,correlation.
The twomenI wanttowriteaboutareW. S. Gosset,
betterknownas Student,who inventedStudent'st
test,and R. A. Fisher,whotookup Gosset'sworkand
extendedand generalized
it so greatly.
I am interested
in the circumstances
thatforcedthemto breaknew
groundwheretheydid.Gossetwas a brewer,
Fishera
mathematics
studentwhen he started.Why should
they have inventedstatisticalmethodsfor experimenters?How did theypinpointthe problemarea
wherehumanthought
was dammedbackand baffled?
Sometimesthe mostimportant
stepin creativework
is simplyto ask the rightquestion.The question
nobodythinksto ask-that mayseemtoo trivial,or
too difficult-butwhich,once asked, must be answered.Gossetaskedthefirstquestionand foundthe
answer,despitehis lack ofmathematics.
Fishermade
it elegantand went on fromthere.Why did they
persistwhileexpertsin the fieldignoredor belittled
theirwork?Mathematicians
onlyslowlyrealizedthat
an intellectualrevolutionwas takingplace through
the new inductiveuses of mathematics.
Even now,
peopleoftenthinkstatisticscan be reducedto deduction-that once you have learnedthe theoryyou
understandstatistics.Gosset and Fisher believed
otherwise.
ofinducTheybelievedthatunderstanding
tive reasoningis acquiredthroughlearningto deal
Joan Fisher Box is theauthorof "R. A. Fisher,theLife
of a Scientist" (Wiley,New York, 1978) and lives at
230 North Hillside Terrace, Madison, Wisconsin
53705.
45
knowingthat practical
with real data inductively,
actionwillbe takenon thebasis ofyourconclusions.
Knowingthetheoryis nottheanswer,butit can help
youfindtheanswerto statisticalproblems.
One ofthe firstthingsyou learnin statisticsis to
betweenthe trueparametervalue of the
distinguish
deviastandarddeviationoaand the sample-standard
did
statisticians
tions. But at theturnofthecentury
not.Theycalledbotha and s thestandarddeviation.
Theyalwaysused such largesamplesthattheirestithe parametervalue,so
matereallydid approximate
to theirresults.But
it did not makemuchdifference
work.
theirmethodswouldnot do forexperimental
You cannotgetsamplesofthousandsofexperimental
largeenoughwouldspreadso
points.Anyexperiment
farin timeor space thatextraneousvariationwould
drownout any effectsof treatmentyou mightbe
lookingfor.
My storybeginswithGosset,forhe came firstin
time.Born in 1876,the oldestof fivechildrenof a
Colonelin the Royal Engineers,Gossetenteredthe
Royal MilitaryAcademy,Woolwich,to become a
RoyalEngineerhimselfbeforebeingrejectedon acwith
countof his pooreyesight.He was verybright,
highideals,and an impishsense of humor.A most
appealing character-quiet, unaffectedlyfriendly,
helpful,patient,loyal-everybodylikedand trusted
worldof statistics,he
him.In the veryquarrelsome
termswitheveryone.He
managedto be on friendly
Afterschoolat
was neveremployedas a statistician.
Winchester
and New College,Oxford,wherehe won
in 1899,he tooka job
a firstclass degreein chemistry
J. F. BOX
46
W S. Gosset,1899
as brewerat Guinness'Breweryin Dublin. He was
Head Brewerwhenhe died in 1937.It was the environmentat Guinness'that made him a statistician.
That is whatwe lookat next.
One hundredyearsago, in 1886,therewas eager
buyingon the London Stock ExchangewhenGuinness' was incorporated
as ArthurGuinnessSon and
Company,Ltd. and sold to the publicforsix million
pounds.Until then,it had been a familyfirm,run
almostexclusively
byGuinnessand one otherfamily.
The firstArthurGuinnessand son had startedbrewofDublinin
ingat St. James'Gate on thesouthwest
1759.Fortyyearslatertheyweregetting
a foothold
in
the London market,sellingthe brownbeer called
withLondonpor"porter"because of its popularity
ters.Later,Guinnessoffered
whattheycalled"a stouterporter"or stout.As this gainedin popularity,
it
came to be knownsimplyas Guinness.All through
thecenturyGuinnesskeptextendingits marketand
to dealwithgrowand extending
thebrewery
building
itwas thelargest
ingtrade.By theendofthecentury,
in theworld,producing
over1.5 millionbulk
brewery
barrelsa yearfordistribution
in Englandand Ireland
and forexportto theendsoftheearth.
Irelandis a smallislandand,all through
thebarleygrowingdistrictsof the south and east, Guinness
commissioned
maltsters
to buylocal barleyand malt
it exclusively
forthe St. James'Gate brewery.
Their
commission
maltsterswereprosperouscitizensof influencein theirownareas.
Afterincorporation,
Cecil Guinness,head of the
firm,
becameChairmanoftheBoardand Christopher
DiggesLa Touche (one oftwobrewerstakenin from
outsidethe familyin the 1870s)was madeManaging
Director.AlthoughGuinnesshimselfwas apparently
removedfromthe runningofthe brewery,
it was he,
withLa Touche'sadvice,whomadeall theimportant
decisions.In particular,Guinness,by thenthe first
Lord Iveagh,and La Touchetogetherbroughtabout
a quiet revolutionaroundthe turnof the century.
Theyhad decidedto makethebrewingscientific.
Up
to thattimethebrewers
had learnedas apprentices
to
knoweveryprocessand to followmeticulously
the
traditional
practices-all theblackmagicofbrewing,
as it were.Now,Guinnessespouseda new approach
and investeda lotin it.
First,theybegan to hirea seriesof the brightest
youngmentheycouldfind-all ofthemnewlygraduated fromOxfordor CambridgeUniversity
withfirst
classdegreesinchemistry-and
toappointthembrewers.Otherbreweries
employedtheoccasionalchemist
in researchlaboratories,but Guinnessbroughtin
in positions
thesechemistsas theirtop management,
previously
occupiedby the Guinnessfamily.As soon
as each new brewercompleted2 yearsas a junior,
learninghis dutiesunderseniorbrewersin everydehe was put in chargeof a sectionof the
partment,
breweryand researchwork.Guinnessbrewerswere
awfullygrand.Stella Cunliffe(1976) recallshow,on
arrivalat St. James'Gateas a newly-hired
statistician
in 1947,she was instructed
howto dressand howto
behavein hernewwork.If she was luckyenoughto
meeta brewerin the corridor,
she was told,she was
on no accountto recognizehimbutto lowerhereyes
untilhe had passed.
So, thenewbrewerswereappointed:1893,Thomas
Case; 1895,Alan McMullen;1897,ArthurJackson;
1899,E. G. Peake; October1899,W. S. Gosset;January 1900,Geoffrey
Phillpotts;and so on. Untilthey
at theGuinnesshousefor
married,
theylivedtogether
unmarried
brewersat St. James'Gate. At work,they
ate togetherin the brewers'diningroom.Offduty,
theyseem to have been veryactive outdoors;they
skied,fished,sailed,golfed,cycled,and walkedin the
WicklowHills, and visitedand read and talkedtogether.In someways,theirlifewas likean extension
ofcollege.
As theygot involvedin research,of course,they
needednewfacilities.
Case and McMullenstartedwith
GUINNESS, GOSSET, FISHER, AND SMALL SAMPLES
chemicalanalysesto tryto identify
and quantify
what
itwasthatgavehopsandbarleytheirbrewing
quality.
Untilthenthecriteriawerequalitative,
e.g.,the"rub"
of hops or the "texture"of barley,whichmightbe
"milky"or "steely."So, in additionto the brewers'
and chemists'laboratories,
a newGuinnessResearch
Laboratory
was openedin 1900,headedby the most
chemistalive,HoraceBrown.
distinguished
brewing
The next questionsconcernedthe raw materials,
barleyand hops. Wherecould theybe got best and
cheapest?What varietiesof barleyand hops,what
cultivation
and manuring,
whatconditionsof drying
and storinggavethebestmaltingvalueofbarleyand
valueofmaltand hops?So, in 1899,
thebestbrewing
Guinnessstarteda program
ofbarleyplotexperiments
throughHenryBennett,theirCommissionMaltster
at Ballinacurra,CountyCork.He selectedthe farms
and supervisedthe experiments
to comparedifferent
varietiesand cultivations
and fertilizers.
The variety
trialsseemedmostpromising
and theseexperiments
withcooperation
wereextended,
fromothermaltsters
and farmersin everybarley-growing
districtin Ireland.The newMinistry
in Irelandwas
ofAgriculture
officially
in chargeand in 1904 appointeda barley
expert,HerbertHunter,who made his headquarters
withHenryBennett.Togethertheytouredthebarley
severaltimesduringthegrowing
season.
experiments
The GrandTour was shortlybeforeharvest-Grand
becausebrewersfromDublinas wellas Bennettand
Hunterattendedthistourwhichlasted2 weeks;they
spentlongdaysin the barleyfieldswiththe farmers
in styleby
and local maltsters(and wereentertained
whiletheysaw forthemthe maltstersafterward),
selvesthe conditionofexperimental
plotsand ofthe
barleythat would soon be harvestedand sent to
St. James'Gate.
In 1901,an experimental
malthousewas built at
St. James'Gate, of a suitablesize formaltingeach
batchof barleyseparately.In 1903,an
experimental
was openedthere.Now,barley
experimental
brewery
grownon experimental
plotscouldbe followedfrom
seedcornto harvest,throughmaltingand brewingto
seriesof
thefinalbeerin a uniqueand comprehensive
observations
(McMullen,1908).
At the same time,enquirieswere goingforward
varietiesand cultivationof hops,and how
regarding
bestto dryand storethem.Manyyearslater,Rupert
Guinness,thesecondLordIveagh,recalled1902when
he had goneon a cycletourofKenthopgardenswith
Case and McMullento see whattheycoulddiscover
about hop culture.They had droppedin at the new
themselves
WyeAgricultural
Collegeand introduced
to theDirector,A. D. Hall. Followingup thisvisitin
1904,thefirstLordIveaghand La TouchevisitedHall
and arrangedto renttwo hop farmsnearby,putting
47
R. A. Fisher,1912
one of Hall's graduatesin as manager.Later,these
addedto,untilGuinfarmswereboughtand gradually
ness was one of the largesthop growersin England.
But it all startedfortheyoungresearchbrewers
to do
researchon whathops to grow,and how,to getthe
bestflavorand the longestlifefortheirbeer,and to
findouthowmuchit cost(Brown,1980).
The lifeofthebeerwasimportant
becauseGuinness
is a naturally
conditioned
beer-it has no additivesor
preservatives
nor,ofcourse,is it pasteurized-andit
has to remainpotablewhileit is exportedto Africaor
the Far East, or storedin the barrelat varyingtembeforereachingtheconsumer.
peratures
Now,can you imaginehalfa dozen energeticand
brightresearchers
givena freehand to explorethe
wholesubjectof brewingfroma conditionof almost
totalignorance?And giventhe experimental
laboratories,barley fields,hop gardens,malthouse,and
fortheirexperiments.
What happensnext?
brewery
Well,ofcourse,theystudiedavailableliterature,
and
foundsomeofituseful.Case readabouta newanalysis
of hops forsoftresincontentand triedit. He found
thequantityofsoftresinin thehopsparalleledGuinness' qualitativeassessmentof hop "condition."
The
48
J. F. BOX
conditionof hops storedat ambienttemperatures
deteriorated
and theirsoftresincontentfellas soft
resinwas convertedto hard resin.If the hops were
cooledin storage,bothhop conditionand softresin
contentfellless. Thus, softresinslookedlikea good
predictorof hop condition,and the brewersbegan
routineanalysisof hop softresincontent.Guinness
foundthis measureveryvaluableforassessinghops
beforepurchase.In particular,Americanhops were
cheaperthanEuropeanand it turnedouttheyhad a
naturally
highsoftresincontent.
in 1902,studieswerepublishedin which
Similarly,
maltingqualityof barleywas shownto dependon
beganroutineanalyses
nitrogen
content.The brewers
ofbarleyfornitrogen,
in additionto theirrecordsof
and theold qualsize ofbarleycorns,
yield,moisture,
and texture.
itativeassessmentsformellowness
and measureReadingled to analysis,experiments,
ments.Theybeganto accumulatedata and, at once,
becausetheirmeasurements
theyranintodifficulties
varied.The effectstheywere lookingforwere not
as theyhad expected,
usuallyclearcutor consistent,
and theyhad no wayofjudgingwhetherthe differoftreatment
oraccident.
encestheyfoundwereeffects
Two difficulties
wereconfounded:
the variationwas
werefew.As a result,for
highand the observations
his hop-cooling
experiexample,whenCase reported
mentsin 1898,he pointedout"theweaklinkbetween
or analysisand thebrewing
examination
value,"that
and effect,and said "mostof
is, betweentreatment
the results . . . require substantiationby irreproacha-
ble figures,"
whichhis obviouslywerenot. He convalue in termsof lifebecluded,"The comparative
tweenthecooledand uncooledhopsmaybe regarded
as of the orderof about 10%" (Brown,
provisionally
1980). He was not stickingout his neck,althougha
is quitea lot.
10% difference
on starting
theirexperiments,
found
Otherbrewers,
thesamedifficulty.
Theyhadnowayoftakingaccount
theirdata.Aftertrying
ofthevariationin interpreting
to correlatefertilizertreatmentswith barleyplot
thateach
yields,E. G. Peake endedup recommending
farmer
shouldfindoutthemanurialneedsofhis own
land, because "the resultswere most irregularand
varied with everyfarm."Peake tried to correlate
springrainfallwithyieldofbarleyfromRothamsted
data thatA. D. Hall also had analyzed.By takinga
different
slightly
periodthanHall had fortherainfall,
in theopposite
Peake managedto discerncorrelations
direction
to thosefoundby Hall, and had to giveup,
concludingthat "the whole subjectof the effectof
weather ... deserves furthercareful investigation"
(McMullen,1908).
In the case of the barleyplots,the variationwas
excessive.First,it was obviousthatalien barleywas
intermixed
in theseed,forperhaps10% ofplantsin a
plotofa short,late-maturing
varietywouldgrowtall
and matureearly,and so on. Next,plantsevenfrom
seed genuinelyof one varietyshowedmanysmaller
variations.Old Irishwas the mostheterogeneous
variety.Unliketheotherswhichhad beenselectedoriginallyfroma singleplantorsmallstandofplants,Old
Irishwas an indigenousIrishvarietythathad never
beenselected.It retainedthevariability
accumulated
throughout
its ancestry,
and madean extremely
unevenand unsatisfactory
crop.
Thiscauseofvariation,
at least,couldbe eliminated.
Barleyis usuallyself-fertilized.
In 1904, therefore,
BennettandHunterbeganpropagation
ofeachvariety
of interestin purelinesfromsinglegrainsofbarley.
In 1907theyhad enoughpure-lineseedto sowall the
experimental
plots,and on the GrandTour everyone
marvelled
at thewonderfully
evengrowth
ofthefields.
But geneticvariability
was a specialcase. One could
noteliminatevariationsofrainfall,
birddamage,soil
chemistry,
temperature
(whichaffected
brewingbeer
as wellas growing
crops),Qranyoftheunrecognized
variablesaffecting
theirdata. Theyneededsomeway
to decidewhichdifferences
to ignoreand whichto
takeseriously.
The youngresearchbrewers
workedwelltogethersomewereveryclosefriends.
Each seemedto fitinto
hisownrolein brewery
affairs.
Andto themit seemed
naturalto take theirnumericalproblemsto Gosset.
He haddonesomemathematics
at Oxfordand seemed
lessscaredofmathematics
thantheywere.(In a report
on the theoryof error,he observed,"It may seem
strangethat reasoningof this naturehad not been
morewidelymadeuse of,butthisis due,first,
to the
populardreadofmathematics.")
He was alwaysready
to listento them,and veryquickto grasptheirconcerns.He wouldalwaysdo his bestto comeup with
an answer,goingback to firstprinciplesand arguing
to a solution(McMullen,1938),and he was
through
notsatisfieduntilhe reachedit. So, statisticalproblemscameto him.He gotholdofAiry'stextbookon
the theoryof errors,and studiedand annotatedhis
In 1903he couldcalculatestandcopyin themargins.
ard errors.In 1904 he wrotea reporton the subject
forthe brewery.
This reportled directly
to his being
sent to consultKarl Pearson about his difficulties,
Professor
Pearsonbeingthegreatnamein biometrics
in thosedays.In 1905,Gossetwas usinga homemade
measureof correlation,
based on examinationof the
difference
betweenE (A + B)2 and E (A - B)2, at the
timehevisitedPearsonandlearnedabouttheproductmomentcorrelation
coefficient
(Pearson,1938).
Onegreatdifficulty
in interpreting
theirexperimental data was thatthe sampleswerealwayssmall.For
instance,the barleyexperimentsstartedwith four
farmseach growingone plot of each variety.The
estimatem ofthe meanbased on a sampleoffouris
GUINNESS, GOSSET, FISHER, AND SMALL SAMPLES
obviously
notexact,and theerrorin theestimates of
This Gosset
thestandarddeviationcannotbe ignored.
For him,the main questionwas exactly
recognized.
how muchwidershouldthe errorlimitsbe to make
by usingthe estiallowanceforthe errorintroduced
of
the
parameters
instead
m
and
s
mates
gt and a.
PearsoncouldnotanswerthatquestionforGossetin
whichwas:whatlevel
1905,northeonethatfollowed,
significant?
be
called
ofprobability
should
tookplaceinJuly1905duringGosset's
The meeting
summerholidayin Englandwhenhe cycledabout20
milesfromhis parents'hometo Pearson's summer
home,and theyhad a longtalk,whichGossetfound
veryhelpful.Later,he recalledthatPearson"wasable
in abouthalfan hourto putme in thewayoflearning
the practiceof nearlyall the methodsthenin use."
He wentbackto Guinnessand practicedthosemethof
ods forthe nextyearas actingBrewer-in-Charge
whereanalysisofthecoltheExperimental
Brewery,
was
lecteddatabecamea majorconcern.The meeting
in fact,thatGuinnessarrangedforGosso successful,
at Uniset to spenda yearin Pearson'sdepartment
versityCollege,London,"to takeup the studyofthe
law oferror,the workingof whichwe have foundof
Gossethad foundhis
greatservicein the brewery."
vocation.
Duringthe academicyear 1906-1907at Pearson's
Gossetworkedouttheexactanswerto his
laboratory,
questionabout the probableerrorof the mean and
values of his criterion
z=
tabulatedthe probability
(m - ,)/s for samples of N = 2, 3, ...,
10. He tried
of the correlation
also to calculatethe distribution
coefficient
by the same methodbut managedto get
theansweronlyforthecase whenthetruecorrelation
is zero.It is almosta miraclethathe got so far,for
his mathematicshad to be helped out by inspired
guesses.
Gosset quietlyreturnedto Guinnessin 1907, to
BrewoftheExperimental
becomeBrewer-in-Charge
7 years.
eryforthefollowing
He was just in time to providethe answer to
McMullen'sprayers.The firstseriesofbarleyexperimentswas comingto a close.The resultsof7 yearsof
to Guinness.
workwereto be analyzedand repoTted
McMullenpreparedthe text,with assistancefrom
of barJacksonon the taxonomyand identification
from
leys,fromPeake on mattersofsoil and weather,
whileGosset
Hunteron geneticsand barleybreeding,
took overthe entirestatisticalanalysis:therewere
barley yields, measurementsand assessmentsfor
and
farmsand districts,
different
varieties,different
different
seasons;thereweremaltingassessmentsand
resultsforthesamebarleylots.Gossetfinally
brewing
in termsof
analyzedbarleyyieldand.qualitytogether
he quotedthe
valueperacre,and at everyopportunity
odds (fromhis newlycalculatedt table) it was the
49
especially
naturalwayforhimto expressprobabilities,
in thenativelandofhorseracing!(McMullen,1908).
Now,theyknewthat
AndGuinnessweredelighted.
Ireland
andtheywanted
barley
for
was
the
best
Archer
1,000
island.
They
discovered
the
to growit all over
barrelsofpure-lineDanish Archerseedwas available
theirComto buy.Theyboughtit all. Then,through
itas seedtochosen
theydistributed
missionMaltsters,
togetherwith300 barrelsof Danish Archer
farmers
a miniseedtheyhad grownin Ireland,guaranteeing
mumyield,all ofit to be boughtbackat harvest.The
nextyeartheyhad some 10,000barrelsto distribute
as seed,whichwas grownand boughtback as before.
Afterthat, they had enoughto distributeto any
farmerwho wantedto growit in the ordinaryway.
mostbarleyseedin Ireland
Formanyyearsthereafter,
was selectedbybrewersat the GrandTour reserving
and stored
thecleanestfieldsforseed,to be harvested
by theirmaltsters.Maltsters
apart and distributed
continuedas seedsmenafterWorld War I when
thehybridSprattArchergavewayto a newvariety,
whichHunter,at his firstattemptat making
Archer,
a hybrid
cross,hadcreatedin 1908.Betweenthewars,
90% ofthebarleygrownin Irelandwas Spratt-Archer,
as was most of what was grownin England too
(Hoctor,1971;Hunter,1882-1959).
In hissparetimein 1908,Gossetpreparedhispapers
on theprobableerrorofthemeanand ofthecorrelaforpublication(Student,1908a and
tion coefficient
1908b).Guinnesshad earlieragreedto permitpubli"Itwasdecided
cation.As theBoardminutesrecorded,
by La Touche that such publicationmightbe made
Theywouldbe
without
thebrewers'namesappearing.
merelydesignated"Pupil" or "Student."So Gosset
Student.He continuedstatistical
tookthepseudonym
researchesin his sparetime,because theywerenot
business.And,in his sparetimealso,
brewery
strictly
he made himselfthe trustedfriendof E. S. Beaven.
Beavenwas a self-mademan and a ruggedindividualist who had no patienceat all withstatisticsand
But Gossetshowedhimselfso genuinely
statisticians.
in Beaven'sworkthatsoonhe had Beaven
interested
went(Beaven
eatingoutofhishandso faras statistics
papers).
Beavenwas a maltsterat Warminster,
Wilts,celewhichwas his passion.In
bratedforbarleybreeding,
inthesubject,
1904,whenGuinnessbecameinterested
MaltsterforGuinBeavenhadbecomea Commission
ness and, in 1919,he becameGuinness'buyerof all
importedbarley,but he alwaysregardedthe barley
breedingas his privateaffair.At that time,he had
alreadyselectedandbredpurelinesofseveralvarieties
and, withhis friendsfromthe new CambridgeUniRowlandBiffinand
Department,
versityAgriculture
T. B. Wood,he had startedto makecrossesbetween
varietiesin the hope of creatingnew and better
50
J. F. BOX
barleys.Theyhadbecomekeengeneticists
and breederssincetherediscovery
in 1900ofMendel'spaperon
hybridization.
In 1904,Guinnesssoughttheiradvice
onthebestdirection
forthebarleyresearchin Ireland.
In 1907,GossetcameintoBeaven'slifejust in timeto
helphimdevelopthe nursery
experiments
neededto
test severalof Beaven's hybridsagainsteach other
and against standardvarieties.In 1912 and 1913,
Gossetevenmanagedto runcooperative
experiments
withHunterand Bennettat Ballinacurra,Biffinat
all testing
Cambridge,and Beaven at Warminster,
theirownagainsttheothers'selectedseed.He didthe
analyseshimselfand, analyzingalwaysin termsof
differences
betweenpairsof neighboring
plots,manfortheseexperiagedto evolvea testof significance
ments-an extensionof Student'stest, producing
muchthesamenumerical
resultsas an F test(Beaven
papers).
So wearriveat Cambridge
in 1912whenR. A. Fisher
was introduced
to Gosset.Fisherwas an undergraduate at Gonvilleand Caius College,Cambridge,
and his
tutorwas F. J. M. Stratton,the astronomer.
Biffin
was a Fellowofthe same Collegeand had interested
Strattonin analyzinga fieldexperiment
forT. B.
Wood.Gossethad methimin theircompany.In April
1912, Fisher's firstpaper was published,"On an
AbsoluteCriterion
forFittingFrequencyCurves,"in
whichheintroduced
function
theideaofthelikelihood
and the methodofmaximumlikelihoodbut,without
the word "likelihood,"his presentationwas rather
Gossetread it and thoughtit "A neatbut
confusing.
as faras I couldunderstand
it,quiteunpracticaland
unserviceableway of lookingat things."In June,
Fishersat hisfinalexaminations
forthemathematics
degree.He mustthen-havetalkedto Strattonabouta
he had foundbetweenthe formulafor
discrepancy
standarddeviationin hisownpaperand in Student's.
Maximumlikelihoodgavea denominator
n insteadof
n - 1 in thisequation.Strattontoldhimto writeto
Gossetabout it. He wroteto introduceFisher,and
Fishersenthisproof.The restofthestoryis givenas
Gossetwroteof it to Pearsonon 12 September1912
(Pearson,1968):
This [proof],Stratton,the tutor,made him send
me and withsomeexertionI masteredit, spotted
the fallacy(as I believe) and wrotehim a letter
inthematter
I hope,an intelligent
interest
showing,
and incidentally
makinga blunder.To thishe repliedwithtwofoolscappagescoveredwithmathematicsof the deepestdye in whichhe proved,by
thattheformulawas,afterall,
usingn-dimensions
,/ (x - m) 2/(n
1) and, of course,exposingmy
his stuffand wrote
mistake.I couldn'tunderstand
and said I was goingto studyit whenI had time.I
actuallytookit up to the lakes withme-and lost
it!Nowhe sendsthisto me [themathematical
proof
-
of Student'sdistribution].
It seemedto me thatif
it'sall rightperhapsyoumightliketo puttheproof
in a note. It's so nice and mathematicalthat it
mightappealto somepeople.
Andhe said,"Wouldyoumindlookingat it forme.I
don'tfeelat homein morethanthreedimensions
even
ifI couldunderstand
it otherwise."
That's Gossetforyou-his humorousaccountofa
chapterof accidents,his frankinabilityto deal with
stiffmathematics,and his generousappreciation
nevertheless
of the "nice and mathematical"
proof
that deservedpublicationif it was "all right."In
contrast,
Pearson,whomighthaveunderstood
it,did
notchooseto publishthe note.(Pearsondid publish
it, however,in 1915,whenFisherincludedit in his
ofthecorrelation
paperon the samplingdistribution
the sample
coefficient,
againderivedby representing
in n-dimensional
space.)
Let us turnnowto Fisherand ask whyhe should
be involvedwithappliedmathematicswhenit was
Afterall, as Stratdespisedbypuremathematicians.
tonwroteon Fisher'sbehalfwhenhe was lookingfor
a job in 1919,Fisher"couldhave been a firstclass
mathematician
hadhe stucktotheropes,buthewould
not."Why?
Fisherwastheseventhchildofa fineartsauctioneer
in theWestEnd ofLondon.Alwaysbrilliantat mathin mathematics
to
ematics,he had wona scholarship
Harrowand laterto Gonvilleand Caius College,Cambridge,wherehe won a firstclass honorsdegreein
mathematics.
UnlikeGosset,he had no contactwith
orexperimenters
experiments
uptothattime.Instead,
he had a consuming
desireto makehimselfuseful,to
servehis countryand humankindin a significant,
practicalway.Pure mathematics,
althoughit maybe
a sourceofthegreatestintellectual
to its
gratification
professor,is not a subjectof immediatepractical
utility.
Fisherhad nearlydecidedto takebiologyinsteadof
mathematics
at Cambridge,
because he could see its
and he was seriouslyinterested
in evoluimportance,
tion and genetics.He was especiallyinterestedin
humangeneticsand that broughthim at once to
statisticalconsiderations
becausefewhumancharactersaredichotomous,
beingcontrolled
bya singlepair
in
of geneslike the charactersMendelinvestigated
gardenpeas-for example,tall or shortplants,green
or yellowpeas. Variablessuchas humanheight,skin
have continuousdistributions
color,and intelligence
and musttherefore
be controlled
by multiplepairsof
ofgenesin human
genes.To dealwiththepopulations
One can
one mustdeal withprobabilities.
heredity,
statewhathappensonlyin statisticalterms.
It seemsthat as soon as he felthe could choose,
Fisherveeredto biologicalapplicationsofmathematics. Having succeeded in formingthe Cambridge
GUINNESS, GOSSET, FISHER, AND SMALL SAMPLES
EugenicsSocietyduringhis secondunderUniversity
graduateyear,he was chairmanoftheundergraduate
in 1911-1912.AttheNovember1911meetcommittee
inghe was the speakeron the subjectof Mendelism
(i.e.,Pearson'sstatistical
(i.e.,genetics)and Biometry
methodsofdealingwithcontinuousvariables),which
he explained,sayingbothwerenecessaryforhuman
geneticresearch.He mentionedMaxwell'stheoryof
gases to illustratethe necessityof speakingof gene
We havenoticed
in termsofprobabilities.
populations
frequency
curves.
that his firstpaperwas on fitting
We have seen how quick he was not onlyto prove
in 1912butto realizetheimporGosset'sdistribution
of
the samplingdistributions
tance of determining
otherstatisticsin commonuse and to go on to find
ofthecorrelation
coefficient
thesamplingdistribution
thereafter.
in 1915and a seriesofotherdistributions
in 1912-1913,he
As a graduatestudentat Cambridge
choseto studythetheoryoferrorswithStrattonand
JamesJeans.
quantumtheorywiththeastronomer,
A veryimportant
earlypaper of Fishertookup a
questionthat had been foughtover for a decade:
was compatable
in fact,humaninheritance
whether,
with Mendelian principles. Biometricianswere
divided:Pearson said not, and rejectedgenetics;
G. Udny Yule thoughtit mightbe compatablebut
couldnotproveit; geneticists
just assumedall inheritancewas Mendelian.Fisherbroughthis mathematical geniusto bear on the problemand provedthat
the inheritance
of continuousvariableswas entirely
consistentwithMendelianprinciples-in fact,that
Mendelianprinciplesmust result in preciselythe
numericalrelationshipsthat existed.The way he
as the resultbecause,in
provedit was as important
orderto deal withintractablesample correlations,
he formulatedthe problem in a differentway
and introducedthe conceptof analysisof variance
(Fisher,1918).
AlthoughFisher's biologicalinterestsstimulated
researches
himto do theseimportantmathematical
even while he was a school teacherduringWorld
War I, it was afterhe became the statisticianat
RothamstedExperimentalStation in 1919 that he
really,as he put it,foundhis feetin research.Papers
funpouredfromhis hand,manyofthemcontaining
damentalnewwork.Between1921and 1926thenumrosefrom13 to 55 papers,and
berofhispublications
he wroteStatisticalMethodsforResearchWorkers
research
is theoldestagricultural
(1925).Rothamsted
with
institution
inGreatBritain.Fisherwaspresented
with
experiments
data-lots of data fromlong-term
wheatand rootsand grassplots and farmrotations.
His firstjob was to analyzethe manurialexperiment
withwheaton Broadbalkfieldwherethesame fertilizershad beenappliedto the sameplotsfor67 years.
morebe donewiththe
He was asked,couldanything
51
data than A. D. Hall had done some yearsbefore?
Besides,therewereabouta dozenstaffwhenFisher
wentthere,all busywithresearchprojects.
The activitiesat Rothamsted,the interestsand
problemsof the staff,the discussionsovera cup of
tea, the data, all werea greatstimulusto Fisher's
His paperswereall a mix
andinventiveness.
ingenuity
theory
andpractice.
ofstatistical
ofnewdevelopments
FounTherewasonebigpaper"On theMathematical
dationsof TheoreticalStatistics"(1922) but even in
dethathe insertedthe firstnonlinearexperimental
sign,whichhe had createdforone ofhiscolleaguesat
The seriesof studiesin cropvariation
Rothamsted.
wereostensibly
appliedpapers."Studiesin CropVaroftheyieldofdressed
iationI" was "Anexamination
grainon Broadbalk"(1921).Thatcontainsorthogonal
polynomialsand the analysisof varianceprocedure
"Studies
and one or twootherstatisticalinnovations.
in CropVariationII: The manurialresponseofdiffertheanalysisof
entpotatovarieties"(1923)introduces
variancetablewiththe newz testand its validityis
oftheplot
statedto be conditionalon randomization
treatments.
Theseideasare all just slippedin without
ofrainfallon the
A paperon "The influence
fanfare.
(1924)includesa short
yieldofwheatat Rothamsted"
data
essayon problemsof analysisof meteorological
of the
and the derivationof the null distribution
in additionto its
multiplecorrelationcoefficient
advertisedcontent,not to mentionthe use of what
in the analfunctions
wouldtodaybe calledtransfer
ysis.Thiswasconceivedas "Studiesin CropVariation
III" buttheprefixwasdroppedwhenthepaperbecame
journal
unsuitableforpublicationin an agricultural
merely.
all thisnewmaterial
WhenFisherbrought
together
in the book StatisticalMethodsforResearch Workers
oftheorybut in
(1925),it was not as a development
could
of
that
terms methods experimenters useintheir
evenif
theycouldunderstand,
work,whoseprinciples
were
them.
Fisher
themathematical
beyond
workings
in thepreface:
setforthhisphilosophy
For severalyearsthe authorhas been workingin
somewhatintimatecooperationwitha numberof
the presentbook
biologicalresearchdepartments;
is in everysensetheproductof thiscircumstance.
Daily contactwiththe statisticalproblemswhich
presentthemselvesto the laboratoryworkerhas
stimulatedthe purely mathematicalresearches
uponwhichare based the methodsherepresented.
to showthatthetraLittleexperienceis sufficient
ofstatistical
ditionalmachinery
processesis wholly
Notonly
unsuitedtotheneedsofpracticalresearch.
does it take a cannonto shoota sparrow,but it
built
The elaboratemechanism
missesthesparrow!
on the theoryof infinitely
large samplesis not
data.Onlyby
accurateenoughforsimplelaboratory
J. F. BOX
52
systematically
tacklingsmall sampleproblemson
theirown meritsdoes it seem possibleto apply
accurateteststo practicaldata. Such at least has
beentheaim ofthisbook.
REFERENCES
BEAVEN,
E. S. PapersdepositedwithGuinnessBarleyResearch
Station,Warminster,
Wilts.
BENNETT, J. H., ed. (1971-1974).CollectedPapers of R. A. Fisher
1-5. Univ.AdelaidePress,Adelaide.
GuinnessSon and
BROWN, J.F. (1980).Guinnessand Hops. Arthur
Co. (ParkRoyal),Ltd.,London.
CUNLIFFE, S. (1976).Interaction.
J. Roy. Statist. Soc. Ser. A 139
1-19.
R. A. (1912)On an absolutecriterion
forfitting
frequency
curves.Messeng.Math. 41 155-160.(See Bennett1 No. 1,
1971.)
FISHER, R. A. (1918). The correlation
betweenrelativeson the
of Mendelianinheritance.
supposition
Trans.Roy.Soc. Edinburgh52 399-433.(See Bennett1 No. 9, 1971.)
I. An examination
FISHER, R. A. (1921).Studiesin cropvariation.
oftheyieldofdressedgrainfromBroadbalk.J. Agric.Sci. 1 1
107-135.(See Bennett1 No. 15,1971.)
FISHER, R. A. (1924).The influence
ofrainfall
ontheyieldofwheat
FISHER,
at Rothamsted. Philos. Trans. Roy. Soc. London Ser. B 213
89-142.(See Bennett1 No. 37,1971.)
FISHER, R. A. and MACKENZIE, W. A. (1923). Studiesin crop
potato
variation.II. The manurialresponseof different
varieties.J. Agric.Sci. 13 311-320.(See Bennett1 No. 32,
1971.)
HOCTOR, D. (1971). The Department's Story: A History of the
DepartmentofAgriculture.Instituteof Public Administration,
Dublin.
H. (1926).The Barley Crop. ErnestBenn,London.
H. (1882-1959).Unpublished
papersdepositedwiththe
History,Univ.of Reading,Reading,
Instituteof Agricultural
Berks.
MCMULLEN, A. (1908). Barley Cultivation in Ireland. Arthur
HUNTER,
HUNTER,
Guinness Son and Co., Ltd., Dublin.
MCMULLEN,L. (1938). William Sealy Gosset 1876-1937: "Student"
as a man. Biometrika30 205-220.
PEARSON, E. S. (1938). William Sealy Gosset 1876-1937: "Student"
as statistician.Biometrika30 220-250.
and
E. S. (1968). Studiesin the historyof probability
betweenW. S. GosXX. Someearlycorrespondence
statistics.
PEARSON,
set, R. A. Fisher,and K. Pearson, withnotes and commentary.
Biometrika55 445-457.
STUDENT (1908a). The probableerrorof a mean. Biometrika
6 1-25.
(1908b). Probable errorof a correlationcoefficient.
Biometrika6 302-310.
STUDENT