Sampling Methods in Forestry and Range Management

214 Pages·1942·6.328 MB·English

by  F. X. Schumaher, R. A. Chapman

IMPERIAL AGRICULTURAL
RESEARCH INSTITUTE, NEW DELHI.
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SAMPLING ME':eHODS IN FORESTRY
AND RANGE MANAGEMENT
· IMPERIAL AGRICULTURAL
RESEARCH INSTITUTE, NEW DELI-II.
DUKE UNIVERSI'l'Y
SCHOOL 01<' FORES'l'RY
BULLETIN 7
SAMPLING METHODS IN FORESTRY
AND RANGE MANAGEMENT
BY
:D'. X. SCHUMACHER
7"I"I)fl~SSOl' of' F'01'(!.~t'j'Y,. School of li'(l1'est1'Y
TJ'uke Uni1Je1"sitv
AND
R A. OHAPMAN
LLs{)etate ,"'ilvicnltwrist, Southern PM'est Experiment Station, FM'est SB?"viee,
llnitecZ Stutes DelJal'tment of Agr'imlltul'e
11358
IIIIIIII~IIIIIIIIIIIIIIIII ~~ IIII
IARI
DURHAM, Nonrru CAnOLINA
JANUARY, 1942
OOPYRIGHT, 1942, BY DUKE UNIVlnnSl'l'Y
PRINTED IN THE UNITED S~I'ATES Ole AMERIOA BY
THE SEEMAN PRINTERY, DURHAM, NORTH CA!Wr~IN A
PREFACE
The concept of sampling error is essentially simple. It implies that
the discrepancy-real, but unknown-between a true magnitude, which
is the subject of inquiry, and the sampling estimate thereof, may be
evaluated precisely.
The practice of forestry is replete with problems of sampling. In
many of them, however, as in timber cruises, the essential simplicity of
the concept of sampling enol' is obscured by failure on the part of forest
ers to recognize that the body of data gathered from a systematic pattern
of strips or line-plots, upon which estimates of timber volumes and values
are commonly ba.Red-and which they have been taught in their college
courses in forest mensuration-does not contain information on sampling
error.! Unquestioning acceptance of the systematic pattern as the only
kind worthy of considemtion has resulted in attempts to extract sampling
error thf1t are more akin to the t1rt of the conjurer than to scientific ltSsay.
The development of mathematical statistics, partiCUlarly of that part
concerning the theory of small samples, is exerting remarkable influence
upon the scientific endeavor of research foresters and range ecologists,
by nlf1king fwaibble experimental methods of logical structure which are
at once Cl111n,blc of yielding efficient estimates of effects, and valid tests
of hypotheses pertaining thereto.
Less apparent, perhaps, but nonetheless genuine, is the growing in~
nuance of mathematical statistics upon the everydn,y work of practicing
foresters and range examiners. Administrative decisions pertaining to
management of :1 forest or range business commonly rest upon esti:mates
of the amount, or condition, of forest or range values. Thus the maxi~
mum number of cattle a range can support without deterioration; or t,he
volume of It given class of timber which may be removed from a forest
(lOmpI,l'trncnt without harm to the residue; these are deduced from esti·
mates of existing magnitudes of forest or range values, arrived at by
means of some planncd sampling procedures.
While each such estimate is obviously encumbered with a real error,
it has not been universally recognized that it is the job of practicing
foresters, 01' rl1nge t.echnicians, to acquire the art of planning-and
executing-suitable sampling procedures, such that (1) the real errol' may
?c
assessed uIlambiguously; and (2) the best estimate is obtainable (and,
. lOne of us (F. X, S.) takes this occasion to indict himself as co-author of a t,ext on
forest mensuration in which systematic cruise patterns !1re the only kinds discussed.
[ 5 ]
6 PREFACE
consequently, the real error is least) consistent with t.he time and funds
available for the sampling work.
It is the purpose of this treatise to discuss this twofold aspect of Uw
problem of sampling, of the kind encountered in the practice of forest.ry.
Such use as is made of mathematics in the following pages 11rcsnp
poses no special training in the subject beyond the modest requirements
of a forestry curriculum. Occasionally, when a needed delIlonstration
seemed to become heavy, 01' to distract attention from the Inn,in theme,
it has been relegated to the Appendix.
We are indebted to Professor E. S. Pearson, of University Colkgo,
London, for permission to reproduce a page of Tippett's Randon Sam
pling Numbers; and to R. A. Fisher, and his publishers, Messrs. Oliver
and Boyd, for permission to reproduee the table of t. But we eannot
adequately express our appreeiation of the work of those mathemn,tieians
and seientists-particularly of Professor Fisher and his ttAsoeiates--to
whose vision and insight the development of small-sample theory is due.
Without the foundation of their labors the present work would not have
been attempted.
We are also deeply indebted to James G. Osborne, Chief (If ForeRt
Measurements, Division of Forest Management Research, United States
Forest Service, for a e1'itica1 reading of the manuscript and many valuable
suggestions.
DURHAM, NORTH CAROLINA F. X. SCHuMAclum
January, 1942 R. A. CHAPMAN
TABLE OF CONTENTS
PART 1. STATISTICAL BACKGROUND
Page
Chapter 1. IN'l'IWDUC'rroN
1.1 The art. of sampling ................................... 15
1.2 The mean and the standard deviation of the sample. . . . . .. 15
1.3 The sample and the population. . . . . . . . . . . . . . . . . . . . . . . .. 18
1.4 The distribution of means of independent observations and
[,he normal curve of error .... . . . . . . . . . . . . . . . . . . . . . .. 23
1.5 V n,riance of the sample and of the popUlation. . . . . . . . . . . .. 25
l.l) Variance of sums and of means of independent observations. 28
1.7 Bstimate of population variance from a sample. . . . . . . . . .. 29
Ch!Lpter II. OBSERVATION A.ND EXPECTATION
2.1 A few points about the normal curvc of error ............. 33
2.2 Calculation of expected frequencies of normally distributed
variates ........................................... 35
2.3 Sn,mple size and the normality of distribution of sample means. 37
2.4 Bstimate of the mean of an infinite population from a large
samplc ............................................ 38
2.5 The probability of discrepancy ..... " ................... , 40
2.6 Small samples and the probability of discrepancy. . . . . . . . .. 41
PART 2. DIRECT ESTIMATES BY SAMPLING
Chapter III. SIMPLER CASES OF SA.MPLING FINITE POPULA'l'lONS
8.1 Infinite and finite populations ........................... 47
3.2 Sampling units ........................................ , 48
3.3 Sampling a small rectangular area. . . . . . . . . . . . . . . . . . . . . .. 48
3.4 The variance of the mean of a random sample from a £inite
population ......................................... 52
3.5 Sampling a small area of irregular boundaries ............ ' 55
3.6 Systema,tic versus random sampling. . . . . . . . . . . . . . . . . . . .. 58
[ 7 ]
8 CONTENTS
Chapter IV. REPRESENTATIVE on STRATIFIED RANDOM SAMPUNG
4.1 The principle of representative sampling ................. (il
4.2 Comparison of representative with unrestricted random
sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. f:i2
4.3 The variance of the mean of a representative set, of samples. ()1
4.4 Disproportional sampling by the representative method. . .. 07
Chapter V. SIMULTANEOUS SAMPIJING 01<' MORg THAN ON1G
POPULATION
5.1 The problem and an illustl'nJ,ion. . . . . . . . . . . . . . . . . . . . . . . .. 71
5.2 Variances and covariances involved ...................... n
5.3 Simult,aneous sampling of more than two populations. . . . .. 77
5.4 Systematic reduction of observations. . . . . . . . . . . . . . . . . . .. 80
Chapter VI. TIm METHOD OF SUB-SAMPLING
6.1 Distinctive feature of the method ....................... 85
G.2 An illustration of the method. . . . . . . . . . . . . . . . . . . . . . . . . .. 85
6.3 Components of sampling errol'. . . . . . . . . . . . . . . . . . . . . . . . .. 8(j
6.4 Analysis of variation among sampling units. . . . . . . . . . . . . .. 88
6.5 Application to an insect population ...................... 94
6.6 Analysis of variance and the sampling errol'. . . . . . . . . . . . . .. 07
6.7 Efficiency of the method ............................... 100
Chapter VII. REPRESENTA'rIVE SAMPLING OJ!' IRREGULAR BLOCKS
7.1 Proportional sampling of blocks of known, but diverse, areas .101
7.2 Proportional sampling of blocks of diverse, but unlmown areas. 102
7.3 The observations and the estimate of the population mean .. 104:
7 A The weighted mean of a sample and the estimate of its variance. 105
7.5 Simplification of computational work with samples of two
random sampling units .............................. 107
7.6 The estimate of total area and its sampling variance ....... 111
7.7 The sampling variance of cover type areas ................ 112
PART 3. INDIRECT ESTIMATES THROUGH REGHESSION
Chapter VIII. THE MEANING AND USE OF REGRESSION IN SAMPLING
8.1 The problem of the present part ......................... IH)
8.2 The regression equation ................................ 119
CONTENTS 9
8.3 A numerical example .................................. 124
8.4 Application of the distribution of t to the regression coefficient. 127
8.5 The variance of Y .................................... . 120
8.6 Thc variance of Y when x is free of enol' ................. 130
8.7 The variance of Y when x is subject to sampling error ..... 131
8.8 The utility of regression in sampling ..................... 135
Chaptcr IX. PURPOSIVID SJ<JLECTION IN SAMPLING
9.1 Exemption of the independent variable from the restrict.ion
of randomization ................................... 136
\).2 Effect on pertinent statistics ............................ 137
n.3 Experimenk11 vcrification ............................... 137
0.1 Limitation to purposive sclection ........................ 139
Clwpter X. CONDITIONED HE<:at]]SSION AN]) THE USJ~ 01" WEIGH'l'S
10.1. The sample census of a forest nursery .................... 141
10.2 Conditioned regression and the weights involved .......... 141
10.3 Application to the forest nursery sample census ........... 145
10.'1 The introduction of a second independent varia.ble ........ 148
10.5 The variance of the conditioned regression curve and its
application ........................................ 153
10.G Certnin remarks concerning regression in sampling ......... 157
Chapter XI. REGRT~SSION IN REPRESEN'l'A'l'tVE SAMl'J,ING
11.1 The problem .......................................... 159
11. 2 The analysis of covariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HH
11.3 '1'he adjusted estimate and its variance ................... 163
11.4 Thc adjustment of ocular estimates of correlated populations. luG
11.5 Variances of the adjusted estimates ...................... 172
11..(; Heconciliat,ion of the conflicting requirements of mapping
and sampling in forest surveys ....................... 175
Chapter XII. ON ClDIt'l'AIN PRACTICAL ASPEC'l'S OF SAMPLING
12.1, Definition of sampling objectives ........................ 178
12.2 Bias ................................................. 178
12.3 Size, shape, and structure of sampling units .............. 180
12.4 The sample ........................................... 183
12.5 'fhe determination of sampling intensity ................. 185
12.G Allocation of costs in double sampling .................... 186