Technical
Note
Why Quadratic Mean Diameter?
Robert O. Curtis and David D. Marshall, Pacific NorthwestResearchStation,
3625-93rd AvenueSW, Olympia WA 98512-9193.
ABSTRACT: Quadraticmeandiameteris themeasureof averagetreediameterconventionally
usedin
forestry,ratherthanarithmeticmeandiameter.Thehistoricalandpracticalreasons
for thisconvention
are
reviewed.West.J. Appl.For. 15(3):137-139.
Average
diameter
isa widely
used
stand
statistic
that
appearsin virtuallyall yieldtables,simulatoroutputs,stand
summaries,
andmuchinventorydata.To mostpeople,"average"is synonymous
with the arithmeticmean,definedas
unitsused(0.005454 for B in squarefeet andQMD in inches;
0.0000785 for B in squaremetersand QMD in centimeters).
In angle-gauge
sampling,it canalsobecalculated
directly
(Buckingham1969) as:
arithmeticmean= Z•= (E xi) / n
wherethexiaretheindividual
measurements
andnisthetotal
number of measurements.
But there are in fact some half dozen different kinds of
averages(= means= measuresof centraltendency),each
appropriateto specificuses.One of theseis the quadratic
mean(KendallandBuckland1967,Iles andWilson 1977),
defined as
quadratic
mean
= (•
/n
whichis the squareroot of the arithmeticmeanof squared
values.Other generallyrecognizedmeanssometimesencounteredin forestry are the geometricmean, harmonic
QMD
=•[ns/ Z(1/ d•
2)]
wherethedi arethediameters
andnsis thenumber
of "in"
treesin the angle-gaugesample.
Pastusageof thephrase"averagediameter"hasoftenbeen
very loose, and unwaryreadersoften take it to meanthe
arithmeticmean,whenin factthevaluegivenisthequadratic
mean.It is thereforegoodpracticefor authorsto be specific
(Curtis1968). The quadraticmeangivesgreaterweightto
largertreesandis equalto orgreaterthanthearithmeticmean
by an amountthat dependson the varianceaccordingto the
relationship
(QMD)2= •2 + s•
mean, median, and mode.
where t• is arithmeticmeandiameterands2 is the variance
Theexpression
of averagestanddiameterconventionally
usedin forestryis not the arithmeticmeanof diameters,but
of diameters.
the quadraticmean,
eters,the differencesare slight. In standswith large diametersanda widerangeof diameterspresentor with strongly
skewed diameter distributions, the differences between
arithmetic mean and quadratic mean diameterscan be
substantial(Figure 1).
Peoplenot stronglygroundedin forestmensurationare
quadratic
mean
diameter
= (•
/n
wheredi isthediameter
atbreast
heightofanindividual
tree,
andn is the total numberof trees.Quadraticmeandiameter
is commonlysymbolizedasQMD, Dq, or Dg. Dg, in which
the subscriptstandsfor "Grundflliche,"German for basal
area,is widely usedin Europeand is the symbolrecommendedbytheInternational
Unionof ForestResearch
Organizations(Van Soestet al. 1959).
QMD is oftencalculated
by theequivalent
equation:
QMD= x/B/ (k* n)
whereB is standbasalarea,n is corresponding
numberof
trees,andk is a constantthat dependson the measurement
NOTE:RobertO.Curtisisthecorresponding
authorandcanbereached
at(360)
753-7669;Fax: (360) 956-2346;E-mail: rcurtis@fs.fed.us.
In standsof smalldiameterandnarrowrangein diam-
often unaware of the distinction between arithmetic and
quadraticmeans.Whenthis distinctionis pointedout,they
naturallywonderhow and why sucha strange"average"
cametobeadopted
andwhyit isstillused.Afterall,it israrely
mentionedin introductorystatisticscourses.
The answeris partly a matter of customand historical
precedent,but QMD alsohascertainpracticaladvantages
that still hold true.
Use of the quadraticmeanof diametersis a very old
practice in forestry, which goes back to 19th century
Germanyand possiblyearlier. It hasbeen standardpractice in the United Statesfrom the earliest days of North
Americanforestry.Most standardU.S. mensurationtexts,
WJAF15(3)2000 137
Median, Arithmetic and Quadratic means
for a •-yr-oklpoplarplantation
Median, ArlthmeUc and Quadratic mean•
forsometypicalinventory
data
10
•Median
di• = 3,50in.
0
I
2
3
4
DBH class - Inchea
5
6
2
4
Figure 1. Median, arithmetic mean, and quadratic mean diametars for stands with (A) small diameters and naarly symmetrical diameter
distribution, and (B) larger diameters and somewhat asymmetrical diamater distribution.
starting with Graves (1908), define average standdiameter asthe diametercorresponding
to thetreeof arithmetic
mean basal area, which is the quadratic mean diameter.
Braathe's (1957) summaryof Europeanthinningliterature
specifically defines average diameter as the quadratic
mean. QMD is commonly used in silviculture research
data summariesand reports. Virtually all normal yield
tables prepared in the United States in the period from
around 1920 through the mid-1960s use quadratic mean
diameters (Schnur 1937, McArdle et al. 1961, Barnes
1962), sometimes referred to in older publications as
"average diameter by basal area." This usageof QMD is
alsocommonin currentstandsimulationprograms(Curtis
et al. 1981, Hann et al. 1997). Reineke's (1933) SDI is
basedon QMD, as are the variousrelative densitymeasuresand standmanagementdiagramsderived from the
Reineke relationship(Curtis 1982, Long et al. 1988).
In the Germany of some 150 or more years ago, there
were a number of so-called mean tree methods in use for
estimatingvolume of wood in forest stands.These also
had somelimited usein the early daysof North American
forestry(Graves 1908, p. 224ff.). The basicidea, in simplest form, was that the forester would select a tree(s)
consideredaverage for the stand, cut it and measureits
wood content, and then multiply by the number of trees.
The obviousdifficulty wasin selectingan averagetree(s),
whosevolumewouldapproximateoverallarithmeticmean
volume / tree. In regular even-agedstands,diameterof the
treeof arithmeticmeanvolumeis generallycloseto thatof
the tree of arithmetic mean basal area (which is also the
tree of quadraticmean diameter).Thus, a basiswas provided for selectingsampletreesfor analysis.
Suchprocedures
arenowancienthistory.Butjustification
for use of QMD also arisesfrom the generalrelationship
betweenstandvolumeandother,directlymeasurable,stand
attributes.
In any reasonably regular stand, there is a general
relationship
volume/ unitarea=f* N * Bmn*H
138
WJAF15(3)2000
wheref= standform factor, which for a given speciesand
standconditionhasonlya verylimitedrangeof variationand
can often be treated as constant.
N
= number of trees / unit area
Bmn= arithmetic
meanbasalarea/ tree,and
H
= some"average"height.
In anexistingstandwe cannotdirectlymeasureeithertotal
stand volume or mean volume/tree, but must estimate these
from measurements
of their components.It is often conve-
nientto describe
stands
in termsof means
of 'these
components:namely,numberof trees,arithmeticmeanbasalarea,
and someaverageheight.
Peopledo notusuallythink in termsof basalareaof a tree
(cross-sectional
area at breastheight). It is much easierto
visualizea treeof 19in. dbhthanoneof 2.0 ft2cross-sectional
area. It is thereforecommonto describestandsby QMD (a
surrogatefor arithmeticmean basal area) rather than by
arithmetic mean basal area. In these terms,
Volume
/ unitarea=f* N * [k* (QMD)2]* H
The correctaverageheightin theseequationsis not the
arithmetic
mean,
butLorey'sheight(HL,named
aftera 19th
centuryGermanforester).This is a weightedmean,
HL: Zbih
i / Zbi = Z d/hi / Z di2
where,bi isthebasalareaof anindividual
treeanddi is the
diameter of an individual tree.
HL is somewhat
inconvenient
to calculate
froma fixed
areasampleor standtable, althoughwith angle-gaugesampling it can be easily obtainedas the arithmeticmean of
heightsof the counttrees.A commonapproximationis the
height corresponding
to QMD, as estimatedby a heightdiametercurve or equationfor the individual stand.(Stand
averageheightis of coursea differentstatisticfrom thetop
heightor dominantheightusedfor otherpurposes,though
highlycorrelatedwith thesein unthinnedstands.)
Expressionsof the aboveform are not commonlyused
todayto calculatestandvolumes,althoughtheyarevalidand
aresometimes
usedin standsimulationprograms.We generally applytree volumeequationsdirectlyandsumover all
trees,ratherthanfirst calculatingthesemeans.But thereis
anotherstrongreasonfor usingQMD. This stemsfrom the
relationship
B = k * N * (QMD)2
be consciousof the difference between quadratic and
arithmeticmeandiameters(which usuallyis not large) and
be specific in defining the value used.
Literature
Cited
BARNES,
G.H. 1962.Yield of even-agedstandsof westernhemlock.USDA
For. Serv.Tech.Bull. No. 1273.52 p.
BRAAmE,
P. 1957.Thinningsin even-agedstands:A summaryof European
literature.Facultyof For., Univ. of New Brunswick,Fredericton.92 p.
BUCKINGHAM,
F.M. 1969. The harmonic mean in forest mensuration. For.
Thisis anexactrelationship.
Therefore,knowledgeof any
two of thevariablesautomatically
confersknowledgeof the
third.In contrast,
thereisnoequivalentexactrelationship
for
the arithmeticmean, and conversions
usingthe arithmetic
meanalsorequireknowledgeof thevariance.It is a greatdeal
easierto makeconsistent
estimatesandprojectionsfor two
variablesthanfor three,andtheexactrelationship
thatexists
whenQMD isusedmarkedlysimplifiesconstruction
of yield
tablesand standsimulators,standprojections,and some
inventorycomputations.
This directconvertibilityalsosimplifiesthe construction
anduseof standmanagement
diagramsbasedon numberof
trees,basalarea, and averagediameter(Long et al. 1988,
Gingrich 1967, Ernst and Knapp 1985). Becauseof this
convertibility,theycanbe expressed
in termsof anytwo of
the three variablesN, basal area, and QMD.
The arithmeticmeanis themeasureof centraltendency
mostwidely usedin generalstatistics,andis essentialto a
few procedures(suchas defininga normalprobability
distribution).But most proceduresin common use in
forestrydo not specificallyrequirethe useof the arithmetic mean. Both the mensurational advantagesmen-
tionedaboveandlong-standing
precedentmakethe quadraticmeanof diametersthepreferred"averagediameter"
for expressingstandattributes.In any case,usersshould
Chron. 45(2):104-106.
CURTIS,
R.O. 1968.Which averagediameter?J. For. 66:570.
CURTIS,
R.O. 1982.A simpleindexof standdensityforDouglas-fir.For.Sci.
28:92-94.
CURTIS,R.O., G.W. CLENDENEN,
AND D.J. DEMARS. 1981. A new stand
simulatorfor coastDouglas-fir:DFSIM user'sguide.USDA For. Serv.
Gen.Tech.Rep.PNW-128.79 p.
ERNST,R.L., ANDW.H. KNApp.1985. Forest standdensityand stocking:
Concepts,
terms,andtheuseof stockingguides.USDA For. Serv.Gen.
Tech.Rep.WO-44. 8p.
GINRICh[GINGRICn],
S.F. 1967. Measuringand evaluatingstockingand
stand density in upland hardwood forests in the central states.For.
Sci. 13:38-53.
GRAVES,
H.S. 1908.Forestmensuration.
Wiley, New York. 458 p.
HANN,D.W., A.S. HESTER,
ANDC.L. OLSEN.1997. ORGANON user'smanual,
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ILES,K., ANDL.J. WILSON.1977. A further neglectedmean. Math. Teach.
70:27-28.
KENDALL,
M.G., ANDW.R. BUCKLAND.
1967.A dictionaryof statistical
terms.
Ed. 2. HafnerPublishingCo., New York. 575 p.
LONG,J.N., J.B. McCARTER,
ANDS.B. JACK.1988. A modified density
managementdiagramfor coastalDouglas-fir. West. J. Appl. For.
3(3):88-89.
MCARDLE,R.E., W.H. MEYER,ANDD. BRUCE.1961 (rev.). The yield of
Douglas-fir in the Pacific Northwest. USDA Tech. Bull. No. 201
(rev.). 72 p.
REINEKE,
L.H. 1933. Perfectinga stand-density
index for even-agedforests.
J. Agric. Res.46(7):627-637.
SC,NUR,
G.L. 1937.Yield,stand,andvolumetablesforeven-aged
uplandoak
forests.USDA For. Serv.Tech.Bull. No. 560. 88 p.
VAN SUEST,
P.A., R. SC•OBER,
ANDF.C. HUMMEL.1959. The standardization
of symbols
in forestmensuration.
IUFRO. 32 p. [Reprinted
1965asTech.
Bull. 15 of the Maine Agric.Exp. Sta., Orono.
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