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Sampling Methods in Forestry and Range Management PDF

214 Pages·1942·6.328 MB·English
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IMPERIAL AGRICULTURAL RESEARCH INSTITUTE, NEW DELHI. ~IG IPC--Si--III-l·()3-;,!;2·B. 1:3-5 (lOO. 17 2),58 .:;10/ " 6;31. If.2-8 : 63,4-,; .- .5 ...,;_,;,,.",:~_:,,z-, :J. .s 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

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