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Mean and Weighted Mean Centers The mean center is a measure of the geographic center of a set of observations. This is analogous to statistical mean. We will use the x,y coordinates to determine the mean center. • Best to use projected coordinates whose units are meters, feet, etc… • Geographic coordinates (lon/lat) can be used but are harder to deal with. The mean center is calculated using the equations: n X Coord X i 1 n n i Y i YCoord i 1 n n 13 X Coord YCoord 165754 159382 K ...173851 170924 13 21808553 2176152 K ...2179391 2173138 13 Mean Center 170924, 2173138 170924 2173138 The weighted mean center is calculated by multiplying the coordinate by the weighting factor, using the equations: n w X i XW i 1 n w i 1 i n i w Y i i YW i 1 n w i 1 i X Coord YCoord (2275)165754 (3522)159382...(1613)173851 8859431281 173215 2275 3522...1613 51147 (2275)21808553 (3522)2176152...(1613)2179391 111105588481 2172280 2275 3522...1613 51147 Weighted Mean Center 173215, 2172280 Standard Distance The standard distance is analogous to the standard deviation. We first determine the mean X and Y coordinates and then calculate the squared coordinate deviates. More dispersed point patterns will have larger standard distances, clustered points will have smaller standard distances. One can also substitute the weighted mean center and observations to calculate a weighted standard distance. This method is sensitive to extreme observations (e.g. a point lying far from the rest). The SD is a radius in map units around the mean center. First calculate the mean center: n 18 3996648 38960860 X 222036 Y 2164492 18 18 Mean Center 222036, 2164492 X 222036 Y 2164492 Xi Yi Xdeviate2 Ydeviate2 215058 2168211 48692484 13829308 215586 2166276 41602500 3181863 217766 2164588 18232900 9173 217872 2162759 17338896 3004059 217344 2161738 22014864 7585740 214741 2159839 53217025 21652477 219666 2165081 5616900 346659 220400 2162020 2676496 6111883 222304 2161352 71824 9860996 224309 2162618 5166529 3512709 226736 2159875 22090000 21318741 232399 2161563 107391769 8580343 229023 2163357 48818169 1288729 225997 2165608 15689521 1244960 230183 2172362 66373609 61933402 224626 2167508 6708100 9094916 222515 2166452 229441 3840729 220123 2169653 3659569 26633627 ∑ 3996648 38960860 485590596 203030315 SD 485590596 203030315 6185.2 meters 18 18 Note that the SD is a radius around the mean center in coordinate units (e.g. meters). This is analogous to 1 standard deviation. Concentric rings could be mapped to display several standard distances. Runs Test for Sequential Nominal Data We are interested in buildings along several streets relative to the downtown area. Buildings coded as black squares have had the same business operating for over 10 years (successful). Those buildings coded as white squares have had more than two businesses operating there in the last 10 years (unsuccessful). With Run A we will perform a two-tailed test. Run A Unlike other tests there is no equation for the runs test unless the sample size of either group is greater than 30. One only needs to count the number of runs (u), a run being a series of the same nominal value when counting from one end of the series to the other. Run A Test 1 (Sample A): Two-tailed Test Ho : The distribution of shops along Elm St. is not different than random. Ha : The distribution of shops along Elm St. is different than random. α = 0.05, 2 n1 = 20 (white, open businesses) n2 = 10 (black, closed businesses) u = 13 uCritical = 9, 20 u = 13 Since 9 < 13 < 20 accept H0 The distribution of successful and unsuccessful businesses along Elm St. is random (u13, p > 0.05). As with all tests of randomness: 2-tailed tests allow us only to state the samples are either random or not random. 1-tailed tests allow us to state the samples are either random or clustered. • Clustered: calculated value < the lower critical value. • Random: calculated value falls between critical values. • Uniform: calculated value > the upper critical value. Run B Test 1 (Sample A): One-tailed Test Ho : The distribution of shops along Walnut St. is random. Ha : The distribution of shops along Walnut St. is not random. α = 0.05 n1 = 12 (white, open businesses) n2 = 14 (black, closed businesses) u=7 uCritical = 9, 19 u=7 Since 7 < 9 reject H0 The distribution of successful and unsuccessful businesses along Walnut St. is clustered (u7, p < 0.05). X2 Contingency Analysis This technique uses contingency tables to calculate the expected frequencies of an event based on the observed frequency distribution. Expected frequencies are determined from the row and column totals from the contingency table. H0 : Abandonment level is independent of road type. Ha : Abandonment level is contingent upon road type. 2 ( f ij fˆij )2 fˆ where ij ˆf ( Rowi )(Column j ) ij n and df ( Rows 1)(Columns 1) where f̂ are the expected frequencies, and f are the observed frequencies. Contingency Table Graded Road Dirt Track TOTAL High Abandonment 3 14 17 Moderate Abandonment 7 9 16 Low Abandonment 10 9 19 TOTAL 20 32 52 Graded Road Dirt Track TOTAL High 3 6.54 14 10.46 17 Moderate 7 6.15 9 9.85 16 Low 10 7.31 9 11.69 19 20 32 52 TOTAL (20)(17) fˆ1,1 6.54 52 (20)(16) fˆ1, 2 6.15 52 (20)(19) fˆ1,3 7.31 52 (32)(17) fˆ2,1 10.46 52 (32)(16) fˆ2, 2 9.85 52 (32)(19) fˆ2,3 11.69 52 ˆf ( Rowi )(Column j ) ij n (3 6.54) 2 (7 6.15) 2 (10 7.31) 2 (14 10.46) 2 (9 9.85) 2 (9 11.69) 2 6.54 6.15 7.31 10.46 9.85 11.69 2 2 1.92 0.12 0.99 1.20 0.07 0.62 2 4.92 df (2 1)(3 1) 2 02.05, 2 5.991 Since 4.92 5.991 accept H 0 Village abandonment levels are not contingent upon road type within the Colchane region of Chile and Bolivia (χ24.92, 0.10 > p > 0.05).