COMPARiSGN OF THE SAMFMNG METHOQS USE IN FORESTRY Thais hr m Begum :3! Ph. D‘ MECHEGAN STATE UNEVERSEW 3cm: u Kuiow 39533 THESXS This is to certify that the thesis entitled Comparison of the Sampling Methods Used in Forestry presented by Don Lee Kulow has been accepted towards fulfillment of the requirements for Ph.D. Forestry degree in UW-o 6/104 Major professorU Date November 26, 1963 O~169 LIBRARY Michigan State University COMPARISON OF THE SAMPLING METHODS USED IN FORESTRY by Don L. Kulow AN ABSTRACT Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY 1963 9 - 3-61" x” L ' APPROVED U. I" acrk/(ijf' [RA v U ABSTRACT COMPARISON OF THE SAMPLING METHODS USED IN FORESTRY by Don L. Kulow In testing the accuracy of most of the field sampling techniques used in forestry, a large number of each kind of sample is needed for a thorough statistical analysis. This requires considerable time and ex- pense for the field work and is further complicated by changing stand con- ditions. To overcome these difficulties, three forest areas of 10.4 acres each were mapped to a scale of 1:120, and samples were drawn from these maps. These areas, of oak-hickory, sugar maple-beech, and ash-elm-red maple, represent the most common forest types found in southern Michigan. A series of 144 sampling methods was tested in each forest. These consisted of six areal plot shapes: (1) circular, (2) triangular, (3) square, (4) rectangular 1:2, (5) rectangular 1:4, and (6) rectangular 1:8. These were each sampled by a series of 1/5-, 1/10—, 1/20-, 1/40-, 1/80-, and l/160-acre plot sizes. Two types of Bitterlich point-samples were examined. One type was adjusted for edge bias, during the sampling process, according to Grosenbaugh's peripheral zone scheme, while the "unadjusted" point-sample was corrected by a formula adjustment during analysis. Both types of samples were tested for basal area factors of 5, 10, 20, 30, 40, and 50. Each of these samples was distributed throughout the forest popu- lations by random, systematic, and multiple-random—start techniques. An arbitrary number of sampling units was assigned to each forest, and sampled by basal area. The expected value, standard deviation, iii standard error, coefficient of variation, and the deviation or difference between the expected value and the true population mean were computed for each sample. To find the most precise sampling method, the coefficient of variation of all the samples was subjected to analysis of variance. As a check on the sampling ability of the various distributions, the deviation of the expected value from the known mean was also analysed. No attempt was made to evaluate the efficiency of any of the methods. The Lansing Woods is characteristic of an upland oak-hickory forest. The samples that gave the most accurate results are: (1) BAF-5 point- sample with an average coefficient of variation of 5.87 percent; (2) the one-fifth acre plots with an average CV of 6.11 percent; (3) the RAF-10 with an average of 7.28 percent; and (4) the one-tenth acre plot with 7.56 percent. The differences among these are insignificant. For unadjusted point-sampling, the random, systematic, and multiple-random start distribu- tions had an average of 11.12, 10.93, and 12.33 percent, respectively. For plot samples, in the same order, this was 13.44, 14.16, and 14.39 per- cent. All of the differences among these could be attributed to chance. However, an analysis of the difference between the expected value and the true mean showed that random distribution was significantly more precise than the other two. The Toumey Woods has the characteristics of a sugar maple-beech type. The most accurate samples with no important differences among them are: (1) BAP-5 point~samp1e with an average coefficient of variation of 4.58 percent; (2) RAF-10 with a CV of 5.66 percent; (3) one-fifth acre plot with 5.74 percent; and (4) one-tenth acre with 7.67 percent. The average coef- ficient of variation for the unadjusted point-samples are: 6.60, 7.55, and 8.70 percent, respectively, for the random, systematic, and multiple- iv random—start distributions. For plot samples this is 18.79, 20.60, and 18.97 percent, in the same sequence. None of the differences within point- sample distributions or within plot-sample distributions is significant, however, the systematic arrangement was highly superior to the others in the analysis of the difference between the expected and the true mean. The Red Cedar Woods is similar in characteristics to the ash-elm-red maple type. The samples that gave the most accurate results are: (1) BAF-lO point—sample with an average coefficient of variation of 5.38 per- cent; (2) BAF-5 with 7.33 percent; (3) the one-fifth acre plot with 8.57 percent; and (4) the one-tenth acre with 10.19 percent. The differences among these were insignificant. For the unadjusted point-samples, the random, systematic, and multiple-random—start distributions had average coefficients of variation of 10.26, 13.43, and 15.56 percent, respectively. The random method proved to be highly superior to either the systematic or the multiple-random—start system. The adjusted point-samples gave significantly less accurate results in all cases except on the Lansing Woods where the timber was relatively small, compared to the other two forests. Differences in the sampling accuracy due to sample unit size were very important. The inverse relationship between the coefficient of variation and the sampling unit size is illustrated by several regression equations. Plot shape was of no consequence in this study. COMPARISON OF THE SAMPLING METHODS USED IN FORESTRY by 5. 75/ Don L? Kulow A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY 1963 8 27074 ’é/Jf' ‘2'. fig 74 ACKNOWLEDGEMENTS To my wife, Jean, I wish to express the deepest and most heartfelt appreciation for the patience, encouragement, and assistance that she so unselfishly gave toward the completion of this paper. By the magnitude of her kindness, she has created an indebtedness that will take a life- time to repay -- an obligation that I gladly accept. My sincerest thanks goes to my guidance committee: Dr. T. D. Stevens, Dr. V. J. Rudolph, Dr. K. J. Arnold, and Dr. D. H. Brunnschweiler, for their advice and support. I am doubly grateful to Dr. Rudolph for accepting the responsibility of committee chairman so late in the program, and for correcting a multitude of errors in the manuscript. I would like to thank Dr. W. C. Percival for his generosity in giv- ing me the time and facilities to complete this work. Special thanks are also due to Mrs. Thelma Rice for compiling the data in a suitable form for analysis. To Dr. T. E. Avery, a friend, I would like to say thank you for the advice, encouragement, and the many lunch-hour consultations that con- vinced me that this project could be done. VITA Don L. Kulow Candidate for the Degree of Doctor of Philosophy Final Examination: Dissertation: Forest Sampling Outline of Studies: Major subjects: Forestry Minor subjects: Statistics, Geography Biographical Items: Born March 13, 1933, Reading, Michigan Undergraduate Studies: Bryan University 1952-54 Michigan State University 1954-1957 B.S. Forestry, 1957 Graduate Studies: Duke University 1959-1960 M. F. Forestry 1960 Michigan State University 1960-1963 Ph.D. Forestry, 1963 Experience: Planting Technician, Florida Forest Service, 1958—1959 Junior Forester, Florida Forest Service, 1959 Research Forester, USFS, Southern Forest Expt. Station, summers of 1960 and 1961 Teaching Assistant, Michigan State University, 1962-1963 Assistant Professor of Forestry, West Virginia University, 1963 to date. Member: Society of American Foresters American Society of Photogrammetry XI Sigma Pi Awards: Scholarship, Duke University Xi Sigma Pi Chapter I II III IV TABLE OF CONTENTS INTRowCTION . O O O O O O 0 REVIEW OF LITERATURE . . . . Early Studies . . . . . Point-Sampling . . . . Distribution of the Sampling Units Number of Sampling Units Sampling Efficiency . . Methods of Study . . . . METHODS AND PROCEDURE . . . . The Study Areas . . . . Field Work . . . . . . . Mapping . . . . . . . . Sampling Methods . . . . Required Size and Types of Sampling Units . The Sampling Process . . RESULTS AND DISCUSSION . . . Lansing Woods . . L . . Toumey Woods . . . . . . Red Cedar Woods . . . . SUMMARY AND CONCLUSIONS . . . LITERATURE CITED 9 o o o o. 0 APPENDIX C O O O O O O O O O Page 10 13 15 17 19 19 21 24 24 27 30 31 31 51 69 89 98 104 LIST OF TABLES Table 1. Distribution of trees by two-inch diameter classes . 2. Statistics for the random, one-fifth acre, circular plots for an arbitrary number of sampling units . . . 3. Dimensions of the circular, triangular, and square plots by plot Size 0 O O O O O O O I O O O O I l O O 4. Dimensions of rectangular plots by plot size . . . . 5. Peripheral zone widths for correcting edge-bias in point-sampling (after Grosenbaugh, 1958) . . . . . . 6. Calculated bias based upon forest area, perimeter and average tree radius (after Raga and Maezawa, 1959) . 7. Parameter summary, Lansing Woods, circular plots . . 8. Parameter summary, Lansing Woods, triangular plots . 9. Parameter summary, Lansing Woods, square plots . . . 10. Parameter summary, Lansing Woods, rectangular 1:2 . 11. Parameter summary, Lansing Woods, rectangular 1:4 . . 12. Parameter summary, Lansing Woods, rectangular 1:8 . . 13. Parameter summary, Lansing Woods, adjusted point- salnpling O I O O O O O O O O O O O I O O O O I O O O 14. Parameter summary, Lansing Woods, unadjusted point- smpling O I I O O I O I O I I O O C O O O O C O I O 15. Summary of the coefficient of variation data for the Lansing Woods,plot samples. . . . . . . . . . . . . . 16. Analysis of variance, Lansing Woods, plot samples . . 17. Average coefficient of variation, Lansing Woods . . . 18. Average coefficient of variation by distributions, LarlSj-ng WOOdS O O O O O O O O O I O O O O O O O O O O 19. Summary of the coefficient of variation data for the Lansing WOods, point-sampling . . . . . . . . . . . . Page 20 25 28 28 28 29 32 33 34 35 36 37 38 39 41 42 42 42 45 Table Page 20. Analysis of variance, Lansing Woods, point-sampling, coefficient of variation . . . . . . . . . . . . . . . . . . . 45 21. Analysis of the difference between the most accurate plot-samples and point samples, Lansing Woods . . . . . . . . . 48 22. Average deviations of the expected value from the known population parameter in percent, Lansing Woods . . . . . . . . 49 23. Analysis of variance deviation of the expected value from the true mean in percent, Lansing Woods, plot salnples O O O O O O O O O I O O I O O O O O O O O O O O O O O O 49 24. Analysis of variance deviation of the expected value from the true mean in percent, Lansing Woods, point- samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 25. Parameter summary, Toumey Woods, circular plots . . . . . . . . 52 26. Parameter summary, Toumey Woods, triangular plots . . . . . . . 53 27. Parameter summary, Toumey Woods, square plots . . . . . . . . . 54 28. Parameter summary, Toumey Woods, rectangular 1:2 . . . . . . . 55 29. Parameter summary, Toumey Icods, rectangular 1:4 . . . . . . . 56 30. Parameter summary, Toumey Woods, rectangular 1:8 . . . . . . . 57 31. Parameter summary, Toumey Woods, adjusted point-sampling . . . 58 32. Parameter summary, Toumey Woods, unadjusted point- Salnpling O O O O O O O O O I O O I O O O O O O O C O O O O O O 59 33. Summary of the coefficient of variation data, Toumey WOOdS, pIOt-SMDleS o o o o o o o o c o o o o o o o o o o o o o 60 34. Analysis of variance, Toumey WOods, plot-samples . . . . . . . 61 35. Average coefficient of variation, Toumey Woods . . . . . . . . 61 36. Average coefficient of variation data by distributions, Toumey WOods . . . . . . . . . . . . . . . . . . . . . . . . . 61 37. Summary of the coefficient of variation data, Toumey WOOdS O O O O I O O O O O O O O O I O O O I O O O O 0 O O O 0 63 38. Analysis of variance, Toumey WOods, point-sampling, coefficient of variation . . . . . . . . . . . . . . . . . . . 64 39. Analysis of variance for the random distribution of point samples, Toumey Woods . . . . . . . . . . . . . . . . . . 65 xi Table Page 40. Analysis of the difference between the most accurate plot- and point-samples, Toumey Woods . . . . . . . . . . . . . 66 41. Average deviation of the expected value from the true population mean, Toumey Woods . . . . . . . . . . . . . . . . . 66 42. Analysis of variance of the deviation of the expected value from the known parameter, Toumey Woods, plot saInpleS O O O O O O O O O O O O O O I O O O O O O O O O O O O O 67 43. Analysis of variance of the deviation of the expected values from the true parameter, Toumey Woods, point- sampling 0 I O O I O O O O 0 O O O O O O O I O O O O O O O O O 68 44. Parameter summary, Red Cedar Woods, circular plots . . . . . . 70 45. Parameter summary, Red Cedar Woods, triangular plots . . . . . 71 46. Parameter summary, Red Cedar Woods, square plots . . . . . . . 72 47. Parameter summary, Red Cedar Woods, rectangular 1:2 . . . . . . 73 48. Parameter summary, Red Cedar Woods, rectangular 1:4 . . . . . . 74 49. Parameter summary, Red Cedar Woods, rectangular 1:8 . . . . . . 75 50. Parameter summary, Red Cedar Woods, adjusted point- sampling 0 O O O O O O O O O O O O I O O O O O O O O O O O O O 76 51. Parameter summary, Red Cedar Woods, unadjusted point— Sarr'ples O O O O O I O O O O O O O O O I O O I O O O O O O O O O 77 52. Summary of the sampling error data for Red Cedar Woods, p10t sarnples I O O O O O O O I O O O O O O O O O O O O O O O 0 78 53. Analysis of variance, Red Cedar Woods, plot-samples . . . . . . 79 54. Analysis of variance, random distribution, Red Cedar Woods, plot-samples . . . . . . . . . . . . . . . . . . . . . . 79 55. Average coefficient of variation for all plot samples in the random distribution, Red Cedar Ibods . . . . . . . . . . 81 56. Summary of the sampling error, Red Cedar Woods, point sampling 0 O O O O O O I O O O O O O O O O O O O I O O O O O O 82 57. Analysis of variance, Red Cedar Woods, point-sampling . . . . . 82 58. Analysis of variance, random distribution, Red Cedar WOOdS , pOint-SMPIGS o o o o o o o o o o o o o o o o o o o o o 83 Table 59. 60. 61. 62. 63. Analysis of the differences between the most accurate plot-sample and point-sample, Red Cedar Woods . . . . Average deviation of the expected value from the true population parameter in percent, Red Cedar Woods . . Analysis of variance of the deviation of the expected value from the true parameter, Red Cedar Woods, plot- Salnples O I O O O O O I O O I O O O O O O O O C O O O xii Page Analysis of the variance of the deviation of the expected value from the true parameter, Red Cedar Woods, point- sampling 0 O O O O O O O O O O O O O O O O I O I O O I O O O 88 Summary of the coefficient of variation parameters for the most accurate plot-samples and point-samples . . . . O O O O 96 1. LIST OF FIGURES Figure Grosenbaugh's peripheral zones for adjusting edge-bias in point-sampling. Trees falling in these zones re— ceived the indicated weight . . . . . . . . . . . . . . . . . Diagram showing row and block arrangement of the one- chain square plots . . . . . . . . . . . . . . . . . . . . . Diagram showing the method of obtaining the rectangular coordinates for each tree. The abscissa a is read from the steel tape, starting at the base line. The ordinate .b is measured with a rangefinder.. . . . . . . . . . . . . The relationship between the coefficient of variation in percent and the reciprocal of the square root of the plot size in square feet. From the equation: CV = 457.57 (plot size)’§, Lansing Woods . . . . . . . . . . . . . . . . The relationship between the coefficient of variation in percent and the reciprocal of the plot radius factor. From the equation: cv = 19.905 - 15.351(Prf)'1, Lansing Woods. . . . . . . . . . . . . . . . . . . . . . . . . . The relationship between the coefficient of variation in percent and the reciprocal of the square root of the plot size in square feet. From the equation: CV = 670.399 (Plot size)" , Toumey Woods . . . . . . . . . . . . . . . . . The relationship between the coefficient of variation in percent and the reciprocal of the square root of the plot size in square feet. From the equation: CV = 649.965 (Plot size)’§, Red Cedar Woods . . . . . . . . . . . . . . The relationship between the coefficient of variation in percent and the reciprocal of the plot radius factor. From the equation: cv = 23.895 - 16.74(Prf)‘1, Red Cedar WOods. . . . . . . . . . . . . . . . . . . . . . . . . 22 23 44 46 62 80 84 CHAPTER I INTRODUCTION The choice of sampling designs used for estimating timber is usually based more on personal preference than on the precision of the method in- volved. Various studies have compared two or more of the favored samp- ling designs, but no attempt has been made to compare all of the major sampling systems on the same forest. In order to compare the precisions of the various sampling tech- niques on an equivalent basis, a large number of each kind of sample is required for thorough statistical analysis. The use of an actual forest stand requires a great deal of field work and is complicated by changes in the stand condition with time. These factors make the determination of the most accurate means of forest sampling by field measurements a very costly procedure. This study provides a comparison among the most important sampling techniques used in forestry by mapping the position of each tree in a given population and drawing samples of basal area, (cross-sectional area of the tree at 4.5 feet in square feet), from this map. Several forest areas, representing the most common combinations of species and site con- ditions in the southern lower peninsula, are studied. A number of sampling units are assigned to each area, and the most frequently used forest samp- ling procedures are applied to these units. The results are analyzed with respect to the accuracy of the sampling method involved. No attempt is made to ascertain the efficiency of any of these methods through cost or time studies. The project is limited to the statistical precision of for- est sampling. CHAPTER II REVIEW OF LITERATURE Studies designed to determine the best method of sampling forest populations have been conducted early in forest management history. With the use of statistical analysis applied to sampling error, tests could be made on the precisions of plots of various sizes and shapes; on systems using different numbers and distributions of sampling units; and on the efficiency of various sampling designs. Early Studies Early studies were confined largely to comparisons between two or more of the most popular types of forest sampling units. These were usually strip samples and plot samples. Wright (1925) compared these on an actual forest stand. He used strips ten chains long and one chain wide. .Tally was kept by two-chain intervals, and these segments were treated as the line plots. Statistically, the line plots gave the best results. Candy (1927) used systematic strips and line plots, and con- cluded that only line plots could give sampling with a determined level of accuracy. Robertson (1927) made the same type of test and also dis- carded strip sampling as the least precise. He favored a large number of small plots, evenly spaced over the area rather than a few large plots. From time to time during the sampling process, he would analyse the re- sults to see if'a sufficient number of units had been taken to obtain his estimates within the required limit of accuracy. This is essentially the same system presented by Freese (1962) based upon a formula by Schumacher and Chapman (1948). In New England, Goodspeed (1934) found that the two methods of sampling gave essentially the same results. By 1932, it was apparent that size and shape of the sampling unit were also important to the results of the sample. Experiments with agri- culture crops indicated that the long narrow plots tended to be the most effective for reducing the variability of yields among samples (Clapham, 1932; Justesen, 1932; Kalamkar, 1932). In California, Basel (1938) took a 100 percent inventory of nine sections of redwood forest by two and one-half acre plots (2.5 x 10 chains). He combined these into different "universes" of plotsl/ of various shapes and sizes, and found that the long narrow plots (2.5 x 40 chains) gave the most accurate results. In a later article (1942), he stated that strip plots would give the best results only when these were of maximum length for the area sampled. Both Hasel (1938) and Yates and Zacopanay (1935) stated that an increase in plot size results in an in- crease in the variance when total plot area is fixed. This has been sup- ported in later studies where variance decreased as plot size increased (Barton, 1956; Freese, 1961, 1962; Johnson and Hixon, 1952; Mesavage and Grosenbaugh, 1956; Meyer, 1948). Ecologists were also interested in the best way to sample forest populations. Although the objectives of ecologists and foresters differ, both methodology and research results overlap to a considerable degree and there is an exchange of ideas in both directions (Lindsey, 1956). l/ "Universe of plots" comes from Palley and O'Regan's (1961) concep— tional forest population in which a forest is thought to be made up of two distinct components: 1), The physical population of trees growing on a particular forest; and 2), The partitioning of these trees into elementary units according to the system of sampling used. The conceptional population or universe of plots is the aggregate of all these elementary units. Many ecological methods of sampling refer only to tree-area rela- tionships (Cottam, 1947; Cottam and Curtis, 1949). However, some of these studies did make use of standard forest sampling methods for mea- suring basal area. Borman (1953) took a complete census of 4.3 acres of the Duke Uni- versity forest by squares of two by two meters. He arranged these into 28 other universes of plots of different shapes and sizes for comparison. Analysis showed that the longest plots (10 x 140 meters) gave the best results providing the long axis crosses the population contours. However, he pointed out that the greater periphery in narrow plots would tend to produce a greater edge—errorg/ than square plots of equal area. In California, Johnson and Hixon (1952) inventoried a forty-acre tract of Douglas-fir by plots measuring one-half chain by one-half chain. These were then combined into seven other universes of plots varying from one-half to one chain wide, and from one to six chains in length. Long narrow plots proved to have the lowest coefficient of variation. This parameter decreased as the plot size increased. Freese (1961) also points out the usefulness of the long narrow plots for reducing variability. Meyer (1948) strongly recommended long narrow plots placed at right angles to the contour of the land. 3/ Errors of omission or inclusion of individuals occurring near the edge of a sampling unit of fixed area. Point-Sampling For the first twenty-five years of forest sampling, the use of the fixed area plot (i.e. circles, squares, rectangles, strips) was firmly fixed in the minds of American foresters. In 1947, a new concept in for- est sampling was introduced in Germany by an Austrian forester named Bitterlich (1947, 1948). This method did away with the necessity of de- termining plot boundaries. The average basal area of a forest could be obtained by simply counting the number of trees whose bole diameter ex- ceeded a selected angle. The method became known as the Bitterlich system or the point-sampling system, and was introduced into this country by Grosenbaugh (1952). He expanded the use of the method to measure other forest characteristics such as board- and cubic-foot volumes (Grosenbaugh, 1955), and later derived its probability theory (Grosenbaugh, 1958). Various types of instruments have been devised to expedite the use of the system (Bitterlich, 1948; Bruce, 1955; Lemmon, 1958; Stage, 1958). The literature is replete with explanations concerning the mechanics and theory of the system (Afanasiev, 1957; Grosenbaugh, 1958; Lemmon, 1958; Palley, 1962), so a complete description will not be given here. With the acceptance of point—sampling as a valid means of estimating forest parameters, a renewed interest in comparative studies was revived. In ecology, Shanks (1954) found the Bitterlich method superior to standard plots for classifying vegetation. Rice and Penford (1955) com- pared point-sampling with the ecologist's paired-tree method and found it to be the most precise means of measuring average basal area. Barton (1956) mapped a twenty-acre forest to scale, and sampled it with point- sampling, 1/5-, 1/10-, and 1/40-acre plots and strips. Point-sampling proved to be the most accurate. Sudia (1954) found that the method ranked third in a comparison with five other methods. He pointed out, however, that the system that works best in one area may not work in an- other. Lindsey gt 31” (1958) mapped 20 acres of Indiana hardwoods, and found point-sampling the fourth best in a comparison of nine methods. In forest research, Husch (1955) tested the accuracy of point- sampling of three different angles of inclusion, namely, 52.09 minutes, 104.18 minutes, and 208.38 minutes for basal area factors of 2.5, 10, and 40, respectively. He arbitrarily selected one hundred points in a forty- acre tract for study. However, he did not allow any tree to appear more than once in the sample, so as the angle decreased —- thus including a larger number of trees -- he reduced the number of sampling points to prevent duplication. This resulted in a biased estimate of the error, for which he did not make corrections (Grosenbaugh, 1955). Under the circumstances, his results showed the 104.18-minute angle (RAF-10) to be the most accurate. ' A study in southern Illinois inventoried two forests of eight and 23 acres by plots two by two chains in size (Deitschmamt, 1956). Basal area was recorded by small and large poles and as sawtimber trees. Deviations from the known parameter varied from 20 to 50 percent. This may have been caused by failure to check boundary trees carefully. Avery (1955) in Arkansas found that estimations in volume by the Bitterlich system ranged from a negative two percent to a positive 20 percent. Afanasiev (1958) also made point-sampling volume comparisons against one-chain strips and one-fifth acre plots. Deviations from the known volumes for point sampling ranged from -22 to 104 percent in individual compartments. For the entire 296-acre forest, the difference between the estimated volume and the actual value was only 1.5 percent. Borgeson 23 31., (1958) found differences between individual point-samples and one-fifth acre plots of -25 to 88 percent for basal area. Trappe (1957) also found the Bitterlich system to be much more variable than fixed area sampling. However, Afanasiev (1958) and Borgeson £3 31., (1958) concluded that point-sampling compares favorably with conventional methods when the time required for sampling is considered. Grosenbaugh and Stover (1957) have attested to the ease of taking Bitterlich plots. On a continuous forest inventory program in southeast Texas, they compared the method with concentric one-quarter acre circular plots. Statistical analysis showed point-sampling to be inferior to plot- sampling, as far as accuracy is concerned, but required much less time to execute. Cox (1961L_in testing three basal area factors (10, 18, and 27), found that the RAF-27 had the same sampling error as one-fifth acre plots, but required the measurements of only 58 trees as compared to 235 trees on the fixed radius system. A large body of empirical evidence has shown point-sampling to be a valid means of estimating basal area (Bell, 1957; Ker, 1957; Malain, 1961; Stage, 1962; Lindsey, 1956; Warren, 1960). This has led to the increased use of the method at all levels of forest sampling (Bryan, 1959). In applying point-sampling, some procedure had to be used to correct for the bias created by the edge of the forest. The point-sampling pro- cedure assumes that each tree in the forest is surrounded by an imaginary ring equal to the product of its diameter and some constant K called the plot radius factor. Grosenbaugh (1958) says that the probability of any tree being included in the sample is equal to the ratio of its K x diameter-area to the total area of the forest. He calls this 8 "probability proportional to size." In cases where the tree is closer to the edge of the forest than its imaginary plot radius, a proportion of its circle will lie outside the forest. Thus: . . bias arises because random or systematic loca- tion of sample points . . . is limited by the tract boundaries, so that the trees with enlarged rings or diameters projecting beyond the boundary have less chance of being sampled than their size . . . . would indicate. To eliminate this bias, the portion of each sample tree ring that lies outside the forest must be added to the sample. Grosenbaugh specifies peripheral zones a little wider than the radius of the maximum tree ring expected. These zones are laid out prior to sampling and are illustrated in Figure 1. Trees in the interior zone will receive a unit weight or will be counted once; trees in the size zones twice; and trees in the corner zones four times. Palley (1960) recognized this bias and attempted to correct for it by simply moving the forest boundaries out to include all of the tree rings. In a later study (1961), he used the weighting system mentioned above. Haga and Maezawa (1959) took exception to both of these proced- ures and outlined a technique for obtaining an approximate value of the bias. According to them, this bias is a function of the average tree radius (r) on the forest, the plot radius factor (p) used,and the area and perimeter of the forest. This bias is negligible on a forest of more than 25 acres. Using a plot radius factor of 50 and an average tree radius of 3.94 inches they constructed a graph showing the relative bias (3) for small tracts of land having various areas and perimeters. The percent bias for other values of p and r can be computed with the follow- ing equation: Corner Side Zone Corner Zone Zone 4 2 4 Side Interior Side Zone Zone Zone 2 1 2 Corner Side Zone Corner Zone Zone 4 2 4 Figure l. -- Grosenbaugh's peripheral zones for adjusting edge- bias in point-sampling. Trees falling in these zones received the indicated weight. 10 Percent Bias = (e) (P/50) (r/3.94) (1) the relative bias where: e P = the plot radius factor r = the average tree radius in inches Distribution of the Sampling Units In distributing the sampling units or points within the forest popu- lation, foresters have persisted in using systematic sampling designs. Accurate estimates of error are based upon purely random samples and no one today has managed to come up with a mathematically sound formula for applying this estimate to systematic techniques (Palley, 1961; Shiue, 1960). When a forester attempts to sample a population systematically, he mistakenly assumes that nature has been kind enough to randomize for him. The main argument for systematic sampling is that the results are more accurate than random methods; however, Finney (1949) showed that when there is a strong marked pattern of variation in the population, systematic sampling may give much less accurate results than random methods. Smith and Ker (1957) state that it is impossible to predict this pattern from a sample. Candy (1927) pointed out the importance of a random sample when he said: Any method of survey for which it is possible to cal- culate the accuracy of the estimate obtained is very much superior to methods in which the accuracy of the estimate is doubtful and not at all calculable. Wright (1925) said that the most important use of statistical techniques in cruising was to estimate the error of the cruise. The use of random as against systematic distribution has been urged by many investigators in sampling (Borman, 1953; Cochran, 1953; Finney, 1948; Hasel, 1938; Meyer, 1948). 11 In spite of this, many researchers believe that random samples are less accurate and more expensive than systematic samples. To support this belief, a number of reports have shown that systematic samples are superior to the random technique with respect to precision. Osborne (1942) found that systematic sampling was six times more accurate than random sampling. Meyer (1948) reported that a systematic arrangement of plots would give better coverage of the population,and Hasel (1938) said that the estimation of the mean would also be much more precise. Mesavage and Grosenbaugh (1956) studied three plot arrangements, (4 x 4 and 2 x 8 systematic and random), on twelve forest tracts of 25 acres each. They found that the systematic design was the most precise on nine of the areas, while the random was best on only three of the twelve. Bourdeau (1953), on the other hand, said that the loss in precision from random samples is slight, and more than offset by the fact that estimates of error can be made. He also believes that random sampling requires little more in the way of field work. It has been pointed out that differences in the precisions of vari- ous sampling designs can be attributed to the erroneous assumption that all universes of plots of a given population have a "normal" distribution (Smith and Ker, 1957). To illustrate this, they arranged the data from a study in California (Johnson and Hixon, 1952) by volume classes for each plot size and showed that the distributions ranged from a "Poisson" for very small plots, to a "rectangular" for plots as large as one acre. Only samples with plot sizes of about one-fifth acre could be applied to ”normal" analysis techniques. Several ways have been suggested for obtaining unbiased estimates of the sampling error when systematic sampling is used. Osborne (1942) 12 mentioned the mean-squared-successive-differences, where the difference between successive sampling units are squared, and the estimate of the standard error is made by the following equation (Moore, 1955): Estimated Standard Error = [(Successive differences)2/n-l/n]% (2) where: n is the number of sampling units. However, Meyer (1956) has pointed out that the exact mathematical distri- bution of this parameter is not known, so tests of significance are not possible. Another possible compromise between random and systematic sampling has been suggested by Schumacher and Chapman (1948) and called "propor- tional sampling of blocks of diverse but unknown areas", or stratifica- tion. With this method, the forest is stratified or divided up into rather homogeneous blocks according to some characteristic such as forest condi— tion, age class, volume class, etc. Each block is treated as a separate population from which at least two random samples are drawn. Mudgett and Gevorkiantz (1934) used this method in studying the reliability of early forest surveys. Recently, stratification has been advocated by Bickford (1961), Freese (1962), and Orr (1959) as a means of increasing the accur- acy of sample estimates and decreasing the number of sampling units required. The multiple-random-start method offered by Shiue (1960) is still another way of solving the random vs. systematic problem. A set of sys- tematic samples with one random start is considered to be a cluster of plots. Two or more such sets of plots can be analyzed by cluster tech- niques, and the problem of estimating the error is solved while keeping 13 some resemblance of a systematic cruise. But, the exact number of random starts to use is not known. Too many can be very inefficient, and too few may lower the precision of the estimate by cluster analysis. Also, the distribution of the individuals within the population so affects the sampling error that the results may be inferior to either systematic or random procedures (Smith and Ker, 1957). Number of Sampling Units Required Early sampling problems in forestry consisted of measuring some ar- bitrary fixed percentage of the forest area. The most common proportion was the ten-percent cruise. With the institution of the nationwide con- tinuous forest inventory program, and the initiation of management on large forest areas, it became apparent that the "proportion of the total" sample was no longer feasible. Preston (1934) objected to the use of a preconceived intensity of cruise and urged the use of statistics to determine the size of the sample. In an attempt to base the number of units needed upon some char- acteristic of the population, Girard and Gevorkiantz (1939) published a table of coefficients of variation that varied with stand density and uniformity. Barton and Stott (1946) drew up a set of pictorial charts giving the proportion of the area needed in a sample for stands of dif- ferent uniformity and density at several levels of accuracy. It was based on the equation: Percent Cruise = 400 (plot size) (CV) (3) (total area) (percent accuracy) + 4(plot size) (CV) where a coefficient of variation (CV) came from the tables of Girard and Gevorkiantz (1939). Meyer (1949) said that the variation in timber volume 14 is affected by volume per acre, size and shape of the sampling units, and the efficiency of the cruising design. He measured 68 one-fifth acre plots on a 54-acre forest, and produced a table giving the number of units needed for six different accuracy levels on areas from 50 to 10,000 acres, and coefficients of variation of 15, 30, and 60 percent. A young dense stand was measured at the CV level of 15 percent while an old growth stand with a few large scattered trees used the 60 percent value. The subjectiveness of the foregoing methods is apparent. As early as 1935, the number of sampling units needed was thought to be a function of the sampling error (Schumacher and Bull, 1935). Sampling error itself is a function of variations in the population, size and shape of the sampling unit, and the number of sampling units used. According to Schumacher and Chapman (1948) this number of plots may be found by: n = (S/Si)2 (4) where: n the number of sampling units required U3 II the standard deviation of the population (usually) estimated from a pre-sample) SE = the accuracy desired (frequently expressed as a plus or minus some proportion of the sample mean) Borman (1953) for example, used a plus or minus ten percent of the mean, with a certainty of being right two out of three times. Freese (1962) has expanded the use of formula (4) with respect to the use of a "t" value for various limits of precision. This system is commonly accepted by biometricians and statisticians as a valid means of arriving at n. Other methods designed to estimate the number of sampling units are the ratio of sample variance to sample range (Snedecor, 1959; Freese, 15 1962) and the range-mean ratio (Allen and Mongren, 1960). The usefulness of the latter has been demonstrated in using the Bitterlich system, but its mathematical theory has not been proven. The most common procedure in comparative analysis research has simply been to select an arbitrary number of samples from a population and base all calculations on this fixed number (Barton, 1956; Grosenbaugh and Stover, 1957; Husch, 1955; Johnson and Hixon, 1952; Lindsey e: 21., 1958). Sampling Efficiency Efficiency of sampling may be defined as getting the greatest amount of information for the least expenditure of time and effort (Lindsey gt 21., 1958). Finney (1948) stated that, the efficiency of one sample with respect to another is expressed as a ratio of their respective precisions. According to Cochran (1953),it is a function of the squared sampling er- rors expressed as a percent. Thus: Efficiency (percent) = (Standard survey error in percent)2 (5) (Other survey error in percent)2 When time has been measured or cost of sampling has been computed, the formula becomes: Efficiency (5) = (Standard survey error in percent)2 (Standard survey cost)2 (Other survey error in percent)2 (Other survey cost)2 (6) The choice of which survey data should be in the denominator is purely arbitrary (Grosenbaugh and Stover, 1957). 16 Using formula 6, Lindsey 33.21:! (1958) found that the Bitterlich method was fourth in efficiency for sampling basal area in a study with eight other methods. Methods called the rangefinder-circle-Bitterlich, the tenth-acre-rangefinder-circle, and the one-fifth acre strip, all ranked ahead of the full Bitterlich method. Johnson and Hixon (1952) in comparing only circular and rectangular plots, found that the one— by three-chain unit was the most efficient. Among only the circular plots, the one-quarter-acre plot was best. Mesavage and Grosenbaugh (1956) tested four rectangular plots of 0.1, 0.2, 0.4 and 0.8 acres,respective1y. They found that coefficient of variation decreased as plot size increased, but that the cost also increased. They concluded that in dense homogene- ous stands, the one-tenth acre or smaller plots would be the most effi- cient, while in sparse stands of large timber, plots larger than 0.8 acre would be best. In studying the efficiencies of sampling distributions, Mesavage and Grosenbaugh (1956) showed the 1:4 systematic grid is more efficient than either the 1:8 grid or the random method. Johnson (1949) tested a series of cluster arrangements and found that taking more than three plots in a cluster was inefficient, and that fewer than this would have given the same precision. Grosenbaugh and Stover (1957) stated that because of the small amount of work required to use point-sampling, it is more efficient than plot sampling. Husch (1955), in evaluating a number of point-sampling angles, found the 208.38-minute angle to be more efficient than the more precise 104.18-minute angle. 17 Methods of Study Most studies of forest sampling techniques have been carried out on an actual forest. The problems inherent in this method have been stated earlier. To avoid this difficulty, several researchers inventoried the entire forest by some basic unit of sampling, such as a rectangle or a square plot. The precision of various sampling distributions, plot sizes, and shapes could then be studied by combining these basic units into uni- verses of plots with other dimensions. Grosenbaugh and Stover (1957) used one-tenth acre squares; Deitschaman (1956) used 2- by 2-chain squares; Hasel (1938) used 2.5- by lO-chain rectangles; and Borman (1953) land Bordeau (1953) used 2-meter squares. An approach that offers much in that it allows the study of all types of plots including the Bitterlich system, is the one that maps each tree in the forest to scale. Samples are then drawn from this fixed popu- lation and analysed. Cottam (1953) constructed an artificial population on paper with a table of random numbers. He later mapped three small for— est areas of less than six acres in order to study sampling efficiency for ecological purposes. The method worked so well that Barton (1956) mapped a 20-acre forest to a scale of 1:36. He used all trees over four inches in diameter and measured their coordinates to the nearest foot. In mapping, however, he grouped the trees by 3-inch diameter classes. Lindsey £3 21., (1958) used this same map in their work in ecology. Palley and O'Regan (1961) have recently applied this mapping tech- nique to the speed of electronic computers. They obtained the coordinate measure to the nearest meter for every tree on a two and one-half acre forest. These were then committed to the memory of the IBM-701 Computer. By inserting a formula for the plot radius factor and the coordinates of 18 the plot centers, they could check the accuracy of any basal area factor. The advantage of this system is that all possible points in the forest can be sampled, and the method is without computing error. Memory systems of these machines are so limited, however, that only very small forests may be used. .‘ 0. 0a As 61‘ in ha .46 T» o.“ CHAPTER I I I METHODS AND PROCEDURE The Study Areas Three forest areas were picked for stucbn to determine the most ac- curate method of sampling to be applied in each one. These are charact- eristic of the most common timber and site conditions in the southern half of Michigan's Lower Peninsula. Each area contains 10.4 acres, measuring eight by thirteen chains (528 by 858 feet), and is centered in a larger tract of timber. Table 1 shows the frequency of trees by two-inch diam- eter classes. The Lansing Woods. -- This study area is located at Maple Rapids, Michigan, and is owned by the Lansing Company. Characterized as a white oak-red oak-hickory type, the stand occupies a well-drained upland site. Associate species are sugar maple, American elm, red maple, basswood, and beech. The average basal area is 78.10 square feet per acre, and the av- erage tree diameter is 11.3 inches. Fires have not occurred in the stand in the past ten years and grazing has been very light. Selection cuttings hae been made from time to time. The last cut was made in April, 1962, and removed some of the white oak on a small part of the study area. The Toumey WOods. -- This relatively undisturbed area is located in the east-central part of the campus. It is an old-growth stand of beech- sugar maple on the moist fertile soils of a morainal hill. Other species in this stand are red and white oaks, American elm, basswood, and black 19 20 Table 1. -- Distribution of trees by two-inch diameter classes. Diameter Class Lansing Toumey Red Cedar inches Woods Woods Woods 6 291 163 136 8 247 124 138 10 192 88 129 12 132 73 113 14 118 85 107 16 84 80 85 18 51 89 65 20 37 70 49 22 10 56 31 24 4 38 ll 26 2 27 15 28 1 21 11 30 1 9 3 32 6 1 34 7 2 36 2 1 38 2 3 40 2 3 42 3 Total 1170 942 906 cherry. The average basal area per acre is 116.92 square feet, and the average tree diameter is 15.3 inches. Fires, grazing and logging have not occurred in this forest. The Red Cedar Woods. -- This is a timbered area along the Red Cedar River on the western boundary of the Michigan State University campus. The stand is a variant of the SAP type 39 and is characterized princi— pally by green ash-elm—red maple. Associate species are basswood, syca- more, cottonwood, and slippery elm. The site is moist to wet, and is an elongated area bordering a stream that frequently floods. Drainage is poor. The average basal area of the study tract is 91.60 square feet per 21 acre, and the average tree diameter is 13.8 inches. Tree distribution shows a distinctive clumping characteristic that does not exist in the other two tracts. Fires, grazing and cuttings have not occurred during the known history of the stand. Field Work The field work consisted of measuring the rectangular coordinates of each tree to the nearest foot and tree diameters to the nearest one-tenth inch. In the Lansing and Red Cedar areas, a base line 13 chains (858 feet) long was run along the northern boundary. Stakes were set at one- chain intervals, and strings were run from these markers, across the for- est, perpendicular to the base line. The distance between the strings was checked frequently with a compass and tape. In the Lansing Woods, this job was facilitated by stakes from a previous study marking the cor- ners of each one-chain square. These squares were numbered in 13 rows with eight blocks per row as shown in Figure 2. In order to obtain the rectangular coordinates of each tree, a steel tape was laid out between successive rows, and the abscissa (a) of each tree was read on the tape, to the nearest foot, at a point where a line from the center of the tree was perpendicular to the tape (Figure 3). The ordinate (b), or the distance from the tree to the tape was measured with a 6-inch base rangefinder. This distance was measured tangent to the circumference of the tree. Where underbrush or other trees obscured the view, the ordinate was obtained with a cloth tape. As each tree location was recorded, the tree was assigned a number for that row and block, and its diameter was measured to the nearest one-tenth inch with a diameter tape. Blocks 1 2 3 4 5 6 7 8 Rows q 10 11 12 13 Figure 2. -- Diagram showing row and block arrangement of the one-chain square plots. 23 Row 2 Row 2 Block 1 BlOCR 1 steel tape Base line a) Row 1 Row 2 Block 1 Block 2 Figure 3. -- Diagram showing the method of obtaining the rectangular coordinates for each tree. The abscissa a k; read from the steel tape, starting at the base line. The ordin- ate 2 is measured with a rangefinder. 24 The same procedure was used for Toumey Woods, except the blocks were only sixty-foot squares. In a previous study, iron stakes had been set at these intervals, and since any subdivision scheme is only an aid in measuring and mapping tree locations, this size block was used. Mapping The forest areas were mapped on graph paper (ten squares per inch) to a scale of 1:120. The location of each tree center was circled in ink and the tree number, diameter, and basal area to three decimal places were placed adjacent to the circle (Figure in Appendix A). Every other tree located on the boundary of the forest was discarded. Each completed map measured 4.4 by 7.5 feet. Sampling Methods Number of Samplinngnits. -- Since comparisonsamong the study areas .were not made, an arbitrary number of sampling units was assigned to each forest. As a guide, the number of points used had to give an es- timate of the population mean of, at least, a plus or minus ten percent of the sampling mean at the 68 percent level of accuracy. This was ap- plied to the one-fifth acre§/ circular plots on a random design. Twenty-five plots were drawn at random from the Lansing Woods. The estimate was well within the required limits; however, only the last plot drawn was discarded to make it easier to apply the number of plots to a systematic grid. Twenty plots were selected at random in Toumey Woods. The results were well within the required limits, so the number was g/ Correspondence with Mr. Harold W. Kollmeyer of the Michigan Depart- ment of Conservation indicated that this is the most common size and shape of plot used in state forest inventory work. 25 reduced to 16 for the Red Cedar Woods. The resulting calculations failed to meet the standard of accuracy, so two additional plots were drawn. The results are presented in Table 2. The same number of plots was used for each distribution under investigation in this study. Table 2. -- Statistics for the random, one-fifth acre, circular plots for an arbitrary number of sampling units. Lansing Toumey Red Cedar Statistics Woods Woods Woods Number of Samples 24 20 18 Mean per Acre (sq. ft.) 77.16 117.34 92.71 Standard Deviation (sq. ft.) 17.35 35.88 29.14 Standard Error (sq. ft.) 3.54 8.02 6.87 Level of Accuracy (percent) 5 7 7 Distribution of the Sample Unit. -- Several methods of distributing the sample units were mentioned in the literature review. Of primary im- portance, of course, is the comparison between the random and systematic distributions. The multiple-random-start (MRS) method is a logical com- promise between the two, and was also used. Cluster arrangements, such as those suggested by Johnson (1948), or stratification, as proposed by Schumacher and Chapman (1948), are not included in this study, because of the small size of the forest. In selecting the location of the random samples, each forest is con- sidered to be made up of a population of plot centers equal to the prod- ucts of the sides of the forest (Palley and O'Regan, 1961). In this study, each woods measures 528 feet by 858 feet for a total of 453,024 possible plot centers. The ordinate and abscissa of the center of each sample was chosen by drawing numbers, with replacement, from a table of random digits. If a point fell closer to the plot boundary than the 26 radius of the one-fifth acre plot (52.7 feet), that point was moved far enough to include all of this unit within the forestré/ These points were numbered and marked on the maps with ink (see Appendix A). In the systematic distribution, the distance between samples was computed by the equation: d2 = A/n (7) where: d = grid distance between plot centers in feet A = total forest area in square feet n = number of units in the sample The grid distances for the Lansing Woods, Toumey Woods, and Red Cedar Woods is 138, 150, and 159 feet respectively. The grid was placed on the map in a random manner, by drawing the coordinates of one of the sample units from a table of random numbers. The other units were lo- cated systematically from this first point according to the prescribed grid. This randomization of the first point conforms with the considera- tions of the multiple-random—starts as suggested by Shiue (1957). The multiple-random start distribution was designed, and analysed, according to Shiue (1957), on the basis of a conceptional forest made up of 104 one-tenth acre square plots. The center of these units serve as the center locationsfor all sampling units. The same number of samples was measured in this distribution as in the random and systematic systems. Four random-starts were used in the Lansing and Toumey Woods and three in the Red Cedar. 3/ This is in accordance with the field sampling procedures used by the U. S. Forest Service in sampling timber on the nationwide forest survey. 27 Size and Types of Sampling Units The size, shape, and dimensions of the sample units used in this study are given in Tables 3 and 4. These sampling units include those most frequently used in forest sampling and inventory work. No attempt was made to evaluate those units designed primarily for ecological or botanical research or to test the accuracies of methods like the Bitter- lich line-sampling system, since it has not been used in forestry work. Strip samples were also omitted, because of the small size of the forests, and past studies indicate their lack of precision. In place of strip samples, rectangular plots with various short-to-long side ratios were tested. In order to analyse certain sampling arrangements by regression tech— niques, as suggested by Palley and O'Regan (1961), the areal plot sizes were decreased in a reciprocal logarithmic fashion. These are 1/5, 1/10, 1/20, 1/40, 1/80, and 1/160 of an acre. All plots of a given shape are concentric for these sizes. In point-sampling, the Six basal area factors used were 5, 10, 20, 30, 40, and 50. Plot radius distances for each of these are given in Ap- pendix B. Both Grosenbaugh's and Haga and Maezawah peripheral adjustment systems were tested (see page 8 of the Literature Review). To weight the trees, according to Grosenbaugh, the peripheral zones of Table 5 were used. These are based on the large tree diameters of 252 42, and 42"for the Lansing Woods, Toumey Woods, and Red Cedar Woods,respectively. The corners of the zones were marked on the maps with pins and moved for suc- cessive basal area factors. These samples are referred to as the adjusted point-samples. 28 Table 3. -- Dimensions of the circular, triangular, and square plots by plot size. Plot Plot Size in Acres Shape 1/5 l/10 1/20 1/40 1/80 1/160 Plot Dimensions in Feet Circular (radius) 53 37 26 19 13 9 Triangular (side) 142 100 71 50 36 25 Square (side) 93 66 47 33 23 17 Table 4. -- Dimensions of rectangular plots by plot size. Short to Long Plot Size in Acres Side Ratio 1/5 1/10 1/20 1/40 1/80 1/160 1:2 66:132 47:93 33:66 23:47 16:33 12:23 1:4 47:187 33:132 23:93 16:66 12:47 8:33 1:8 33:264 23:187 16:132 12:93 8:66 6:47 Table 5. -- Peripheral zone widths for correcting edge—bias in point- sampling (after Grosenbaugh, 1958). Toumey and Red Basal Area Factor Lansing Woods Cedar WOods (feet) (feet) 5 97 184 10 69 115 20 49 82 30 40 67 40 34 58 5O 31 52 29 At the completion of the sampling by weighting, the process was re- peated, giving each tree a count of one, regardless of its position in the forest. The percent bias was then calculated according to Haga and Maezawa. Average tree radii for the Lansing Woods, Toumey Woods, and Red Cedar Woods are 5.65, 7.65, and 6.90 inches,respectively. The resulting bias is given in Table 6 and is always positive. These values were ap- plied directly to all estimates in this method of sampling. These samples are referred to as the unadjusted point-samples. Table 6. -- Calculated bias based upon forest area, perimeten and average tree radius (after Haga and Maezawa, l959)a. Basal Area Plot Radius Lansing Toumey Red Cedar Factor Factorb Woods Woods Woods feet/inches (percent) (percent) (percent) 5 3.889 .22 .30 .27 10 2.750 .16 .21 .19 20 1.944 .11 .15 .14 30 1.588 .09 .12 .11 40 1.375 .08 .11 .10 50 1.230 .07 .10 .09 aThese values were obtained from equation (1) with an "e" value of 2.0 based upon a forest area of 4.209 hectares and a perimeter of 844.9 meters (10.4 acres, with 2,772 feet of perimeter). bRadius of the tree ring in feet per inch of tree diameter. 30 The Sampling Process In drawing the plot samples from the maps, successive areal plot sizes were drawn to scale on an acetate sheet to form a template. The center of the template was placed over each sample center, and the basal area Within each plot size outline was recorded as in Appendix A. For drawing point-samples from the maps, the plot radius distances from Appendix B were drawn on acetate strips. A pin secured the zero end of the strip to the sample point being measured. By rotating the strip about the pin and observing the relationship of each tree's diameter to the diameter and plot radius factors on the acetate strip, the tree was H H t A . . or out.‘ Tree diameters on the acetate strips were either counted "in interpolated to the nearest one-tenth inch. If there was any doubt con- cerning the count of an individual, the distance between it and the sam- ple point was checked by the Pythagorean theorem. Approximately fifty such checks were required in this study. I. Within each forest, three distributions were tested. Six plot shapes and two types of point-samples were sampled in each distribution. These were; circular, triangular, square, rectangular 1:2, rectangular 1:4, rectangular 1:8, and the adjusted and unadjusted point-samples. Six plot sizes were applied to each plot shape and six basal area factors were tested in each type of point-sample. The combination of these items yields 144 separate samples (3 x 8 x 6) in each forest. CHAPTER IV RESULTS AND DISCUSSION One hundred and forty—four methods of sampling involving six plot sizes, eight plot types or shapes and three cruising designs were mea- sured and analysed on each forest. The basic analysis involved the use of five parameters for each sampling method. These are: The Expected Value. -- The average basal area per acre in square feet. This is not used in the strict statistical sense. Standard Deviation. -- Standard deviation of the sample expressed in square feet per acre. Standard Error. -- Standard error of the mean expressed in square feet per acre. The Coefficient of Variation. -- This is the standard error expressed as a percent of the mean and is not a strict statistical parameter. It may be referred to as sampling error (after Grosenbaugh and Stover, 1957). The accuracy of a sampling system is inversely proportional to its CV. Comparative analysis among the sampling methods used is carried out by an analysis of variance on this parameter. Deviation from the True Mean. -- This is the deviation of the expected value from the true mean, expressed as a percent of the true mean. It is recorded as either a positive or negative number and is of use only where the population is completely inventoried. Lansing Woods Plot Samples. -- The parameter summaries for all samples taken in this forest are presented in Table 7 to Table 14. The most accurate sampling method is the one with the lowest coefficient of variation (CV). In 31 Table 7. -- Parameter summary, Lansing Woods, circular plots. 32 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 1/40 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Deviae tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 86.56 113.44 23.15 26.74 120.96 163.36 33.34 27.56 +54.88 74.40 84.32 17.21 23.13 -4.74 92.32 80.16 16.36 17.72 +18.21 108.56 109.12 22.27 20.51 +39.00 58.80 54.16 11.06 18.81 -24.71 85.44 11.04 +9.40 96.36 59.48 12.14 12.60 +23.38 61.20 45.36 9.26 15.13 -21.64 71.08 22.02 4.49 6.32 -8.99 84.18 8.92 +7.78 69.00 37.54 11.10 -11.65 79.88 25.26 5.16 6.46 +2.28 86.20 32.64 +10.37 82.03 36.13 +5.03 77.16 4.59 -1.20 77.21 22.61 5.97 -1.14 +2.46 33 Table 8. -- Parameter summary, Lansing Woods, triangular plots. Method of Sampling Unit Size in Acres Distribution Parameter 1/160 1/80 1/40 1/20 1/10 1/5 Expected Value Sq. Ft. 62.08 84.80 87.68 78.28 78.20 75.33 Standard Devia- tion Sq. Ft. 106.08 81.76 55.56 32.74 12.71 14.65 Random Standard Error Sq. Ft. 21.65 16.69 11.34 6.68 2.59 2.99 Coefficient of Variation Percent 34.87 19.68 12.94 8.53 3.31 3.97 Deviation from True Mean Percent -20.51 +8.58 +12.27 + .23 +.13 -3.55 Expected Value Sq. Ft. 119.36 92.40 96.68 96.96 91.52 80.94 Standard Devia— tion Sq. Ft. 188.80 86.80 65.52 43.42 32.54 25.88 Systematic Standard Error Sq. Ft. 38.54 17.72 13.37 8.86 6.64 5.28 Coefficient of Variation Percent 32.29 19.18 13.83 9.14 7.26 6.53 Deviation from True Mean Percent +52.83 +18.31 +23.79 +24.15 +17.18 +3.64 Expected Value Sq. Ft. 91.20 63.04 58.36 72.08 79.62 79.33 Standard Devia- tion Multiple Sq. Ft. 103.84 50.00 37.92 44.60 44.45 38.38 Random Standard Error Starts Sq. Ft. 21.20 10.21 7.74 9.10 9.07 7.83 Coefficient of Variation Percent 23.24 17.62 13.26 12.62 11.39 9.87 Deviation from True Mean Percent +16.77 -19.28 -25.27 -7.71 +1.95 +1.57 34 Table 9. Parameter summary, Lansing Woods, square plots. Method of Sampling Unit Size in Acres Distribution Parameter 1/160 1/80 1/40 1/20 1/10 1/5 Expected Value Sq. Ft. 85.12 98.40 84.92 79.12 76.52 76.39 Standard Devia- tion Sq. Ft. 113.12 92.00 61.84 28.44 24.71 18.89 Random Standard Error Sq. Ft. 23.09 18.78 12.62 5.80 5.04 3.86 Coefficient of Variation Percent 27.13 19.08 14.86 7.33 6.59 5.05 Deviation from True Mean Percent +8.99 +25.99 +8.73 +1.31 -2.02 —2.19 Expected Value Sq. Ft. 120.96 93.92 95.08 88.30 85.38 76.62 Standard Devia— tion Sq. Ft. 163.36 92.20 66.12 45.28 33.66 22.84 Systematic Standard Error Sq. Ft. 33.34 18.82 13.50 9.24 6.87 4.66 Coefficient of Variation Percent 27.56 20.04 14.20 10.46 8.05 6.08 Deviation from True Mean Percent +54.88 +20.26 +21.74 +13.06 +9.32 -l.89 Expected Value Sq. Ft. 75.68 59.44 68.12 68.10 72.30 82.12 Standard Devia- tion Multiple Sq. Ft. 81.60 49.76 45.96 35.28 30.89 33.85 Random Standard Error Starts Sq. Ft. 16.66 10.16 9.38 7.20 6.30 6.91 Coefficient of Variation Percent 22.01 17.08 13.77 10.57 8.71 8.41 Deviation from True Mean Percent -3.10 —23.89 -12.78 -12.80 -7.43 +5.15 Table 10. -- Parameter summary, Lansing Woods, rectangular, l: 35 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 1/40 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 73.60 107.68 21.98 29.86 -5.76 113.28 132.64 27.08 23.90 +45.04 59.52 89.12 18.19 30.56 85.12 75.76 15.46 18.16 +8.99 104.32 106.48 21.74 20.84 +33.57 78.32 63.28 12.92 16.50 +.28 83.56 53.08 10.84 12.97 +6.99 92.52 66.72 13.62 14.72 +18.46 68.16 47.68 14.27 -12.73 70.74 35.68 10.29 -9.42 90.92 9.16 +16.41 75.60 36.60 7.47 9.88 -3.20 71.46 19.19 3.92 5.49 -8.50 79.69 29.88 6.10 83.84 35.72 7.29 +7.35 78.00 5.51 79.35 24.02 4.90 6.17 +1.60 79.10 25.62 5.23 +1.28 Table 11. -- Parameter summary, Lansing Woods, rectangular 1:4. 36 Method of Distribution Random Systematic Multiple Random Starts Parameter Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Sampling Unit Size in Acres 1/160 114.08 144.96 29.59 25.94 +46.07 88.64 113.76 23.22 26.19 +13.50 72.37 96.32 19.66 27.16 -7.34 1/80 84.64 87.28 17.81 21.04 +8.37 94.48 99.28 20.26 21.44 +20.97 79.04 83.20 16.98 21.48 +1.20 1/40 67.08 14.21 -14.11 94.00 64.20 13.10 13.94 +20.36 98.52 64.08 13.08 13.28 +26.14 1/20 72.28 -7.45 80.02 42.02 8.58 10.72 +24.58 99.18 55.74 11.38 11.47 +26.99 1/10 82.16 29.21 5.96 +5.20 82.00 32.86 8.18 +4.99 84.10 27.47 5.61 +7.68 1/5 77.63 23.04 6.05 6.28 -6.84 75.50 20.02 4.09 -3.33 Table 12. -- Parameter summary, Lansing Woods, rectangular, 1:8. 37 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 1/40 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 93.28 128.48 26.23 28.12 +19.44 64.96 89.44 18.26 28.11 -16.81 68.48 121.92 24.89 36.35 90.40 99.52 20.31 22.47 +15.75 82.48 64.00 13.06 15.83 +5.61 93.20 88.40 18.04 19.36 +19.33 82.64 49.20 10.04 12.15 +5. 81 89.64 61.56 12.57 14.02 +14.77 110.44 64.44 13.15 11.91 +41.41 84.82 10.02 +8.60 87.14 44.96 9.18 10.53 +11.57 105.54 55.32 11.29 10.70 +35.13 82.40 25.48 5.20 +5.50 84.75 35.37 7.22 +8.51 86.32 36.97 7.55 +10.52 77.12 17.16 3.50 4.54 —1025 81.16 22.52 4.60 5.67 +3.92 77.44 18.58 4.89 -.84 Table 13. 38 -- Parameter summary, Lansing Woods, adjusted point-sampling. Method of Distribution Parameter 50 iBasal Area Factor (BAF)* 40 30 20 10 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia— tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia— tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation percent Deviation from True Mean Percent 83.50 62.00 12.66 15.16 6.91 98.00 70.00 14.29 14.58 +25.48 66.50 65.50 13.37 20.10 -14.85 90.00 60.40 12.33 13.70 15.24 101.60 63.60 12.98 12.77 +30.09 73.20 13.39 -6.27 97.50 58.80 12.00 12.31 24.84 101.40 57.60 11.76 11.60 +29.83 80.10 11.82 +2.56 100.00 49.40 10.08 10.08 28.04 103.40 52.40 10.70 10.35 +32.39 82.40 40.60 8.29 10.06 +5.50 122.50 69.90 14.27 11.65 56.85 115.80 47.60 9.72 8.93 +33.42 151.25 69.80 14.25 9.42 93.66 61.90 12.63 +70.81 113.60 38.80 7.92 6.97 +45.45 *The number associated with the basal area factor (BAF) indicates the square feet of basal area per acre for every tree counted. Table 14. 39 -- Parameter summary, Lansing Woods, unadjusted point-sampling. Method of Distribution Parameter Basal Area Factor (BAR) 50 4O 30 .20 10 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 83.50 62.00 12.66 15.16 +6.91 98.00 70.00 14.29 14.58 +25.48 66.50 62.00 12.67 19.05 -8.51 85.00 58.00 11.84 13.93 +8.83 101.60 63.60 12.98 12.77 +30.09 68.40 14.33 -12.42 88.80 48.90 9.98 11.24 +13.70 70.10 57.60 11.76 11.91 +26.38 77.40 77.60 11.60 50.40 10. 29 11.23 +17.28 73.40 58.80 12.00 16.35 -6.08 75.80 -2.94 81.70 +4.61 77.90 7.12 —026 77.50 18.80 3.84 4.95 -.77 76.90 29.15 5.95 7.74 -1.54 40- this woods, it is a one-tenth acre, triangular plot on the random distri- bution, and has a CV of 3.31 percent. The causes of variation in CV are plot distribution, treatment (plot shape, size, and interaction), and an undefined quantity called error, so it is possible that this sample is the best because of chance. To analyse this possibility, the CV of all samples were summarized in Table 15 and subjected to analysis of variance techniques. The sums of squares (SS), degrees of freedom (df) and the mean square, associated with each source of variation are given in Table 16. According to the "F" test, there is no significant difference among the random, systematic,or multiple-random-start (MRS) distributions ex- cept that arising by pure chance. Any one of the three methods yields approximately the same results. Plot shape also has little influence on the magnitude of the sampling error. The preference for one plot shape over another would have to be a matter of efficiency -- i.e., circular plots are easier to establish than angular plots (Lindsey et_al., 1958), and have less perimeter for a given area. The interaction between plot shape and plot size is also negligible. Plot size, on the other hand, is a very strong source of variation and is highly significant. The sampling error increases sharply as the plot size decreases. The average sampling errors by plot sizes are shown in Tablesl7 and 18. The five degrees of freedom and the associate sums of squares for plot size are partitioned into certain comparisons among the various plot sizes (Appendix C). The results showed that each suc- ceeding increase in plot size significantly decreased the coefficient of variation, except for the one-fifth and one-tenth acre units; either of these could have been used with the same results. 41 Table 15. -- Summary of the coefficient of variation data for the Lansing Woods plot samples. Plot Plot Plot Size Shape Plot Shape Size* Random Systematic MRS Total Total Circular 160 26.74 27.56 23.13 77.43 80 17.72 20.51 18.81 57.04 40 11.04 12.60 15.13 38.77 20 06.32 08.92 11.10 26.34 10 06.46 07.73 09.00 23.19 5 04.59 05.97 08.45 19.01 Total 72.87 83.29 85.62 241.78 Triangular 160 34.87 32.29 23.24 90.40 80 19.68 19.18 17.62 56.48 40 12.94 13.83 13.26 40.03 20 08.53 09.14 12.62 30.29 10 03.31 07.26 11.39 21.96 5 03.97 06.53 09.87 20.37 Total . 88728 88780 259.53 Square 160 27.13 27.56 22.01 76.70 80 19.08 20.04 17.09 56.21 40 14.86 14.20 13.77 42.83 20 07.33 10.46 10.57 28.36 10 06.59 08.05 08.71 23.35 5 05.05 06.08 08.41 19.54 Total 80.04 86.39 80.56 246.99 Rect. 1:2 160 29.86 23.90 30.56 84.32 80 18.16 20.84 16.50 55.50 40 12.97 14.72 14.27 41.96 20 10.29 09.16 09.88 29.33 10 05.49 07.65 08.69 21.83 5 05.51 06.17 06.61 18.29 Total 82.28 82.44 86.51 251.23 Rect. 1:4 160 25.94 26.19 27.16 79.29 80 21.04 21.44 21.48 63.96 40 14.21 13.94 13.28 41.43 20 07.19 10.72 11.47 29.38 10 07.25 08.18 06.67 22.10 5 06.05 06.28 05.42 17.75 Total 81.68 86.75 85.48 253.91 Rect. 1:8 160 28.12 28.11 36.35 92.58 80 22.47 15.83 19.36 57.66 40 12.15 14.02 11.91 38.08 20 10.02 10.53 10.70 31.25 10 06.31 08.52 08.75 23.58 5 04.54 05.67 04.89 15.10 Total 83.61 82.68 91.96 258.25 All Plot 160 172.66 165.61 162.45 500.72 Shapes 80 118.15 117.84 110.86 346.85 40 78.17 83.31 81.62 243.10 20 49.68 58.93 66.34 174.95 10 35.41 47.39 53.21 136.01 5 29.71 36.70 43.65 110.06 Total 483.78 509.78 518.13 1,511.69 *p'lnf civn in nnraa rlonnminofn-n nh117 42 Table 16. —- Analysis of variance, Lansing Woods, plot samples. Source df SS MS F Distribution 2 17.8300 8.9150 1.87 N.S Plot Shape 5 12.7520 2.5504 - N.S Plot Size 5 6137.4834 1227.4969 254.91 * * Interaction 25 97.7112 3.9084 - N.S Error 70 337.0884 4.8155 Total 107 6602.8650 N.S. Non-significant * * Highly significant (99 percent level) Table 17. -- Average coefficient of variation, Lansing Woods. Plot Samples Point-Samples Plot Shape CV Plot Size CV BAF Adjusted CV Unadjusted CV (percent) (acres) (percent) (percent) (percent) Circular 13.43 1/160 27.82 50 16.61 16.35 Triangular 14.42 1/80 19.27 40 13.28 13.67 Square 13.72 1/40 13.51 30 11.91 11.75 Rect. 1:2 13.96 1/20 9.72 20 10.16 12.35 Rect. 1:4 14.11 1/10 7.56 10 9.66 7.28 Rect. 1:8 14.35 1/5 6.11 5 8.65 5.87 Table 18. -- Average coefficient of variation by distributions, Lansing Woods. Samples Random Systematic MRS (percent) (percent) (percent) Plot 13.44 14.16 14.39 Adjusted Point 12.05 11.19 11.88 Unadjusted Point 10.20 10.67 12.78 43 The relationship between plot size and the corresponding coefficient of variation is shown in Figure 4. The CV from Table 17 was plotted over the reciprocal of the square root of the plot size. The resulting pat- tern of points can be described by the equation: cv = 457.570(p)‘§ (a) where: CV the coefficient of variation in percent P plot size in square feet 457.57 the regression coefficient derived from the data The correlation coefficient between the independent and dependent varia- bles is 99.92 percent and is highly significant. Under conditions simi- lar to this study, an increase in the plot size will increase the accuracy of the estimate. The limitation on this would be the loss in efficiency for large plots. Point-Sampling.-- The summary of the CV data for point-samples is given in Table 19. The analysis of variance of these data “is given in Table 20. Here, the manner of distributing the sample points, and the . method of sampling, either with or without peripheral adjustments (type), is of no consequence in this forest. However, the choice of basal area factors used, strongly influences the magnitude of the CV. Table 17 gives the average CV for each basal area factor. Comparison among these factors show that all differences are highly significant except between RAF-5 and RAF-10. The trend of the CV with respect to the basal area factors may be shown by plotting CV over the plot radius factor for each BAF. The resulting pattern of points in Figure 5 may be described by the equation: 30 25 20 15 10 Coefficient of variation in percent 0 .02 .04 .06 .08 (Plot size in sq. ft.)-5 Figure 4. -- The relationship between the coefficient of variation in percent and the reciprocal of the square root of the plot size in square feet. From the equation: cv = 457.57 (plot size)‘§, Lansing Woods. 44 45 Table 19. -- Summary of the coefficient of variation data for the Lansing Woods, point-sampling. Distribution BAF Plot Type Plot Type BAF Random Systematic MRS Total Total Adjusted 50 15.16 14.58 20.10 49.84 Point- 40 13.70 12.77 13.39 39.86 Samples 30 12.31 11.60 11.82 35.73 20 10.08 10.35 10.06 30.49 10 11.65 08.39 08.93 28.97 5 09.42 09.47 06.97 25.86 Total 72.32 67.16 71.27 210.75 Unadjusted 50 15.16 14.85 19.05 49.06 Point- 40 13.93 12.77 14.33 41.03 Samples 30 11.24 11.91 12.10 35.25 20 09.47 11.23 16.35 37.05 10 06.44 08.30 07.12 21.86 5 04.95 04.93 07.74 17.62 Total 61.19 63.99 76.69 201.87 All Plot 50 30.32 29.43 39.15 98.90 Types 40 27.63 25.54 27.72 80.89 30 23.55 23.51 23.92 70.98 20 19.55 21.58 26.41 67.54 10 18.09 16.69 16.05 50.83 5 14.37 14.40 14.71 43.48 Total 133.51 131.15 147.96 412.62 Table 20. -- Analysis of variance, Lansing Woods, point-sampling, coef- ficient of variation. Source df SS MS F Distribution 2 13.8040 6.9020 - N.S Treatment 11 364.3711 33.1246 11.99 * * Type 1 2.1903 2.1903 - N.S BAF 5 337.0892 67.4178 24.40 * * Interaction 5 25.0916 5.0183 — N.S Error 22 60.7799 2.7627 Total 35 438.9550 N.S. Non-significant. * * Highly significant. 46 25 1 20 +2 : o o e o a : -v-t g 15 o H +9 m o: o d > w 0 10 .p g: 0 .3 ‘\\~ 0 o H ‘H cH m \\\\\\\\ o L) 5 0 .2 .4 .6 .8 1.0 ‘ (Plot radius factor)-1 Figure 5. -- The relationship between the coefficient of variation in percent and the reciprocal of the plot radius factor. From the equation: CV = 19.905 — 15.351(Prf)'1, Lansing Woods. 47 cv = a + b(Prf)'1 (9) where: CV = coefficient of variation in percent Prf = plot radius factor in feet per inch of tree diameter a = constant b = regression coefficient Following solution, the equation becomes: CV = 19.905 - 15.351 (Prf)'1 (10) with a highly significant correlation coefficient of 97.89 percent. A decrease in the BAF will result in significant decreases in the sampling error. Comparison of Plot-Samples with Point-Samples. -- The most accurate areal plot is the one-fifth acre size with an average coefficient of. variation of 6.11 percent (Table 17). Plot shape and method of distribu- tion is of no importance, in that any sampling method would have given essentially the same results with this size plot. The most accurate point-sampling factor to use is the BAF-5 with an average CV of 7.25 per- cent (Table 17). The "t" test on the pooled variances is employed to analyse the difference between these two samples (Table 21). The calculated "t" value of 1.48 on 22 degrees of freedom does not exceed the tabular value at the 95 percent level of accuracy (Freese, 1963), so the difference between these two types of samples may be attri- buted to chance and either may be used with the same expected precision. The difference between the one-tenth acre plot (CV = 7.56) and the BAF-10 (CV = 8.47) is also unimportant. The calculated "t" is 1.39 on 22 de- grees of freedom. 48 Table 21. -- Analysis of the difference between the most accurate plot sample and point-sample, Lansing Woods. Method No. of Observations df Ave. CV SS (percent) One-fifth acre 18 17 6.11 38.38 BAF-5 6 5 7.25 20.63 Sum and differences 22 1.14 59.01 Deviation from the True Population Mean. -- The possibility exists that comparisons among distributions may not be valid. All sample es- timates of error are based upon an infinite population, while in system- atic sampling, this idea is not entirely true. Although the location of the first sample has a very large number of possible centers, the fixed grid forces the establishment of all other units throughout the popula- tion, and tends to sample the full range of the variation. Grouping of the plots does not occur. In random sampling, on the other hand, the plots can, and often do, sample certain sections of the forest much more intensely than the total number of plots warrants. Perhaps random sam- ples are as good as or better than systematic or MRS because they measure only a part of the population variance. To check this, the deviation of the expected values from the true population mean can be examined. Table 22 lists the averages for each plot shape and plot size. These averages were compiled from the absolute values of Tables 7 to 14. Average devia- tions for the random, systematic and MRS distributions are 8.70 percent, 18.53 percent, and 12.47 percent, respectively, (from Tables 7 to 14). The closeness of the random estimate to the true value indicates that this method did an adequate job of sampling the population. Table 23 shows that these differences are highly significant. 49 Table 22. -- Average deviations of the expected value from the known population parameter in percent, Lansing Woods. Adjusted Unadjusted Point- Point- Mean Mean Sample Sample Plot Shape Deviation Plot Size Deviation BAF Deviation Deviation (percent) (area) (percent) (percent) (percent) Circular 13.76 1/160 22.64 50 15.75 13.63 Triangular 14.32 1/80 17.35 40 17.20 17.11 Square 13.08 1/40 17.77 30 19.08 13.66 Rect. 1:2 11.42 1/20 13.34 20 21.98 8.00 Rect. 1:4 13.65 1/10 6.44 10 46.18 2.60 Rect. 1:8 13.17 1/5 2.37 5 69.97 1.50 Table 23. -- Analysis of variance,deviation of the expected value from the true mean in percent, Lansing Woods, plot samples. Source df SS MS F Distribution 2 1770.2601 885.1300 7.84 * * Treatment 35 6749.6629 1928.4751 17.09 * * Plot Shape 5 89.0740 17.8148 - N.S. Plot Size 5 5227.5647 1045.5129 9.27 * * Interaction 25 1433.0242 57.3210 - N.S. Error _19 7899.0903 112.8441 Total 107 16,419.0133- N.S. Non-significant * * Highly significant 50 A cursory inspection of the values in Table 22 indicates a lack of correlation between plot shape and the amount of deviations from the true parameter. This is to be expected, as is the fact that plot size in- creases result in closer estimates of the true mean. However, the differ- ences between the two methods of sampling by Bitterlich are startling. By using Grosenbaugh's peripheral zone technique, the error from the true value ranges from about 16 percent for a BAF-50 to nearly 70 percent for BAF-5. Review of these values in Tables 7 and 14 shows that this is a positive situation. The unadjusted technique shows the same trend as the areal plot sizes, and gives the closest estimate of any method when a BAF-5 is employed. Table 24, however, shows that there is no signifi- cance between either sample unit types or among the BA factors. Treat- ment is highly significant, but the interaction took so much of the sums of squares, that partitioning resulted in non-significance of the three sources under treatment. This is to be expected since variation increased in the adjusted plots as the BAF decreased. Interaction would normally claim a large portion of the sum of squares in this case. Under these circumstances, it is not advisable to use the peripheral method of adjust- ment. Table 24. Analysis of variance,deviation of the expected value from the true mean in percent, Lansing Woods, point-samples. Source df SS MS F Distribution 2 1773.3293 886.6646 - N.S. Treatment 11 27,473.6989 2,4976.0899 13.13 * * Type 1 4465.3579 4465.3579 2.35 N.S. BAF 5 2037.6916 407.5383 - N.S. Interaction 5 5760.3535 1152.0707 - N.S. Error gg 41,844.1632 1902.0438 Total 35 55,881.6975 N.S. Non-significant * * Highly significant 51 Toumey Woods Plot-Samples. -- The parameter summaries for all samples taken in this forest are presented in Tables 25 to 32. To analyse the differences between sampling methods, the coefficient of variations are listed in Table 33, and subjected to the analysis of variance in Table 34. Average values for the coefficient of variation for sample size, shapes, and dis— tributions are given in Tables 35 and 36. These were compiled directly from Table 33. According to the analysis of variance, plot size is the only factor that contributes significantly to the variation in sampling error. One method of distribution is as good as another and, as far as accuracy is concerned, plot shape also has no significant influence on the variation. The five degrees of freedom allocated for plot size may be partitioned into five comparisons as in the Lansing Woods. However, it is enough to show that the difference between the one-fifth and One-tenth acre plots is insignificant, and the difference between these two and the one- twentieth acre plot is significant. 1/5 vs 1/10 (138.09 - 103.49)2/2(18) = 33.2544 (11) F 33.2544/16.2348 = 2.05 (not significant) (1/5 + 1/10) vs 1/20 [2(216.28)-(l38.09+103.49)]2/6(18) = 337.7163 (12) F 337.7163/16.2348 = 20.80 (Highly significant) The "F" test is based upon one and 70 degrees of freedom. Either of the two largest plots may be used for the best results. The relationship between plot size and CV may be shown by plotting the CV over the reciprocal of the square root of the plot size (Fig. 6). Following regression, the relationship may be expressed as: Table 25. -- Parameter summary, Toumey Woods, circular plots. 52 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 1/40 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 76.96 153.44 34.31 44.58 -34.18 114.22 225.60 50.45 44.17 -2.31 92.63 176.80 39.53 42.67 -20.77 126.72 145.68 32.58 25.71 +8.38 103.31 127.04 28.41 27.50 -11.64 209.92 242.40 54.20 25.82 +79.54 89.84 77.80 17.40 19.37 -23.16 112.97 108.88 24.35 21.55 -3.38 146.79 140.20 31.35 21.36 +25.55 104.86 56.10 12.54 11.96 -10.31 124.80 67.38 15.07 12.07 + 6.74 120.19 78.38 17.53 14.58 +2.80 114.70 33.95 7.59 6.62 -1.90 136.56 35.10 7.85 5.75 +16.80 132.37 45.42 10.16 +13.21 117.34 35.88 124.28 28.38 5.11 +6.29 124.23 35. 41 7.92 6.37 +6.25 53 Table 26. Parameter summary, Toumey Woods, triangular plots. Method of Samplinngnit Size in Acres Distribution Parameter 1/160 1/80 1/40 1/20 1/10 1/5 Expected Value Sq. Ft. 140.64 104.40 88.56 84.88 111.49 118.71 Standard Devia— tion Sq. Ft. 231.68 134.64 63.32 41.32 27.28 29.37 Random Standard Error Sq. Ft. 51.81 30.11 14.16 9.24 6.08 6.57 Coefficient of Variation Percent 36.83 28.84 15.99 10.88 5.45 5.53 Deviation from True Mean Percent 20.29 -10.71 -24.25 -27.40 -4.64 1.53 Expected Value Sq. Ft. 110.15 96.64 104.34 123.53 133.84 123.29 Standard‘Devia- tion Sq. Ft. 227.86 123.84 114.92 53.60 41.34 33.68 Systematic Standard Error Sq. Ft. 50.84 27.69 25.70 11.98 9.24 7.53 Coefficient of Variation Percent 46.15 28.65 24.63 9.70 6.90 6.11 Deviation from True Mean Percent -5.79 -17.34 —10.76 5.65 14.47 5.45 Expected Value Sq. Ft. 124.77 183.60 169.70 127.61 132.78 80.24 Standard Devia- tion Multiple Sq. Ft. 327.68 199.44 140.16 70.28 46.18 31.16 Random Standard Error Starts Sq. Ft. 73.27 44.60 31.34 15.71 10.33 6.97 Coefficient of Variation 58.72 24.29 18.47 12.31 7.78 8.69 Deviation from True Mean Percent +6.71 57.03 45.14 9.14 13.56 -31.37 Table 27. Parameter summary, Toumey Woods, square plots. 54 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 1/40 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 119.84 194.88 43.58 36.36 +2.50 113.28 226.24 50.59 44.66 3.11 110.40 193.92 43.36 39.27 141.52 142.16 31.79 22.46 +21.04 88.00 131.52 29.41 33.42 24.73 186.24 207.72 46.45 24.94 59.29 95.64 75.00 16.77 17.53 -18.20 128.00 118.08 26.41 20.63 144.32 134.48 30.07 20.83 23.43 115.26 48.18 10.77 -056 130.96 63.86 14.28 10.90 12.01 119.74 65.32 14.61 12.20 24.11 114.42 35.20 7.87 6.88 -2.14 128.29 5.77 9.72 137.44 54.43 12.17 8.85 17.55 119.35 21.87 4.89 4.10 2.08 130.07 31.13 6.96 11.25 127.64 40.73 9.11 7.14 Table 28. —- Parameter summary, Toumey Woods, rectangular 1:2. 55 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 (*1/40 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 35.04 81.92 18.32 52.28 70.03 81.92 197.92 44.26 54.03 29.94 128.33 197.92 44.26 34.49 9.76 119.28 155.28 34.72 29.11 120.56 188.80 42.22 35.02 202.56 255.28 57.08 28.18 73.25 104.40 76.68 17.15 16.43 10.71 103.52 109.56 24.50 24.54 11.46 130.11 114.88 25.69 19.74 11.28 100.70 22.48 4.99 13.87 128.98 63.12 14.11 10.94 10.31 128.57 91.06 20.36 15.83 9.96 111.93 116.01 26.31 5.88 5.25 4.27 121.38 »48.52 10.85' 3.81 130.89 61.55 13.76 10.51 11.95 31.74 6.12 .78 122.22 27.01 6.04 122.00 Table 29. -- Parameter summary, Toumey Woods, rectangular 1:4. 56 Method of Distribution Random Systematic Multiple Random Starts Parameter Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value 'Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Sampling Unit Size in Acres 1/160 61.84 114.72 25.65 41.48 -47.11 98.42 170.88 38.21 ‘ 38.82 -15.82 109.92 160.96 35.99 32.74 -5.99 1/80 76.23 109.52 24.49 32.13 -34.80 119.52 170.64 38.16 31.93 +2.22 121.84 105.76 23.65 19.41 +4.21 1/40 102.65 93.48 20.90 20.36 -12.20 118.40 104.92 23.46 19.81 +1.26 150.12 148.44 33.19 22.11 +28.39 1/20 107.02 50.02 11.18 10.45 - 8.47 114.86 72.96 16.31 14.20 -1.76 123.00 82.32 18.41 14.97 +5.20 1/10 113.60 45.35 10.14 9.81 -2.84 117.01 7.61 110.09 40.36 9.02 8.19 -5.84 1/5 110.53 20.20 4.52 -5.46 123.26 25.55 5.71 +5.42 122.28 36.06 8.06 +4.58 Table 30. -- Parameter summary, Toumey Woods, rectangular 1:8. 57 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 T/4o 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 52.82 108.48 24.26 45.93 54.82 71.08 148.64 33.24 46.76 39.21 104.16 140.80 31.48 30.22 10.91 81.21 113.20 25.31 31.17 30.54 94.02 112.32 25.12 26.72 19.59 128.48 158.48 35.44 27.58 9.89 77.52 77.44 17.32 22.34 33.70 105.02 123.96 27.72 26.39 10.18 168.56 135.16 30.22 17.93 44.17 109.77 76.56 17.12 15.60 6.11 111.65 62.06 13.88 12.43 147.70 85.40 19.10 12.93 26.32 109.70 6.17 111.83 48.94 10.94 9.78 4.35 129.86 46.97 10.50 8.08 11.07 101.06 13.56 116.49 25.93 5.80 4.98 .37 111.50 4.89 58 Table 31. -- Parameter summary, Toumey Woods, adjusted point-sampling. Method of Basal Area Factor (BAF) Distribution Parameter 50 40 30 20 10' 5' Expected Value Sq. Ft. 120.00 126.00 142.50 145.00 168.50 235.75 Standard Devia- tion Sq. Ft. 49.50 70.00 64.50 62.80 80.00 81.00 Random Standard Error Sq. Ft. 11.07 15.66 14.42 14.04 17.89 18.11 Coefficient of Variation Percent 9.22 12.42 10.11 9.68 10.61 7.68 Deviation from True Mean Percent +2.63 +7.76 +21.88 +24.02 +44.11 +Mn.63 Expected Value Sq. Ft. 152.50 164.00 165.00 178.00 202.00 274.75 Standard Devia- tion Sq. Ft. 104.50 89.20 82.80 92.60 106.80 111.35 Systematic Standard Error » Sq. Ft. 23.37 19.95 18.51 20.71 23.88 24.90 Coefficient of Variation Percent 15.32 12.16 11.21 11.63 11.82 9.06 Deviation from True Mean Percent 30.43 40.27 41.12 52.24 72.77 134.99 Expected Value Sq. Ft. 137.50 174.00 180.00 186.00 207.50 256.75 Standard Devia- tion Multiple Sq. Ft. 107.50 150.40 137.40 132.60 127.10 121.10 Random Standard Error Starts Sq. Ft. 24.04 33.63 30.72 29.65 28.42 27.08 Coefficient of Variation Percent 17.48 19.32 17.06 15.94 13.69 10.54 Deviation from ' True Mean Percent 17.60 48.82 53.95 59.08 77.47 119.59 Table 32. Parameter summary, Toumey Woods, unadjusted point-sampling. 59 Method of Distribution Random Systematic Multiple Random Starts Parameter Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion ~ Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Basal Area Factor (BAF) 50 105.00 10.19 115.00 58.50 13.08 11.37 1.64 110.00 57.50 12.86 11.69 -5092 40 110.00 40.80 9.12 8.29 5.92 122.00 52.80 11.81 124.00 58.00 12.97 10.46 +6.05 30 117.00 38.70 .07 115.50 7.61 1.21 126.00 57.30 12.81 10.17 +7.76 g0 116.00 35.80 8.00 6.90 .79 115.00 1.64 127.00 43.60 10 121.00 27.30 6.10 5.04 3.49 119.00 28.20 1.78 121.00 36.00 8.05 6.65 +3.49 119.00 22.90 5.12 4.30 119.25 20.75 4.64 3.89 1.99 112.50 27.95 6.25 5.55 -3.78 60 Table 33. -- Summary of the coefficient of variation data, Toumey Woods, plot-samples. Plot Plot Plot Distribution Size Shape Plot Shape Size* Random Systematic MRS Total Total Circular 160 44.58 44.17 42.67 131.42 80 25.71 27.50 25.82 79.03 40 19.37 21.55 21.36 62.28 20 11.96 12.07 14.58 38.61 10 06.62 05.75 07.68 20.05 5 06.83 05.11 06.37 18.31 Total 115.07 116.15 118.48 349.70 Triangular 160 36.83 46.15 58.72 141.70 80 28.84 28.65 24.29 81.78 40 15.99 24.63 18.47 59.09 20 10.88 09.70 12.31 32.89 10 05.45 06.90 07.78 20.13 5 05.53 06.11 08.69 20.33 Total 103.52 122.14 130.26 355.92 Square 160 36.36 44.66 39.27 120.29 80 22.46 33.42 24.94 80.82 40 17.53 20.63 20.83 58.99 20 09.34 10.90 12.20 32.44 10 06.88 05.77 08.85 21.50 5 04.10 05.35 07.14 16.59 Total 96.67 120.73 113.23 330.63 Rect. 1:2 160 52.28 54.03 34.49 140.80 80 29.11 35.02 28.18 92.31 40 16.43 24.54 19.74 60.71 20 04.99 10.94 15.83 31.76 10 05.25 08.94 10.51 24.70 5 06.12 04.94 06.46 17.52 Total 114.18 138.41 115.21 367.80 Rect. 1:4 160 41.48 38.82 32.74 113.04 80 32.13 31.93 19.41 83.47 40 20.36 19.81 22.11 62.28 20 10.45 14.20 14.97 39.62 10 09.81 07.61 08.19 25.61 5 04.09 04.63 06.59 15.31 Total 118.32 117.00 104.01 339.33 Rect. 1:8 160 45.93 46.76 30.22 122.91 80 31.17 26.72 27.58 85.47 40 22.34 26.39 17.93 66.66 20 15.60 12.43 12.93 40.96 10 08.24 09.78 08.08 26.10 5 05.56 04.98 04.89 15.43 Total 128.84 127.06 101.63 357.53 All Plot 160 257.46 274.59 238.11 770.16 Shapes 80 169.42 183.24 150.22 502.88 40 112.04 137.55 120.44 370.01 20 63.22 70.24 82.82 216.28 10 42.25 44.75 51.09 138.09 5 32.23 31.12 40.14 103.49 Total 7576755 7711—749 m ‘ 2100.91 *Plot size in acres, denominator only. 61 Table 34. -- Analysis of variance, Toumey Woods, plot samples. Source df SS MS F Distribution 2 71.2182 35.6091 2.19 N. S. Treatment 35 18,313.8007 523.2514 32.23 * * Plot Shape 5 49.8658 9.9732 - N. S. Plot Size 5 17,992.2804 3598.4561 221.65 * * Interaction 25 271.6545 10.8662 - N. S. Error 70 1136.4365 16.2348 Total 107 19,521.4554 N. S. NOn-significant * * Highly significant Table 35. -- Average coefficient of variation, Toumey Woods. Plot Sampling Point-Sampling Plot Shape CV Plot Size CV BAF Adjusted CV Unadjusted CV (percent) (acres) (percent) (percent) (percent) Circular 19.43 1/160 42.79 50 14.01 10.24 Triangular 19.77 1/80 27.94 40 14.63 9.48 Square 18.37 1/40 20.56 30 12.79 8.33 Rect. 1:2 20.43 1/20 12.02 20 12.42 7.35 Rect. 1:4 18.85 1/10 7.67 10 12.04 5.66 Rect. 1:8 19.86 1/5 5.74 5 9.09 4.58 Table 36. -- Average coefficient of variation data by distributions, Toumey Woods. Samples Random Systematic MRS (percent) (percent) (percent) Plot 18.79 20.60 18.97 Adjusted Point 9.95 11.87 15.67 Unadjusted Point 6.60 7.55 8.70 45 40 35 30 25 20 15 Coefficient of variation in percent 10 Figure .02 .04 .06 .08 (Plot size in sq. ft.)'2 6. -- The relationship between the coefficient of variation in percent and the reciprocal of the square root of the plot size in square feet. From the equation: CV = 670.399 (Plot size)'2, Toumey Woods. 62 where: P cv = 670.399(P)-% plot size in square feet CV = coefficient of variation in percent 670.399 = calculated regression coefficient 63 (13) The correlation coefficient between the two variables is 99.70 percent. It may be concluded that an increase in plot size will significantly in- crease the accuracy of the estimate according to Equation 13. Due con- siderations must be given to time and cost, however. Point-Sampling. -- A summary of the coefficient of variation data for point-sampling is given in Table 37. ling method are listed in Table 35 and 36. The average CV for each samp- Table 37.--Summary of the coefficient of variation data, Toumey Woods. Distribution Plot Type Plot Type BAF Random Systematic MRS Total Total Adjusted 50 09.22 15.32 17.48 42.02 Point- 40 12.42 12.16 19.32 43.90 Sampling 30 10.11 11.21 17.06 38.38 20 09.68 11.63 15.94 37.25 10 10.61 11.82 13.69 36.12 5 07.68 09.06 10.54 27.38 Total 59.72 71.20 94.03 224.95 Unadjusted 50 07.67 11.37 11.69 30.73 Point- 40 08.29 09.68 10.46 28.43 Sampling 30 07.39 07.61 10.17 25.17 20 06.90 07.47 07.68 22.05 10 05.04 05.29 06.65 16.98 5 04.30 03.89 05.55 13.74 Total 39.59 45.31 52.20 137.10 All Plot 50 16.89. 26.69 29.17 72.75 Types 40 20.71 21.84 29.78 72.33 30 17.50 18.82 27.23 63.55 20 16.58 19.10 23.62 59.30 10 15.65 17.11 20.34 53.10 5 11.98 12.95 16.09 41.02 Total 99.31 116.51 146.23 362.05 64 .Analysis of variance in Table 38 shows the method of plot distribu- tion, as well as the treatment, to be highly significant. Separation of the two degrees of freedom into comparisons among distributions shows the random method to be superior to either of the other systems on one and 22 degrees of freedom (Equations 14 and 15). Tableifi3. -- Analysis of variance, Toumey Woods, point-sampling, coeffi- cient of variation. Source df SS MS F Distribution 2 93.9057 46.9529 19.27 * * Treatment 11 342.8057 31.1642 12.79 * * Error 22 53.6176 2.4372 Total 35 490.3290 * * Highly significant. MRS vs (R+S) [2(146.23)—(99.31+116.51)32/6(12) = 81.5790 (14) F = 81.5790/2.4372 = 33.47 *w=(High1y significant, 99 percent level) R vs s = (116.51 - 99.31)2/2(12) = 12.3267 (15) F = 12.3267/2.4372 = 5.06* (Significant, 95 percent level) To compare the differences in treatment for plot type and BAF size, the analysis of variance is applied only to the random distribution (Table 39). The other two methods are considered statistically inaccurate. In analysing the variance for the random distribution, the basal area factor proved to be insignificant in its contribution to the varia- tion, so a regression between BAF and CV was not calculated. In timber 65 Table 39. -- Analysis of variance for the random distribution of point- samples, Toumey Woods. Source df SS MS F Type 1 33.7681 33.7681 35.55 * * BAF 5 20.0098 4.0020 4.21 N.S. Error 5 4.7496 .9499 Total 11 58.5275 N.S. Non-significant * * Highly significant as large as that in the Toumey Woods, any BAF would have produced essen- tially the same results. Plot type, on the other hand, with its differ- ence between Grosenbaugh's peripheral adjustment and the unadjusted samp- ling, shows high significance. The average CV for the adjusted method is 59.72 percent,while for the unadjusted, it is only 39.59 percent. Samp- ling by the adjusted method is not recommended. Comparison of Plot Samples with Point-Samples. -- The most precise plot sample is the one-fifth acre unit, (shape is of no consequence) with an average CV of 5.74 percent (Table 35). In the unadjusted point- sampling method, the BAF-5 has the lowest CV with 4.58 percent (Table 35). To test the difference between these two, the data of Table 40 are sub- jected to the "t" test by pooling the variance. The calculated "t" value is only 1.639, and does not exceed the tabular value of 2.093 at the 95 percent level for 19 degrees of freedom. It is concluded that the dif- ference between the accuracy of the one-fifth acre areal plot and the BAF-5 point-sample is insignificant, and either may be used with the same results. This, of course, ignores efficiency. 66 Table 40. -- Analysis of the difference between the most accurate plot and point-samples, Toumey Woods. Method Observations df Ave. CV SS (percent) 1/5 acre 18 17 5.74 22.9779 BAF-5 3 2 4.58 1.4954 19 1.16 24.4733 Deviations of the Expected Value from the True Mean. -- To see if each distribution adequately sampled the population, the deviation of the expected value from the known population mean was subjected to the analysis of variance in the same manner as the sampling error. Table 41 shows the averages of these deviations. Table 41. -- Average deviation of the expected value from the true population mean, Toumey Woods. Plot Sam ling Point-Sampling Adjusted Unadjusted Plot Shape Deviation Plot Size Deviation BAF Deviation Deviation (percent) (acres) (percent) (percent) (percent) Circular 15.20 1/160 20.81 50 16.89 5.92 Triangular 17.29 1/80 26.07 40 32.28 5.44 Square 14.22 1/40 19.76 30 38.98 3.01 Rect. 1:2 15.85 1/20 10.29 20 45.11 3.68 Rect. 1:4 10.65 1/10 8.55 10 64.78 2.92 Rect. 1:8 18.34 1/5 6.52 5 118.74 2.52 67 From Tables 25 to 32 the average deviation of the expected values from the known population mean is 15.85, 9.57, and 20.33 percent for the ran- dom, systematic, and MRS distributions,respectively. The larger deviation by the random method indicates that perhaps this distribution of plots did not sample the full variation within the forest. This question is purely academic, since, under normal circum- stances, the true population parameter would not be known, and any guesses as to its magnitude would be no better than a random estimate. In the present case, this phenomenon can be attributed to chance. An- other random sample, or the same samples applied to another population, would produce different results. Table 42 presents the analysis of variance for areal plots, and proves the differences in distributions to be significant at the 95 Table 42. -- Analysis of variance of the deviation of the expected value from the known parameter, Toumey Woods, plot samples. Source df SS MS F Distribution 2 2102.3787 1051.1894 4.63 * Treatment 35 10,639.6805 303.9909 - N.S. Plot Shape 5 673.2927 134.6585 - N.S. Plot Size 5 5577.0439 1115.4088 4.92 * * Error 70 15,885.1831 226.9312 Total 35 43,403.9729 N.S. Non-significant * Significant (95 percent level) * * Highly significant 68 percent level. Inspection of the values in Table 41 shows that the de- viations decrease as plot size increases. This is highly significant ac- cording to the "F" test. Table 43 presents the analysis for point- sampling. With the large variances introduced by the peripheral adjust- ments, all aspects are important at the 99 percent level. Table 43. -- Analysis of variance of the deviation of the expected values from the true parameter, Toumey Woods, point-sampling. Source df SS MS F Distribution 2 1714.0902 857.0451 * * Treatment 11 40,992.4155 3726.5832 * * Type 1 21,506.2225 21,506.2225 * * BAF 5 9114.1206 1822.8241 * * Interaction 5 10,372.0724 2074.4145 * * Error 22 697.4672 31.7031 * * Total 35 43,403.9729 * * Highly significant In spite of the differences, the best procedure would call for the use of a random distribution of either one-fifth acre plot samples, or BAF-5 point-samples, since these are statistically the most accurate, ac- cording to the analysis of the coefficient of variation data. The use of peripheral adjustments is not recommended. 69 Red Cedar Woods Plot Samples.-- The parameter summaries for all samples taken in this forest are presented in Tables 44 to 51. The coefficients of varia- tion are listed in Table 52 and subjected to the analysis of variance in Table 53. According to this analysis, the differences in methods of dis- tribution are highly significant. From Table 52 the differences between distributions produced a highly significant "F" value of 31.27. The same type of test between the systematic and the MRS shows no important differences. Therefore, the random distribution with the low- est CV of 20.07 percent, will be used for the test on the bases of plot shape and size. The others are discarded as statistically inaccurate. From Table 54, the analysis of variance, it is evident that the plot shape has no bearing on the variation in sampling error. A plot of any shape will give essentially the same results. The size of the plots, on the other hand, ‘is very important. Comparisons among plot sizes were carried out in the same manner as for the Lansing Woods (Appendix C) and the results showed that the one-fifth, one-tenth, and one-twentieth acre plots would give approximately the same accuracy. Plots smaller than these were significantly less precise. The mathematicalrelationship between plot size and the coefficient of variation is illustrated in Figure 7. The equation was derived from a regression of the data of the two variables from Table 55. The resulting equation is: Table 44. —- Parameter summary, Red Cedar Woods, circular plots. 70 Method of Distribution Random Systematic Multiple Random Starts Parameter Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value . Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Sampling Unit Size in Acres 1/160 181.44 277.12 65.31 36.00 +98.08 146.88 344.80 81.26 55.32 +60.35 140.80 265.92 62.67 44.51 +5.37 1/80 117.92 166.08 39.14 33.20 +28.73 90.56 174.72 41.18 45.47 -l.13 127.68 165.04 38.90 30.47 +39.39 ‘1/40 108.68 89.88 21.18 19.49 +18.65 81.44 100.76 23.75 29.16 -11.09 98.80 97.64 23.01 23.29 +7.86 1/20 89.72 41.90 11.01 -2.05 86.38 88.06 20.75 24.02 -5.70 110.88 92.94 21.90 19.75 -21.05 1/10 87.76 30.26 8.13 -4.19 91.46 49.44 11.65 12.74 -.15 102.96 56.83 13.39 13.00 +12.40 1/5 92.71 29.14 6.87 +1.21 81.94 10.53 -10.54 90.54 44.94 10.59 11.70 -1.16 Ta Table 45. -- Parameter summary, Red Cedar Woods, triangular plots. 71 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 1/40 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia— tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 134.72 191.84 45.21 33.56 +47.07 115.20 181.60 42.80 37.15 +25.76 122.08 248.32 58.52 47.93 +33.27 128.48 137.60 32.43 25.24 +40.04 99.52 175.68 41.40 41.60 +8.65 97.20 140.32 33.07 34.02 +6.11 106.08 99.40 23.43 22.09 +15.81 88.00 116.00 27.34 31.07 -3.93 143.24 143.00 33.70 23.53 +56.37 101.38 47.30 11.15 11.00 +10.68 87.86 70.76 16.68 18.98 -4.08 105.60 92.40 21.78 20.63 +15.28 88.29 6.82 -3.61 82.62 52.89 12.47 15.09 -9.80 97.02 73.78 17.39 17.92 +5.92 90.12 31.16 7.34 -l.61 88.52 -3.36 94.66 44.71 10.54 11.13 +3.34 Table 46. -- Parameter summary, Red Cedar Woods, square plots. 72 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 '1/40 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 171.84 271.52 63.99 37.24 +87.60 162.40 349.44 82.36 50.72 +77.29 106.72 226.88 53.47 50.11 +16.51 112.32 161.60 38.09 33.91 +22.64 102.72 175.92 41.46 40.37 +12.14 111.04 150.00 35.35 31.84 +21.22 118.44 76.72 18.08 15.27 +29.30 82.76 107.36 25.30 30.58 -9.65 98.08 99.68 23.49 23.95 +7.07 94.98 42.90 10.11 10.65 +3.69 85.12 75.26 17.74 20.84 -7.07 95.94 79.18 18.66 19.45 +4.74 89.49 90.74 50.12 11.81 13.02 -0.94 97.69 52.93 12.47 12.77 +6.65 92.84 33.02 7.78 8.38 +1.35 85.90 34.32 85. 71 34.23 8.07 -6.43 Tab Ra 5351 Hui Re St Table 47. -- Parameter summary, Red Cedar Woods, rectangular 1:2. 73 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 1/40 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 175.36 116.96 27.56 15.72 +91.44 78.72 180.64 42.57 54.08 -14.06 120.96 238.72 56.26 46.52 +32.05 130.48 168.08 39.61 30.36 +42.44 111.36 178.56 42.08 37.79 +21.57 93.60 136.40 32.15 34.35 +2.18 89.08 88.40 20.83 23.38 -2.75 85.40 96.24 22.68 26.56 -6.77 113.48 96.96 22.85 20.14 +23.89 93.14 42.78 10.08 10.82 +1.68 96.52 83.48 19.67 20.38 +5.37 109.76 76.68 18.07 16.41 +19.82 88.24 -3.67 87.83 45.98 10.84 12.34 -4.11 99.56 46.06 10.85 10.90 +8.69 92.32 +.79 9.56 -2.80 97.16 48.89 11.52 11.86 +6.07 Table 48. -- Parameter summary, Red Cedar Woods, rectangular 1:4. 74 Method of Distribution Parameter Sampling Unit Size in Acres 1/160 1/80 1/40 1/20 1/10 1/5 Random Systematic Multiple Random Starts Expected Value . Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 148.96 304.48 71.76 48.17 +62.62 34.72 52.80 12.44 35.83 -62.10 90.40 185.44 43.71 48.35 -1.31 123.68 167.12 39.39 31.85 +35.02 81.44 132.16 31.15 38.29 -11.09 81.92 126.48 29.81 36.39 -10.57 96.04 88.96 20.97 21.83 +4.85 91.56 116.56 27.47 30.00 134.56 152.20 35.87 26.66 +46.90 97.94 59.46 14.01 14. 30 +6.92 84.20 70.30 16.57 19.68 -8.08 112.96 80.52 18.98 16.80 +23.32 107.10 44.55 10.50 9.80 +16.92 99.22 57.29 13.50 13.61 +8.32 100.74 60.10 14. 16 14.05 +9.98 100.10 35.31 8.32 +9.28 97.90 42.53 10.02 10.24 +6.88 10.74 -1.92 Table 49. -- Parameter summary, Red Cedar Woods, rectangular 1:8. 75 Method of Distribution Random Systematic Multiple Random Starts Parameter Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia— tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Sampling Unit Size in Acres 1/160 116.48 219.36 51.70 44.38 +27.16 21.44 41.44 9.77 45.57 -76.59 59.20 160.00 37.71 63.70 -35.37 1/80 106.72 175.44 41.35 38.75 +16.51 19.12 28.16 6.64 34.73 -79.13 92.24 139.84 32.96 35.73 +.70 ’1/40 103.52 90.92 21.43 20.70 +13.01 64.08 79.28 18.68 29.15 -30.04 92.68 86.44 20.37 21.98 +1.18 1/20 114.08 95.52 23.22 21.14 +24.54 87.92 66.98 15.79 17.96 -4.02 96.94 93.52 22.04 22.73 +5.83 1/10 91.57 66.87 15.76 17.21 -.03 88.74 45.41 10.70 12.06 -3.12 90.36 68.04 16.03 17.74 -1.35 1/5 87.82 41.16 9.70 11.04 -4.13 90.54 43.72 10.30 11.38 -l.16 92.60 38.94 9.91 +1.09 76 Table 50. -— Parameter summary, Red Cedar Woods, adjusted point-samples. Method of Distribution Random Systematic Multiple Random Starts Parameter Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Basal-Area Factor (BAF) 50 133.50 92.50 21.80 16.33 +45.74 125.00 113.00 26.63 21.31 +36.46 119.50 116.50 27.46 22.98 +30.46 40 148.80 98.80 23.28 15.65 +62.44 122.40 100.40 23.66 19.33 +33.62 133.20 128.00 30.17 22.65 +45.41 30 138.30 99.30 23.40 16.92 +50.98 123.30 89.70 21.14 17.15 +34.61 143.40 127.80 30.12 21.01 +56.66 20 145.60 76.60 18.05 12.40 +58.95 133.40 95.00 22.39 16.79 +45.63 140.00 96.80 22.82 16.30 +52.84 10 169.40 62.60 14.75 +84.93 150.60 79.70 18.78 12.47 +64.41 152.20 102.20 24.09 15.83 +66.16 249.20 113.70 26.80 10.75 £U2.05 205.00 91.25 21.51 10.49 4423.80 211.40 146.00 34.41 16.28 -HflO.79 77 Table 51. -- Parameter summary, Red Cedar Woods, unadjusted point—samples. Method of Distribution Random Systematic Multiple Random Starts Parameter Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia— tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent Expected Value Sq. Ft. Standard Devia- tion Sq. Ft. Standard Error Sq. Ft. Coefficient of Variation Percent Deviation from True Mean Percent 50 111.00 67.50 15.91 14.33 +21.18 100.00 76.50 18.03 18.03 +9.17 94.50 84.00 19.80 20.95 +3.16 Basa1_Area Factor (BAF) 40 115.60 61.20 14.42 12.78 +26.20 100.00 70.40 16.59 16.59 +9.17 97.60 78.00 18.38 18.84 +6.55 30 101.40 54.60 12.87 12.69 +10.70 95.10 61.20 14.42 15.17 +3.82 99.90 78.30 18.45 18.47 +9.06 20 98.80 +7.86 92.20 50.40 11.88 12.88 98.80 58.40 13.76 13.93 +7.86 10 97.20 22.20 +6.11 82.80 33.70 7.94 9.59 -9.61 98.40 30.60 +7.42 87.80 -4.15 81.40 34.40 8.11 9.96 -7.20 -11.13 78 Table 52. -- Summary of the sampling error data for Red Cedar Woods, plot samples. Plot Plot Plot Distribution Size Shape Plot Shape Size * Random Systematic MRS Total Total Circular 160 36.00 55.32 44.51 135.83 80 33.20 45.47 30.47 109.14 40 19.49 29.16 23.29 . 071.94 20 11.01 24.02 19.75 054.78 10 08.13 12.74 13.00 033.87 5 07.41 10.53 11.70 029.64 Total 115.24 177.24 142.72 435.20 Triangular 160 33.56 37.15 47.93 118.64 80 25.24 41.60 34.02 100.86 40 22.09 31.07 23.53 076.69 20 11.00 18.98 20.63 050.61 10 06.82 15.09 17.92 039.83 5 08.14 09.84 11.13 029.11 Total 106.85 153.73 155.16 415.74 Square 160 37.24 50.72 50.11 138.07 80 33.91 40.37 31.84 106.12 40 15.27 30.58 23.95 069.80 20 10.65 20.84 19.45 050.94 10 08.23 13.02 12.77 034.02 5 08.38 09.42 09.41 027.21 Total 113.68 164.95 147.53 426.16 Rect. 1:2 160 15.72 54.08 46.52 116.32 80 30.36 37.79 34.35 102.50 40 23.38 26.56 20.14 070.08 20 10.82 20.38 16.41 047.61 10 09.97 12.34 10.90 033.21 5 08.13 09.56 11.86 029.55 Total 98.38 160.71 140.18 399.27 Rect. 1:4 160 48.17 35.83 48.35 132.35 80 31.85 38.29 36.39 106.53 40 21.83 30.00 26.66 078.49 20 14.30 19.68 16.80 050.78 10 09.80 13.61 14.05 037.46 5 08.31 10.24 10.74 29.29 Total 134.26 147.65 152.99 434.90 Rect. 1:8 160 44.38 45.57 63.70 153.65 80 38.75 34.73 35.73 109.21 40 20.70 29.15 21.98 071.83 20 21.14 17.96 22.73 061.83 10 17.21 12.06 17.74 047.01 5 11.04 11.38 09.91 032.33 Total 153.22 150.85 171.79 475.86 All Plot 160 215.07 278.67 301.12 794.86 Shapes 80 193.31 238.25 202.80 634.36 40 122.76 176.52 139.55 438.83 20 78.92 121.86 115.77 316.55 10 61.12 78.86 86.38 226.36 5 51.41 60.97 64.75 177.13 Total 722.59 955.13 910.37 2588.09 *Plot size in acres, denominator only. 79 Table 53. -— Analysis of variance, Red Cedar Woods, plot-samples. Source df SS MS F Distribution 2 845.7373 422.8686 16.17 * * Treatment 35 16,715.0288 477.5723 18.26 * * Error 70 1830.8146 26.1545 Total 107 19,391.5807 * * Highly significant Table 54. -- Analysis of variance, random distribution, Red Cedar Woods, plot-samples. Source df SS MS F Plot Shape 5 _296.3909 59.2782 2.38 N. S. Plot Size 5 4046.3616 809.2723 32.44 * * Error .25 623.6091 24.9444 Total 35 4966.4198 N.S. Non-significant * * Highly significant Coefficient of variation in percent Figure 7. 35 30 25 20 15 10 O 9 C 0 .02 .04 06 .08 (Plot size in sq. ft.)-; -- The relationship between the coefficient of variation in percent and the reciprocal of the square root of the plot size in square f et. From the equation: CV = 649.965 (Plot size)’ , Red Cedar Woods. 80 81 Table 55. -- Average coefficient of variation for all plot samples in the random distribution, Red Cedar Woods. Plot Samples Point-Samples Plot Shape CV Plot Size CV BAF Adjusted CV Unadjusted CV (percent) (acres) (percent) (percent) (percent) Circular 19.21 1/160 35.84 50 16.33 14.33 Triangular 17.80 1/80 32.22 40 15.65 12.78 Square 18.95 1/40 20.46 30 16.92 12.69 Rect. 1:2 16.40 1/20 13.15 20 12.40 9.02 Rect. 1:4 22.38 l/10 10.19 10 8.71 5.38 Rect. 1:8 25.54 1/5 8.57 5 10.75 7.33 CV = 649.965(P)-§ (16) where: CV = the coefficient of variation in percent P the plot area in square feet 649.965 = the regression coefficient derived from the data The relationship between the two variables is highly significant as shown by a correlation coefficient of 99.48 percent. An increase in the plot size results in a corresponding decrease in the CV. The question of efficiency would, of course, limit the maximum size of the sample plots. Point-Sampling. -- The CV data are summarized in Table 56. Analysis of variance (Table 57) shows the differences in sample distribution to be highly significant. Comparisons among distributions indicate that the random method is best. Analysis is carried out in the same manner as on the plot sample (Table 58). Plot type, consisting of samples with or without peripheral adjustments, is , of course, highly significant. Plots 82 Table 56. -- Summary of the sampling error, Red Cedar Woods, point- sampling. Distribution BAF Plot Type Plot Type BAF Random Systematic MRS Total Total Adjusted 50 16.33 21.31 22.98 60.62 Point- 40 15.65 19.33 22.65 57.63 Samples 30 16.92 17.15 21.01 55.08 20 12.40 16.79 16.30 45.49 10 08.71 12.47 15.83 37.01 5 10.75 10.49 16.28 37.52 Total 80.76 97.54 115.05 293.35 Unadjusted 50 14.33 18.03 20.95 53.31 Point- 40 12.78 16.59 18.84 48.21 Samples 30 12.69 15.17 18.47 46.33 20 09.02 12.88 13.93 35.83 10 05.38 09.59 11.20 26.17 5 07.33 08.31 09.96 25.60 Total 61.53 80.57 93.35~ 235.45 All Plot 50 30.66 39.34 43.93 113.93 Types 40 28.43 35.92 41.49 105.84 30 29.61 32.32 39.48 101.41 20 21.42 29.67 30.23 81.32 10 14.09 22.06 27.03 63.18 5 18.08 18.80 26.24 63.12 Total 142.29 178.11 208.40 528.80 Table 57. -- Analysis of variance, Red Cedar Woods, point-sampling. Source df SS MF Distribution 2 182.5303 91.2652 Treatment 11 503.6107 45.7828 Error 22 27.3910 1.2450 Total 35 713.5320 * * Highly significant. 83 Table 58. -- Analysis of variance, random distribution, Red Cedar Woods, point—samples. Source df SS MS F Plot Type 1 30.8161 30.8161 * * BAF 5 117.4380 23.4876 * * Error 5 1.3537 .2707 Total 11 149.6078 * * Highly significant with adjustments have an average sampling error of 16.30 percent while those without the adjustment are in error by 13.08 percent (Table 56). An inspection of the data in Table 55 shows approximately the same variation in CV among adjusted point—samples and among unadjusted point- samples for the different basal area factors. Comparative testsshow no significant differences between the BAF-5 and BAF-10 (t = .0003). Either will produce the same precision. The difference between the BAF-20 and lthe combined BAF-5 and BAF—10 is highly significant with a "t" of 135. The relationship between the basal area factor used and the accuracy of the estimate may be shown by plotting the CV over the plot radius fac- tor for each BAF. The resulting pattern of points is illustrated in Figure 8 and may be described mathematically by the regression equation: cv = a + b(Prf)-l (17) where: CV the coefficient of variation in percent Prf = the plot radius factor in feet per inch of tree diameter~ a constant derived from the data 9) ll 0‘ II regression coefficient derived from the data 25 ’ 20 15 10 Coefficient of variation in percent 84 Figure 8. .2 .4 .6 .8 110 (Plot radius factor)"1 -— The relationship between the coefficient of variation in percent and the reciprocal of the plot radius factor. From the equation: cv = 23.895 — 16.74(Prf)'1, Red Cedar Woods. C)" n) 85 Following solution by regression techniques, the equation becomes: 1 cv = 23.895 - 16.74 (Prf)- (18) The constant,a,and the regression coefficient,b, both had an "F" value greater than the tabular value at the 99 percent level (63 and 1200, respectively). The correlation coefficient between the two variables is 96.92 percent and is highly significant. Equation 18 may be used to pre- dict the coefficient of variation for various BAF sizes. Comparison of Plot-Samples with Point-Samples. -- According to Table 52, the plot system with the smallest average coefficient of variation is the one-fifth acre plot on the random distribution. This has a total CV of 51.41 percent for an average of 8.57 percent. From Table 56, the most accurate point-sample is the BAF-5 on the random distribution, with a to- tal CV of 18.08 percent for an average CV of 3.01 percent. The differ- ence between these is highly significant according to the calculated "t" value of 4.49 on the pooled variances of Table 59. For the best results in sampling this type of forest, a Bitterlich sample, with a BAF-5, on a random distribution should be used. Table 59. -- Analysis of the differences between the most accurate plot sample and point-sample, Red Cedar Woods. Sampling Method Observations df Ave. CV SS (percent) One-fifth acre 6 5 8.57 7.93 BAF-5 point-sample 2 l 3.01 5.85 Sum and difference 6 5.56 13.78 M— (BL th: 1)‘ Ta Ci Tr Sq Re Re Re t1 44 is Ih Du 86 Deviation of the Expected Value from the True Parameter. -- To de- termine how well a sampling method measured the variance in this forest, the average deviation of the expected value from the known mean is ana- lysed. Table 60 lists these averages in percent as compiled from the ab- solute values of Tables 44 to 51. Table 60. -- Average deviation of the expected value from the true popu- lation parameter in percent, Red Cedar Woods. Plot Samples J_ Point-Samples 4#_ Adjusted Unadjusted Plot Shape Deviation Plot Size Deviation BAF Deviation Deviation (percent) (acres) (percent) (percent) (percent) Circular 18.28 1/160 47.44 50 37.55 11.17 Triangular 16.37 1/80 22.18 40 47.16 13.97 Square 17.93 1/40 16.06 30 47.71 7.86 Rect. 1:2 16.12 ‘1/20 9.66 20 52.47 5.46 Rect. 1:4 18.12 1/10 5.68 10 71.83 7.97 Rect. 1:8 18.05 1/5 3.85 5 142.21 7.90 From Table 60, it is clear that the shape of the plot has little in- fluence on the deviations from the known mean. However, the deviations do vary inversely with sample unit size; going from nearly 50 percent for the smallest plots to less than four percent for the largest. From Tables 44 to 51 these averages by distribution are: random, 21.78 percent; sys- tematic, 16.75 percent; and MRS, 13.95 percent. The random distribution was the least accurate of all, while the systematic was the best. This illustrates the tendency for random samples to sample some areas more than others, while the systematic grid, scatters the plots evenly through- out the population. Analysis of variance (Table 61) shows that the 87 Table 61. -- Analysis of variance of the deviation of the expected value from the true parameter, Red Cedar Woods, plot samples. Source df SS MS F Distribution 2 1117.5174 558.7587 - N.S. Treatment 35 25,854.8409 738.7097 - N.S. Plot Shape 5 83.9942 16.7988 - N.S. Plot Size 5 23,546.9551 4709.3910 6.11 * * Interaction 25 2223.8916 88.9557 - N.S. Error 70 59,911.4276 770.1633 Total 108 80,883.7859 N.S. Non-significant * * Highly significant “ difference in distribution is not significant. Plot size is the only im- portant influence in areal samples. In point—sampling, the unadjusted type shows the same trend as the areal plots, while the adjusted points are just the opposite, with loss- es in accuracy of over 100 percent in going from the BAF-50 to the BAF-5. Analysis of variance (Table 62) shows that all sources of variation are significant, including the differences in distributions. It should be noted that the unadjusted point-samples with a BAF-5 compares favorably to one-fifth acre plot-sample (7.90 percent vs 3.85 percent, Table 60). Consideration of time and cost would probably dictate the use of point- samples, and for precise estimations of the CV, the random distribution should be used. 88 Table 62. -- Analysis of the variance of the deviation of the expeCted value from the true parameter, Red Cedar Woods, point— sampling. Source df SS MS F Distribution 2 1450.7081 725.3540 * * Treatment 11 52,361.9979 4760.1816 * * Type 1 29,614.4767 29,614.4767 * * BAF 5 10,771.4065 2154.2813 * * Interaction 5 11,976.1147 2395.2230 * * Error 22 1544.5294 70.2059 Total 35 55,357.2354 * * Highly significant. CHAPTER V SUMMARY AND CONCLUSIONS Forest sampling methods used in this country are usually the ones that have empirically given the best results. Comparative studies among a few of the more popular designs have led to the use of some in prefer— ence to others. Investigation of a large number of methods is limited by cost and time for field work, and the problem of changing forest condi- tions. To overcome these difficulties, three forest areas were mapped to scale on large size graph paper and measured by a series of 144 sampling designs. These were a combination of six areal plot shapes with six plot sizes; two point-sampling or Bitterlich procedures with six basal area factors; and three methods of plot distributions -- random, systematic, and multiple-random—starts (MRS). The precision of each design was tested on the basis of its coefficient of variation by analysis of vari- ance techniques. No attempt was made to determine sampling efficiency with respect to time and cost. The Lansing Woods This tract of timber has the characteristics of an upland oak— hickory type. The average tree diameter is 11.3 inches. Some fires, grazing, and cuttings have occurred in the stand. The individual tree distribution is quite uniform. 89 90 Plot Size. -- Analysis of variance on the coefficient of variation statistics showed that the one-fifth and one-tenth acre plots were the most accurate by a significant margin of any plots tested (6.11 percent and 7.56 percent, respectively). The relationship between plot size and its coefficient of variation can be accurately expressed by the function: cv = 457.570(P)‘§ (8) where: CV = coefficient of variation in percent P = plot size in square feet 457.570 = the regression coefficient derived from the data Plot Shape. -- This factor had no more influence on the variation in coefficient of variation than could be any shape would have produced the same Point-Sampling. -- Coefficient of cedures varied directly with the basal attributed to chance. A plot of results. variation for‘the Bitterlich pro- area factor used. Following Palley and O'Regan (1961) this relationship is expressed by: cv = 19.905 - 15.351 (Prf)_1 (10) where: CV = coefficient of variation in percent Prf = plot radius factor in feet per inch of tree diameter 19.905 = derived regression constant -15.351 = derived regression coefficient According to the formula, the smallest give the lowest error. On this forest possible basal area factors will , the BAF-5 and BAF-10 gave the best results with coefficients of variation of 7.25 percent and 8.47 per- cent,respectively. The difference between these is of no importance. A 91 "t" test between plot and point-samples showed that either could be used with the assurance of the same expected precision. Point-sampling with or without the peripheral adjustments for edge bias showed the same level of sampling error. Distributions. -- The average coefficient of variation for the random, systematic, and MRS distributions are 13.44 percent, 14.16 percent, and 14.39 percent, respectively. None of these differences is significant. To test the reliability of a distribution to sample the population completely, the differences between the expected values and the true popu- lation mean were analysed. The random distribution showed an average dif- ference of only 8.70 percent as compared to 18.53 percent and 12.47 per- cent for the systematic and the multiple-random-start methods. This was highly significant and indicates that the random distribution was superior to either of the others. For point-sampling, the differences among distri- butions were insignificant as far as the difference between the expected and the true population was concerned. Toumey Woods This woodlot is a relatively undisturbed area of old-growth timber on a moist fertile morainal hill. The species composition is comparable to the sugar maple—beech type of southern Michigan. The average tree diameter is 15.3 inches. The distribution of the individual trees in the forest is uniform. Plot Size. -- The two larger plot sizes, (one-fifth and one-tenth acre) gave the most precise estimates of the mean with coefficients of variation of 5.74 and 7.67 percent. Smaller plots produced significantly less accurate results. The relationship between plot size and 92 the coefficient of variation can be expressed by the equation: cv = 670.399 (P)'§ (13) where: CV = coefficient of variation in percent P = plot size in square feet 670.399 = regression coefficient derived from the data Plot Shape. -- No significant differences were observed among the six plot shapes tested. Neglecting the questions of efficiency, a plot of any shape yields the same results. Point-Sampling. -- The range between point-sampling with peripheral adjustments (12.49 percent), and without this adjustment (7.62 percent) is highly significant. The large trees in this forest and the wide peri- pheral zones used, over-adjusted the estimate for each successively smaller basal area factor. Variations among basal area factors were not important. For all practical purposes, a BAF of any size would have re- sulted in estimations of the same precision. Differences between all point-samples and the best plot samples were also small enough to be ig- nored. The choice between these two methods would be a matter of efficiency. Distributions. -- Among distributions, the random arrangement gave the lowest coefficient of variation (18.79 percent vs 20.60 and 18.97 percent for the systematic and MRS, respectively. However, none of these differences were significant. Analysis of the differences between the estimated and the true mean for both the random and the systematic distribution showed that the systematic was significantly better than the random for both points and plots (Points: 2.10 percent vs 3.71 percent; and plots: 9.57 percent vs 15.85 93 percent). This may result from the tendency of random samples to cluster in certain areas as compared to the rigid dispersal of the systematic method. The Red Cedar Woods This study area occupies the low, poorly drained sites characteristic of the forest type ash-elm-red maple. It is a relatively undisturbed area with an average tree diameter of 13.8 inches. Unlike the Lansing and Toumey Woods, the trees in this forest show a tendency for clustering. Plot Size. -- The one-fifth acre and the one-tenth acre plots gave the significantly lowest coefficient of variation (9.84 and 12.58 percent), of the six plot sizes tested. Unless considerations of efficiency are important, the use of smaller plots is not recommended as indicated by the equation: cv = 649.965 (In-5 (16) where: CV = the coefficient of variation in percent P = plot size in square feet 649.965 = derived regression coefficient Plot Shape. -- This factor did not contribute an appreciable amount to the size of the coefficient of variation. Point—Sampling. -- The unadjusted point-samples result in much more‘ 11ccurate estimates than the adjusted point-samples (13.08 percent vs 16.30 percent). Differences in basal area factors also produce signifi- <2ant differences in the sampling error as expressed in the formula: 94 SE = 23.895 - 16.740 (Prf)_1 (18) where: CV = the coefficient of variation in percent Prf = the plot radius factor in feet per inch of tree diameter 23.895 regression constant -16.740 regression coefficient The BAF-5 with an average sampling error of 8.54 percent and the BAF-10 with an error of 8.72 percent are the most accurate Bitterlich point samples tested. The "t" test between the best point-samples and best plot samples, showed the Bitterlich system to be the more precise of the two (3.01 per- cent vs 8.57 percent). Distribution. -- The random distribution of sampling units produced the lowest average significant coefficient of variation of all three methods (20.07 percent vs 26.53 and 25.29 percent for the systematic and MRS, respectively). A test of the three distribution methods, based on the differences between the expected value and the true mean, showed no significant dif- ferences among methods. One method of sample unit distribution estimated the true mean as well as any other, for all practical purposes. Conclusions In each study area, the shape of the sample unit had no significant Gaffect on the magnitude of the coefficient of variation. A plot of any Sllape, within a given size, would yield approximately the same results. “naere the forest terrain is steep, or the changes in the forest condition aIVB abrupt, plot shape may become important. If efficiency is considered, tlle use of circular plots would perhaps be preferred, because of 95 their minimum perimeter for a given plot size, and ease of establishment (Johnson and Hixon, 1952; Lindsey 23 31., 1958). The unadjusted point-samples gave more precise results than the adjusted point-samples, except in one forest. In the Lansing Woods, with its comparatively small tree diameter and relatively narrow peripheral zones, the two methods of point-sampling were approximately equal. In analysing the differences between the expected value and the true mean, the adjusted point-samples gave estimates that were as much as 100 per- cent too high. Until the relationship between tree size, BAF, and edge- bias is thoroughly investigated, the adjustment point-sampling method has little to recommend it, unless the trees are small. As might be expected, the larger the size of the sample units, the more precise the estimate. In all three forests, the one-fifth and one- tenth acre plots were significantly more accurate than the smaller units. The difference between these two is negligible. On two of the areas, either the BAF-5 or the BAF-10 would give more accurate estimates than basal area factors in which fewer trees were sampled. In the Toumey Woods, with its even distribution of large trees, all basal area factors gave approximately the same results. Several regression equations showed the inverse relationships between sample unit size and the coefficient of ‘Variation. The limit on the maximum sampling unit to use, would be a Inatter of efficiency. According to Table 63, the most precise point-samples were more ac- <1urate than the comparable plot samples in each forest. This difference vVas of significance in the Red Cedar Woods. Various studies (Bell, 1957; Bergeson, 1958; Deitschman, 1956; Grosenbaugh and Stover, 1957) have 96 demonstrated the effectiveness and ease of using point-samples. Under the assumption that point-samples require less time to measure, and con- sidering their precision in this study, they should be employed for the most efficient results. Table 63. -- Summary of the coefficient of variation parameters for the most accurate plot samples and point-samples. Sample Lansing Woods Toumey Woods Red Cedar Woods* (percent) (percent) (percent) BAF-5 5.87 4.58 7.33 BAF-10 7.28 5.66 5.38 l/5-Acre 6.11 5.74 8.57 1/10-Acre 7.56 7.67 10.19 *Random distribution only. In the three study areas, the random distribution proved to be stat- istically equal to, or better than, the systematic and multiple-random- start methods, and, in the case of the Red Cedar Woods, this superiority was highly significant. Analysis of the difference between the expected mean and the true mean showed the random method to be significantly best on the Lansing Woods, while the systematic was superior on the Toumey area. On the Red Cedar Woods, these differences could be attributed to Chance. The multiple-random-start arrangement offered little to recom- lnend it in either analysis. Because it is impossible to determine variation patterns in botani- ‘381 populations, and because nature fails to randomize properly, the ran- ‘3Cnn sample distribution should be used at every opportunity. It is the or11y arrangement of plot that affords a precise estimate of the sampling 97 error. In this study, it yielded results as good or better than the sys- tematically placed samples. Although the three forests are distinctive with respect to species, site, tree size, and distribution, nearly the same sampling methods are applicable to each area. Considering the work of other investigatiors on sampling efficiency, and the accuracy of the method involved in this study, the unadjusted BAF-5 or BAF-10 point-sample should be used on a random distribution in forests with these characteristics. 98 LITERATURE CITED Afanasiev, M. 1957. The Bitterlich method of cruising -- why does it work? Jour. Forestry 55: 216-217. 1958. Some results of the use of the Bitterlich method of cruising in an even age stand of longleaf pine. Jour. Forestry 56: 341. Allen, R. H., and E. W. Mongren. 1960. Range-mean ratio of basal area as an indication of Bitterlich sampling intensity in lodgepole pine. Colorado State Univer- sity College of Forestry and Range Management, Res. Note 13, 2 pp. Avery, T. E. 1955. Gross volume estimation using plotless cruising in southern Arkansas. Jour. Forestry 53: 206-207. Barton, W. B., and C. B. Stott. 1946. Simplified guide to intensity of cruise. Jour. Forestry 44: 750-754. Barton, J. D. 1956. Comparison of four forest sampling techniques. Unpublished Ph.D. thesis. Purdue University. Bell, J. F. 1957. Application of the variable plot method of sampling forest stands. Oregon State Board of Forestry, Res. Note 30, 22 pp. Bickford, A. C. 1961. Stratification for timber cruising. Jour. Forestry 59: 761- 763. Bitterlich, W. 1947. Die Winkelzahlprobe. Allegemeine Forest-und Holzwirtschaftlicke Zeitung 58: 94-96. 1948. Die Winkelzahlprobe. Allegemeine Forest—und Holzwirtschaftlicke Zeitung 59(1/2): 4-5. Borgeson , A. E. 1958. A field test of the Bitterlich variable plot cruising method in Maine. Maine University Forestry Dept., Tech. Note 48, 4 pp. BOrman , F. H. 1953. The statistical efficiency of sample plot size and shape in forest ecology. Ecology 34: 474-487. 99 Bourdeau, P. F. 1953. A test of random vs systematic ecological sampling. Ecology 34: 499-512. Bruce, D. 1955. A new way to look at trees. Jour. Forestry 53: 163-167. Bryan, M. B. 1959. Improving accuracies of volume tables. Topic 5-7 San Fran- cisco Forest Survey Meeting. Southeast Forest Expt. Sta., un- published memo, 32 pp. Cmfly,&.m 1927. Accuracy of methods of estimating timber. Jour. Forestry 25: 164-169. Clapham, A. R. 1932. The form of the observational unit on qualitative ecology. Jour. Ecology 20: 192-197. Cochran, W. G. 1953. Sampling techniques. John Wiley & Son, Inc., New York. 330 pp. Illus. Cottam, G. 1947. A point method for making rapid surveys of woodlands. Bull. Ecol. Soc. of America 28: 60. , and J. T. Curtis 1949. A method of making rapid surveys of woodlands by means of pairs of randomly selected trees. Ecology 30: 101-104. , and 1956. The use of distance measures in sampling. Ecology 37: 451-460. 1953. Some sampling characteristics of a population of randomly dis- persed individuals. Ecology 34: 741-757. ijx, P. 1961. A test of variable plot cruising in mixed stands on Lafour State Forest. Calif. Dept. of Nat. Res., Division of Forestry, State Forest Note 5. Deitschman, G. H. 1956. Plotless timber cruising tested in upland hardwoods. Jour. ' Forestry 54: 844-845. E“:Lnney, D. J. 1948. Random and systematic sampling in timber surveys. Forestry 22: 64-99. 1949. An example of periodic variation in forest sampling. Forestry 23: 91—111. 100 Freese, F. 1961. Relation of plot size to variability: an approximation. Jour. Forestry 59: 679. 1962. Elementary forest sampling. Agriculture Handbook 232, USDA, Forest Service, 91 pp. Gerard, J. W., and S. R. Gevorkiantz. 1939. Timber cruising, USDA, Forest Service, Mimeograph, 160 pp. Goodspeed, A. A. 1934. Modified plot methods of timber cruising applications in Southern New England. Jour. Forestry 32: 43-46. Grosenbaugh, L. R. 1952. Plotless timber estimates -- new, fast, easy. Jour. Forestry 50: 32-37. 1955. Better diagnosis and prescription in southern forest manage- ment. USFS Occasional Paper 145, 27 pp., illus. 1955. Comments on "Results of an investigation of the variable plot method of cruising." Jour. Forestry 53: 734. , and W. S. Stover. 1957. Point-sampling compared with plot-sampling in Southern Texas. Forest Science 3: 2-13. 1958. Point-sampling and line-sampling: probability theory, geometric implications, synthesis. USFS Occasional Paper 160, 34 pp., illus. Haza, T. and K. Maezawa. 1959. Bias due to edge effect in using the Bitterlich method. For- est Science 5: 370-376. Hasel, A. A'. 1938. Sampling errors in timber surveys. Jour. Agr. Res. 57: 713-736. 1942. Estimation of volume in timber stands by strip sampling. Ann. Math. Stat. 13: 179-206. Husch, B. 1955. Results of an investigation of the variable plot method of cruising. Jour. Forestry 53: 570-574. cJohnson, F. A., and H. J. Hixon. 1952. The most efficient size and shape of plot to use for cruising in old growth Douglas-fir timber. Jour. Forestry 50: 17-20. 101 1948. Statistical aspects of timber-volume sampling in the Pacific Northwest. Jour. Forestry 47: 292-294. Justesen, H. S. 1932. Influence of size and shape of plots on the precision of field experiments with potatoes. Jour. Agr. Sci. 22: 365-372. Kalamkar, R. J. 1932. Experimental error and the field plot technique with potatoes. Jour. Agr. Sci. 22: 373-383. Ker, J. W. 1957. Observations on the accuracy and utility of plotless cruising. British Columbia. University, faculty of forestry, Research Note 15, 2 pp. Lemmon, P. E. 1958. Aids for using wedge prisms. Jour. Forestry 56: 767-768. Lindsey, A. A. 1956. Sampling methods and community attributes in forest ecology. Forest Science 2: 287-296. , J. D. Barton, and S. R. Miles. 1958. Field efficiencies of forest sampling methods. Ecology 39: 428-444. Malain, R. M. 1961. A test of variable plot cruising in young growth redwood. California Dept. of Natural Resources, Division of Forestry; Note 7, 7 pp. Mesavage, C., and L. R. Grosenbaugh. 1956. Efficiency of several cruising designs on small tracts in North Arkansas. Jour. Forestry 54: 569-576, illus. Meyer, H. A. 1948. Cruising by narrow strips. Penn. State Forestry School, Re- search Paper 12, 3 pp. “ 1949. Cruising intensity and accuracy of cruises. Jour. Forestry 47: 646-649. 1956. Calculation of the sampling error of a cruise from the mean— square-successive-differences. Jour. Forestry 54: 341. 1Moore, P. G. 1955. The properties of the mean square-successive-differences in samples from various populations. Jour. American Stat. Assoc. 102 Mudgett, B. D., and S. R. Gevorkiantz. 1934. Reliability of forest surveys. Jour. American Stat. Assoc. 29: 257-281, illus. Orr, T. 1959. Timber stand maps, plotless cruising and business machine computations as elements of a timber survey method. Jour. Forestry 57: 567-572. Osborne, J. G. 1942. Sampling errors of systematic and random surveys of cover type areas. Jour. American Stat. Assoc. 37: 256-264. Palley, M. N., and L. A. Horwitz. 1961. Properties of some random and systematic point sample esti- mators. Forest Science 7: 52-65. , and W. G. O'Regan. 1961. A computer technique for the study of forest sampling methods. Forest Science 7: 282-293. Preston, J. F. 1934. Better cruising methods. ’Jour. Forestry 32: 876-878. Rice, E. L., and William T. Penford. 1955. An evaluation of the variable radius and paired-tree method in the black-jack-post oak forest. Ecology 36: 315-320. Robertson, W. M. 1927.1 The line plot system: its use and application. Jour. Forestry Schumacher, F. X., and H. Bull. 1932. Determination of error for a forest survey. Jour. Agr. Re- search 45: 741-756. , and R. A. Chapman. 1948. Sampling methods in forestry and range management. Duke Univ. School of Forestry. Bull. 7: 222 pp. illus. Shanks, R. E. 1954. Plotless sampling trials in Appalachian forest types. Ecology 35: 237-244. Shiue, Co ' 1960. Systematic sampling with multiple random starts. Forest Science 6: 42-50. Smith, J. H. G., and J. W. Ker. 1957. Some distributions encountered in sampling forest stands. Forest Science 3: 137-144. Snedecor, G. W. 1959. Statistical methods. The Iowa State College Press, Ames, Iowa. 534 pp., illus. 103 Stage, A. R. 1958. 1962. An aid for comparing variable plot radius with fixed plot radius cruise designs. Jour. Forestry 56: 593. A field test of point-sample cruising. USFS, Intermountain Forest and Range Expt. Sta., Research Paper 67, 17 pp. Sudia, T. W. 1954. Trappe, J. 1957. Warren, W. 1960. Wright, W. 1925. Yates, F., 1935. A comparison of forest ecological sampling techniques with use of a known population. Unpublished Ph.D. thesis, Ohio State Univ. M. Experience with basal area estimation by prism in lodgepole pine. Pacific Northwest Forest and Range Expt. Sta., Research Note 145. G Measurement of stand basal area with optical wedges. Forest Research Institute, Technical Paper 33, Wellington, New Zealand, 12 pp. G. Variation in stand as factor in accuracy of estimates. Jour. Forestry 23: 600-607. and J. Zacopanay. The estimation of the efficiency of sampling with special re- ference to sampling for yield in cereal experiments. Jour. Agr. Sci. 25: 545-577. APPENDI X 104 APPENDIX A. Example 1. -- Record Sheet for Sampling Basal Area on Red Cedar Woods. Circular Plots on a Random Distribution Plot 1 Plot 2 Plot 3 Size BA Acc. BA Size BA Acc. Size BA Acc. BA 1/160 -- 0 1/160 -- 0 1/160 -- 0 1/80 -- 0 1/80 -- 0 1/80 1379 1/40 245 245 1/40 1344 184 1563 1/20 894 ' 852 2196 1/40 1467 3030 3409 4548 1/20 1396 3592 1/20 307 1/10 979 1/10 625 1163 4500 1227 6754 442 1/10 2664 1/5 734 1948 238 1131 1069 568 672 894 8570 3947 601 1/5 1485 908 1414 11306 275 2181 2248 1277 -16283 721 1/5 275 291 1362 413 291 18211 223 1344 1310 245 2053 1558 648 21384 LPlot 4 Plot 5 Plot 6 Size BA Acc. BA Size BA Acc. BA Size BA Acc. BA 1/160 1006 965 1/160 -- 0 1/160 1948 1948 432 2403 1/80 -- 0 1/80 -- 1948 1/80 524 2927 1/40 1227 1227 1/40 1614 3562 14/40 812 1/20 1748 1/20 1632 772 4511 1651 4626 1485 6679 1/20 ‘ 307 4818 1/10 210 1/10 -- 6679 1/ 10 747 190 1/5 367 950 825 153 1227 1147 1379 472 442 1211 524 482 7922 503 601 1/5 625 825 535 462 315 1211 11135 245 760 1/ 5 472 349 315 672 260 3168 1131 2712 203 APPENDIX A., Example 1, Con't. Plot 4 Size (1/5) Plot 7 Size BA Acc. BA Plot 5 Size 1558 147 1867 2688 2475 2337 1709 799 965 1948 BA 29903 Acc. BA 1/160 1/80 1/40 1/20 1/10 1/5 672 1131 1277 747 472 636 1024 950 747 472 579 1558 812 524 432 238 636 1039 1344 184 147 524 965 1006 994 1115 403 3080 4299 5959 8128 12271 20628 (1/5) Plot 8 Size 1/160 1/80 1/40 1/20 1/10 1/5 105 Plot 6 BA Acc. BA Size BA Acc. BA 1244 (1/5) 1227 216 9079 26184 852 299 825 216 1728 432 18387 Plot 9 BA Acc. BA Size BA Acc. BA -— 0 1/160 -— O —- 0 1/80 —— 0 601 601 1/40 1748 3142 3743 432 2181 1179 3359 1709 7633 1/20 601 3960 184 1/10 1054 908 2117 734 147 7278 299 1/5 323 772 1100 171 11701 2292 2712 1244 2270 556 965 18740 APPENDIX A., Example 1, Con't. Plot 10 Size BA Acc. BA 1/160 1/80 1/40 1/20 1/10 1/5 Plot 13 Size 472 482 1651 2605 2605 2605 3206 6216 12610 Acc. BA 1/160 1/80 1/40 1/20 1/10 1/5 291 291 291 1461 7386 17716 Plot 11 Size BA Acc. BA 1/160 994 994 1/80 394 340 165 1893 1/40 1260 2810 5963 1/20 299 6262 1/10 936 7198 1/5 442 2074 9714 Plot 14 Size BA Acc. BA 1/160 -— 0 1/80 252 252 1/40 -- 252 1/20 3801 4053 1/10 1084 697 394 965 7193 1/5 4876 613 4337 503 908 579 799 1576 1115 880 825 2451 153 190 26998 106 Plot 12 Size BA Acc. BA 1/160 852 853 1/80 -- 852 1/40 2181 3033 1/20 238 852 2074 6197 1/10 323 299 6819 1/5 3631 299 1260 994 165 340 936 442 14886 Plot 15 Size BA Acc. BA 1/160 3888 2737 6625 1/80 1449 8074 1/40 -- 8074 1/20 332 482 8888 1/10 223 1100 367 4035 14613 1/5 545 6305 545 1195 648 1115 165 25131 APPENDIX A., Example 1, Con't. 107 Plot 16 Plot 17 Plot 18 Size BA Acc. BA Size BA Acc. BA Size BA Acc. BA 1/160 —— 0 1/160 1179 1/160 -- 0 1/80 -- 0 432 1611 1/80 223 223 1/40 697 1/80 -- 1611 1/40 -- 223 203 1/40 601 2212 1/20 1414 1637 1179 2079 1/20 —- 2212 1/10 165 1/20 159 2238 1/10 1748 1163 1/10 1344 1244 1260 203 2117 1115 1039 1054 1195 1100 147 8522 210 6745 291 1/5 556 1/5 2885 1867 2270 684 672 8754 965 349 1/5 1503 590 307 697 852 13755 1748 880 852 13755 376 1244 590 492 332 1163 358 291 349 14723 3355 1887 908 323 21497 108 APPENDIX A. Example 2. Plot summary sheet and parameter computation table for circular plots on the random distribution, Red Cedar. Plot Size in Acres Plot No. 1/160 1/80 1/40 1/20 1/10 1/5 1 0 o 0.245 4.548 6.754 11.306 2 0 0 2.196 3.592 8.570 21.384 3 0 1.563 3.303 4.500 16.283 18.211 4 2.403 2.927 4.511 4.818 11.135 29.903 5 0 0 1.227 4.626 7.922 18.387 6 1.948 1.948 3.562 6.629 6.679 26.184 7 3.080 4.299 5.959 8.128 12.271 20.628 8 0 0 0.601 3.743 7.633 11.701 9 0 0 3.359 3.960 7.278 18.740 10 2.605 2.605 2.605 3.206 6.216 12.610 11 0.994 1.893 5.693 6.262 7.198 09.714 12 0.852 0.852 3.033 6.197 6.819 14.886 13 0.291 0.291 0.291 1.461 7.386 17.716 14 0 0.252 0 4.053 7.193 26.998 15 6.625 8.074 8.074 8.888 14.613 25.131 16 0 0 2.079 2.238 8.754 21.497 17 1.611 1.611 2.212 2.212 8.522 13.755 18 0 0.223 0.223 1.637 6.745 14.723 Total 20.409 26.538 48.900 80.748 157.971 333.747 Plot Mean 1.134 1.474 2.717 4.486 8.776 18.542 Mean per Acre 181.44 117.92 108.68 89.72 87.76 92.71 :xp1ots)2 74.126 112.364 218.694 436.848 1541.995 6765.601 Variance 2.999 4.308 5.050 4.389 9.154 33.967 Standard Deviation 1.732 2.076 2.242 2.095 3.026 5.828 Standard Deviation Per Acre 277.12 166.08 89.88 41.90 30.26 29.14 109 04/03/310 I, . , ‘ 1. _\\ ‘- _§_ oI/m/no 50 I60 0 "Int/on \\ 0 "/"' 0 01/13/26: \ n/utlgur \\ o also/u. \‘\\ \_\ O/TJh‘IS ’ \ .\ QDZ/IOJ/lu! \\.~ ~ ‘ \ MO *, a \ Gmfml’..7 g'l').o. \\ '7 0? 3.01307 . 1 ' ‘ 8 ‘ .\ 1 0.11160! 11m 1 , 3 ., [ix/7.2:! . . .7 . 2 b Q/IOJ/IJ‘IO \‘ yup“ ‘ 120 x ‘3. I I, y \5\_,,~ / .1 I‘ \ \ R \ O 3/153/1277 05 me ‘~ \ “\~ 0 I/10.9/ so: ‘ 3 n // o 3/N.o/ ' a) . /// ot/u.u/10u ’/ owl/"3 O‘skna.-9/_9o0-f ‘0 ouo/tor/uss o o/n.o /uo 0.2.. 87:173. 00/12/30? ‘0 Oa/ur/om 01/03/0000 Ov/uo/nu LEGEND 0 Ofi/N/ln 0 not out" show/2333 /'4 o/ on L MI noun go 0" - ' m m mun ulcuu) 01/40/912 Out/l!.t/Ilt O us/u/m ’23—. m an. “.- n.) 04 9‘ t “I?“ ”ll? €11,239." 5/ on i mun-uno- (um alto/nu 8 nun noun ”9w mat/"fl“ an “an 0 20 4O 60 00 APPENDIX A . contain three decimal places. IOO IZO -- Section of the Red Cedar Woods with the circular plot system on random sampling unit three. Basal area values Factors (BAF) APPENDIX B. -- Plot Radius in Feet by Tree Diameters and Basal Area 110 Tree Diameter Basal Area Factors Inches BAF-5 BAF-10 BAF-20 BAF-30 BAF-40 BAF-50 ----------- plot radius feet - - - - - - - - - - - 5 19.4 13.8 9.7 7.9 6.9 6.2 6 23.3 16.5 11.7 9.5 8.3 7.4 7 27.2 19.3 13.6 11.1 9.6 8.6 8 31.1 22.0 15.6 12.7 11.0 9.8 9 35.0 24.8 17.5 14.3 12.4 .11.1 10 38.9 27.5 19.4 15.9 13.8 12.3 11 42.8 30.3 21.4 17.5 15.1 13.5 12 46.7 33.0 23.3 19.1 16.5 14.8 13 50.5 35.8 25.3 20.6 17.9 16.0 14 54.4 38.5 27.2 22.2 19.3 17.2 15 58.3 41.3 29.2 23.8 20.6 18.5 16 62.2 44.0 31.1 25.4 22.0 19.7 17 66.1 46.8 33.0 27.0 23.4 20.9 18 70.0 49.5 35.0 28.6 24.8 22.1 19 73.9 52.3 36.9 90.2 21.1 23.4 20 77.8 55.0 38.9 31.8 27.5 24.6 21 81.7 57.8 40.8 33.3 28.9 25.8 22 85.5 60.5 42.8 34.9 30.3 27.1 23 89.4 63.3 44.7 36.5 31.6 28.3 24 93.3 66.0 46.7 31.1 33.0 29.5 25 97.2 68.8 48.6 39.7 34.4 30.8 26 101.1 71.5 50.5 41.3 35.8 32.0 27 105.0 74.3 52.5 42.9 37.1 33.2 28 108.9 77.0 54.4 44.5 38.5 34.4 29 112.8 79.8 56.4 46.1 39.9 35.7 30 116.7 82.5 58.3 47.6 41.3 36.9 31 120.5 85.3 60.3 49.2 42.6 38.1 32 124.4 88.0 62.2 50.8 44.0 39.4 33 128.3 90.8 64.2 52.4 45.4 40.6 34 132.2 93.5 66.1 54.0 46.8 41.8 35 136.1. 96.3 68.0 55.6 48.1 43.1 36 140.0 99.0 70.0 57.2 49.5 44.3 37 143.9 101.8 72.0 58.8 50.9 45.5 38 147.8 104.5 73.9 60.3 52.3 46.7 39 151.7 107.3 75.8 61.9 53.6 48.0 40 155.5 110.0 77.8 63.5 55.0 49.2 41 159.4 112.8 79.7 65.1 56.4 50.4 42 163.3 115.5 81.6 66.7 57.8 51.7 111 APPENDIX C Example of the Analysis of Variance and Plot Comparisons Table 15 on page 41 gives the summary of the coefficient of varia- tion data for the Lansing Woods. Handling of these data by the analysis of variance is as follows: Correction term 108 (i x)2 108 1511.692 108 Total sum of squares Distribution 55’ Treatment SS Error SS 108 2 :E| x — C. T. (26.742 + 17.722 + . . . + 4.892) - C.T. 27,762.1859 - C. T. 6602.8650 on 107 degrees of freedom (df) 3 i (distribution totalsz) - C. T. Obs. per distribution (483.782 + 509.782 + 518.132) - C.T. 36 17.8300 on 2 df. 3 1? (Plot size totalsz) - C.T. LObs. per total (77.432 + 57.042 + . . . + 15.102) - C.T. 3 6247.9466 on 35 df. Total SS - Distribution SS - treatment SS 337.0884 on 70 df. 112 APPENDIX C. -- Analysis of variance Source df SS MS F Distribution 2 17.8300 8.9150 1.85 N. S. Treatment 35 6247.9466 178.5128 37.07 * * Error _19 337.0884 4.8155 Total 107 6602.8650 N.S. Non-significant * * Highly significant Treatment is significant, so the sum of squares and degrees of freedom making up this source may be partitioned into the respective factors: plot size, plot shape, and size-shape interaction. This is carried out in the following way: 6 p101: type 85 = 2(241.782 + 259.532 + . . . + 258.252) - C.T. 18 12.7520 on 5 df. 6 2600.722 + 346.852 + . . . + 110.062) - C.T. 18 Plot size SS 6137.4834 on 5 df. Treatment SS - Plot type SS - Plot size SS Interaction 97.7112 on 25 df. 113 APPENDIX C. -- Analysis of variance. Source df SS MS F Distribution 2 17.8300 8.9150 1.85 N.S. Treatment 35 6247.9466 178.5138 37.07 * * Plot type 2 12.7520 2.5504 - N.S. Plot size 5 6137.4834 1227.4969 254.91 * * Interaction 25 92.7112 3.9084 - N.S. Error _19 337.0884 4.8155 Total 107 6602.8650 N.S. Non-significant * * Highly significant The five degrees of freedom for plot type and their associate sums of squares may be partitioned evenly among several orthogonal comparisons ac- cording to: APPENDIX C. -- Orthogonal comparisons of plot sizes. Plot Size - Acres .Algebraic _1_ .1_ .1_ _1_ _1_ .1 Sum of Co- Plot Comparisons 160 80 40 20 10 5 efficients 1/5 vs 1/10 0 0 0 0 + - 0 1/5 + 1/10 vs 1/20 0 0 0 +2 — - 0 1/5 + 1/10 + 1/20 vs 1/40 0 0 +3 - - - 0 1/5 + 1/10 + 1/20 + 1/40 vs 1/80 0 +4 - - — - 0 Others vs 1/160 +5 — - - - — 0 Product of Coefficients 0 0 0 0 + - = 0 If the comparisons check orthogonally then the sum of squares of the comparisons will equal the sum of squares for the plot size in the analysis of variance, and the test of significance can be based on the error term with one and 70 degrees of freedom. 114 eqnefimaaman magma: * * penoamaeman-eoz .m.z vmmv.bmH® mmawSUm wo sum fiance * * mumm.mmae Amavom\mflAem.oflmvuame.oomvmu u omH\H me naeseo * * mmea.mnea AmavomxmmAmfl.eemvuamm.eemveg u ow\H n» oe\H + om\H + oH\H + n\H * * Hemm.mme AmfivmflxmfiAmo.HmeV-AoH.memvmg u oexfl me o~\H + ofixfi + m\H * * oflmm.me AmavexmmAeo.oemvnanm.eeavmg u omxfi n> oH\H + n\H .m.z mm.n ence.me Amflvm\maeo.ofiauflo.emav n oH\H n> nxfl m mm manpoQOHQ nemfiaquoo pon .mU003 MQHmGNwH was. .HOH mGOmHHNQEOU #OHAH HGOH‘ ll .0 NHQZWAHQ< 115 The distributions totals from Table 18 are random, 483.78; system- atic, 509.78; MRS, 518.13. The possibility exists that the random distri- bution may be significantly smaller than the other two. The two degrees of freedom and the sums of squares for distribution may be partitioned into two comparisons -- each with one and seventy degrees of freedom. (518.13-509.78)2 72 Systematic vs MRS F = .9684 Nos. Random vs (Systematic + MRS) '2(483.78-(509.78 + 518.137l2 6(36 16.8617 ’11 II 16.8617 ==350 N.S. 4.8155 There are no significant differences among these distributions. Point-Sampling. -- The same type of analysis was applied to the point-sampling data from Table 19. There were no significant differences. R0034 use 02%“ ‘1 J g 2y.“ .1“ . T: RIEs ..WWEggijfijfliuwfliflig’ixfilwx