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ELEMENTARY FOREST SAMPLING FRANK FREESE Southern Forest Experiment Station, Forest Service Asticulture Handbook No. 232 December 1962 U.S. Department of Agriculture 0 Forest Service Reviewed and approved for reprinting, November 1976. I should like to express my appreciation to Professor George W. Snedecor of the Iowa State University Statistical Laboratory and to the Iowa State University Press for their generous permission to reprint tables 1, 3, and 4 from their book Statistical Methods, 5th edition. Thanks are also due to Dr. C. I. Bliss of the Connecti- cut Agricultural Experiment Station, who originally prepared the material in table 4. I am indebted to Professor Sir Ronald A. , Fisher, F.R.S., Cambridge, and to Dr. Frank Yates, F.R.S., Roth- amsted, and to Messrs. Oliver and Boyd Ltd., Edinburgh, for permission to reprint table 2 from their book Statistica Tables for Biological, Agricultural, and Medical Research. I?EANK FltEESE Southern Forest Experiment Station Forukbytha Sucwintondant of Doatmwb, U.S. twommmt Printing offia Washington. D.C. 20402 - Prior $1.60 25% dbcountJtwodon~d100ormorotoonerddraa stock MO. ool -ooo-Olesl -2/Cata#og No. A 1.7&232 Thus la a minimum chuga 04 $1.00 ior aach ma&l order ii , CONTENTS P*go Basic concepts .-___________-_--___.-.-----------.--.....-------..--.-----.--------- ma-em-e 1 Why sample? __._____-___..._--._.--.-------.--._--.___-________-_________ -----.--.-- 1 Populations, parameters, and estimatzs -...____._.____..___.__-_--.---..--- 2 Bias, accuracy, and precision ___.___.______._.___---.--..-.---.---..----------- 3 Variables, continuous and discrete ___._..____________..___------------------- 6 Distribution functions __________._____ ________.____._.__._____._.__.___.-_._------ 6 Tools of the trade .__.____.__--________________________-__--.___._------.-.--.--.--------- 6 Subscripts, summations, and brackets __ . ___ ________.._________.__._._____--- 6 Variance -________ _ _ _ ____ __ _ _ _ _ _ _ _ _ _ _ _ _ . __ __ .._..____..__..___.__.___.._.-.--- _____._ - 9 Standard errors and confidence limits _____ . ________ _ _ _ _ . __ __ _ __ _ _ _ ___ . _ _ ,-- 10 Expanded variances and standard errors .__.__________.___._____._____..--- 12 CoefRcient of variation ___.__.______-___--._.__--_.-__________._.____-______--_--- 13 Covariance _______.___________.-_.______.____-....________..______________-_._._-...-- 14 Correlation coefficient ..__-___..__________________--______..___________.____..--__15 Independence ____________.______.____. _._---_._________._.--.---.-.-------.__..___.-- 16 Variances of products, ratios, and sums _______________._____________._____ 17 Transformations of variables ____ __ ____. ________________._________________I____ 19 Sampling methods for continuous variables .______________.__._---.-------------- 20 Simple random sampling ____ _-----_-_._____..___-.___-__..-___-..-.--.__-__----- 20 Stratified random sampling . __ . ._____._______.__.-----.------------------.--.--- 28 Regression estimators. ______.___._-___________--__-..--__-.-.._---__.-------.----- 36 Double sampling ____________.-_-____.__..-_---.--.----.------..___--__-_.__...-__._ 43 Sampling when units are unequal in size (including pps sampling) . _ 47 Two-stage sampling .___.__________.________________________-.---.---------------.5570 Two-stage sampling with unequal-sized primaries ._.___.________._______ Systematic sampling ____.__._________I__---..-.---.---------.----..--.__. ____ __ ___ 60 Sampling methods for discrete variables ____._________________-..-----.---.---.-.- 61 Simple random sampling-classification data _____________._______________61 Cluster sampling for attributes ____________.___.___.-------.-----.______-_____- 6”; Cluster sampling for attributes-unequal-sized clusters ._________..---- Sampling of count variables ________.___________---.--.---.-.--- ~~_~.-~~~---~~ -- 68 Some other aspects of sampling _________ __.______. _________ ______-___----._ _..-. _-__ 70 Size and shape of sampling units _________________._-._-- -------.------------- ‘77 Estimating changes __________.____.__..__________________._--.--. -.-------- --__--- Design of sample surveys ________________.___-..-.-..-..---------------.--------- 75 iii Referencea for additional reading . 70 Practice problems in subscript and summation notation . 79 Tables . 82 1. Ten thousand randomly assorted digits ................................. 82 2. The distribution of t .......................................................... 86 3. Confidence intervals for binomial distribution.. ....................... 87 4. Arcsin transformation ....................................................... 89 ELEMENTARY FOREST SAMPLING This is a statistical cookbook for foresters. It presents some sampling methods that have been found useful in forestry. No attempt is made to go into the theory behind these methods. This has some dangers, but experience has shown that few foresters will venture into the intricacies of statistical theory until they are familiar with some of the common sampling designs and computations. The aim here is to provide that familiarity. Readers who attain such familiarity will be able to handle many of the routine sam- pling problems. They will also find that many problems have been left unanswered and many ramifications of sampling ignored. It is hoped that when they reach this stage they will delve into more comprehensive works on sampling. Several very good ones are listed on page 78. BASIC CONCEPTS Why Sample? Most human decisions are made with incomplete knowledge. In daily life, a physician may diagnose disease from a single drop of blood or a microscopic section of tissue; a housewife judges a watermelon by its “plug” or by the sound it emits when thumped; and amid a bewildering array of choices and claims we select toothpaste, insurance, vacation spots, mates, and careers with but a fragment of the total information necessary or desirable for complete understanding. All of these we do with the ardent hope that the drop of blood, the melon plug, and the advertising claim give a reliable picture of the population they represent. In manufacturing and business, in science, and no less in fores- try, partial knowledge is a normal state. The complete census is rare-the sample is commonplace. A ranger must advertise timber sales with estimated volume, estimated grade yield and value, esti- mated cost, and estimated risk. The nurseryman sows seed whose germination is estimated from a tiny fraction of the seedlot, and at harvest he estimates the seedling crop with sample counts in the nursery beds. Enterprising pulp companies, seeking a source of raw material in sawmill residue, may estimate the potential tonnage of chippablt material by multiplying reported production ::A;;“, of conversion factors obtamed at a few representative However desirable a complete measurement may seem, there are several good reasons why sampling is often preferred. In the first place, complete measurement or enumeration may be impossible. The nurseryman might be somewhat better informed if he knew 1 2 AGRICULTURE HANDBOOK 232,U.S.DEpT. OF AGRICULTURE the germinative capacity of all the seed to be sown, but the de- structive nature of the germination test precludes testing every seed. For identical reasons, it is impossible to measure the bend- ing strength of all the timbers to be used in a bridge, the tearing strength of all the paper to be put into a book, or the grade of all the boards to be produced in a timber sale. If the tests were permitted, no seedlings. would be produced, no bridges would be built, no books printed, and no stumpage sold. Clearly where test- ing is destructive, some sort of sampling is inescapable. In other instances total measurement or count is not feasible. Consider the staggering task of testing the quality of all the water in a reservoir, weighing all the fish in a stream, counting all the seedlings in a SOO-bednursery, enumerating all the egg masses in a turpentine beetle infestation, measuring diameter and height of all the merchantable trees in a lO,OOO-acreforest. Obviously, the enormity of the task would demand some sort of sampling procedure. It is well known that sampling will frequently provide the essen- tial information at a far lower cost than a complete enumeration. Less well known is the fact that this information may at times be more reliable than that obtained by a loo-percent inventory. There are several reasons why this might be true. With fewer observa- tions to be made and more time available, measurement of the units in the sample can be and is more likely to be made with greater care. In addition, a portion of the saving resulting from sampling could be used to buy better instruments and to employ or train higher caliber personnel. It is not hard to see that good measure- ments on 5 percent of the units in a population could provide more reliable information than sloppy measurements on 100 percent of the units. Finally, since sample data can be collected and processed in a fraction of the time required for a complete inventory, the infor- mation obtained may be more timely, Surveying 100 percent of the lumber market is not going to provide information that is very useful to a seller if it takes 10 months to complete the job. Populations, Parameters, and Estimates The central notion in any sampling problem is the existence of a population. It is helpful to think of a population as an aggregate of unit values, where the “unit” is the thing upon which the obser- vation is made,
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