Arboriculture & Urban Forestry 41(6): November 2015 was varied within the formula for standard error with the known population variance for each city. Variation in Sample Estimates To illustrate how random samples can yield vary- ing estimates of population totals, five block seg- ments in Syracuse, New York, were randomly selected 30 different times to estimate the total street tree population. Then 10 block segments were randomly selected 30 different times, fol- lowed by 15 block segments, 20 block segments, and so on, until 100 block segments were randomly selected 30 times. Individual sample estimates along with average street tree population estimates and standard errors from the 30 samples were graphed and contrasted against the actual street tree population total. Plotting the 30 samples gives a visual idea of how much the individual estimates will vary. A larger sample size is not needed as RSE does not need to be estimated and is calculated using the exact formula for RSE for any sample size. Ratio Estimates Street tree data from one city with street length information (Syracuse, New York) were used to illustrate how the RSE may differ between simple random sample estimates and ratio estimates using known block lengths. For each block seg- ment in the city, the number of trees was counted, No. Buffalo, New York Lansing, Michigan Livonia, West Virginia Syracuse, Delaware z blocks Totalz 5,954 5,381 3,357 Michigan Parkersburg, 876 5,122 New York Wilmington, 2,132 Total number of trees. y Number of block segments sampled to reach a 10% relative standard error. x Percent of block segments needed to reach a 10% relative standard error. w Average number per block. v Standard error of average number per block. 10% RSEy 67,593 145 57,902 116 37,854 89 1,734 279 34,149 185 12,586 98 Street trees %x 2.4 2.2 2.7 349 as well as the average and standard error of trees per block using simple random and ratio esti- mates based on block length. To illustrate how RSE varies with sample size between the simple random and ratio estimate approaches, the number of block segments sampled was var- ied within the formula for standard error with the known population variance for each city. RESULTS For five out of the six cities analyzed, the per- cent of block segments needing to be sampled to attain a 10% RSE was between 2.2% and 4.6%, with total number of block segments sampled between 89 and 185 (Table 1; Table 2). How- ever, for the smallest city analyzed, Parkersburg, West Virginia, nearly 32% of the block segments need to be sampled to attain a 10% RSE, which required 279 block segments to be sampled. When analyzing subsets of the popula- tion (e.g., removals, Norway maples), the number and percent of block segments needing to be sampled to attain a 10% RSE increased. For removals, the percent of block segments needing to be sampled ranged between 9.7% and 58.8%; for Norway maple it ranged between 6.0% and 72.8% (Table 1). Sample size affected the variability of the esti- mate. Relative standard error decreased at differing Table 1. Street tree, removal, and Norway maple statistics for analyzed U.S. cities. City Avg.w 11.4 10.8 11.3 31.8 2.0 3.6 4.6 6.7 5.9 SEv 13.8 11.7 10.7 4.0 9.2 6.0 Totalz 3,119 0 471 132 2,345 263 10% RSEy 578 na 814 515 654 770 Removals %x 9.7 na 24.2 58.8 12.8 36.1 Avg.w 0.5 0.0 0.1 0.2 0.5 0.1 SEv 1.3 0.0 0.5 0.5 1.3 0.4 Totalz 10% RSEy 15,823 523 9,788 323 10,330 277 107 8,357 573 638 329 654 Norway maple %x 8.8 6.0 8.3 Avg.w 2.7 1.8 3.1 72.8 0.1 6.4 1.6 30.7 0.3 SEv 6.4 3.4 5.3 0.6 3.1 0.8 ©2015 International Society of Arboriculture
November 2015
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