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E.2 Wilcoxon Rank Sum Test

Contents
E.2.1 Sample sizes for Wilcoxon Rank Sum Test;
E.2.2 Critical values for the Wilcoxon Rank Sum Test test;
E.2.3 Rejecting null hypothesis based on Wilcoxon Rank Sum Test;
E.2.4 Power of the Wilcoxon Rank Sum Test;
E.2.5 Spreadsheet Formulas for the Wilcoxon Rank Sum Test

E.2.1 Sample sizes for Wilcoxon Rank Sum Test

Table E.3 Sample sizes for Wilcoxon Rank Sum test
(Number of measurements to be performed in the reference area and in each survey unit)

(α,β) or (β,α)
0.01 0.01 0.01 0.01 0.01 0.025 0.025 0.025 0.025 0.05 0.05 0.05 0.1 0.1 0.25
Δ/σ 0.01 0.025 0.05 0.1 0.25 0.025 0.05 0.1 0.25 0.05 0.1 0.25 0.1 0.25 0.25
0.1 5452 4627 3972 3278 2268 3870 3273 2646 1748 2726 2157 1355 1655 964 459
0.2 1370 1163 998 824 570 973 823 665 440 685 542 341 416 243 116
0.3 614 521 448 370 256 436 369 298 197 307 243 153 187 109 52
0.4 350 297 255 211 146 248 210 170 112 175 139 87 106 62 30
0.5 227 193 166 137 95 162 137 111 73 114 90 57 69 41 20
0.6 161 137 117 97 67 114 97 78 52 81 64 40 49 29 14
0.7 121 103 88 73 51 86 73 59 39 61 48 30 37 22 11
0.8 95 81 69 57 40 68 57 46 31 48 38 24 29 17 8
0.9 77 66 56 47 32 55 46 38 25 39 31 20 24 14 7
1 64 55 47 39 27 46 39 32 21 32 26 16 20 12 6
1.1 55 47 40 33 23 39 33 27 18 28 22 14 17 10 5
1.2 48 41 35 29 20 34 29 24 16 24 19 12 15 9 4
1.3 43 36 31 26 18 30 26 21 14 22 17 11 13 8 4
1.4 38 32 28 23 16 27 23 19 13 19 15 10 12 7 4
1.5 35 30 25 21 15 25 21 17 11 18 14 9 11 7 3
1.6 32 27 23 19 14 23 19 16 11 16 13 8 10 6 3
1.7 30 25 22 18 13 21 18 15 10 15 12 8 9 6 3
1.8 28 24 20 17 12 20 17 14 9 14 11 7 9 5 3
1.9 26 22 19 16 11 19 16 13 9 13 11 7 8 5 3
2 25 21 18 15 11 18 15 12 8 13 10 7 8 5 3
2.25 22 19 16 14 10 16 14 11 8 11 9 6 7 4 2
2.5 21 18 15 13 9 15 13 10 7 11 9 6 7 4 2
2.75 20 17 15 12 9 14 12 10 7 10 8 5 6 4 2
3 19 16 14 12 8 14 12 10 6 10 8 5 6 4 2
3.5 18 16 13 11 8 13 11 9 6 9 8 5 6 4 2
4 18 15 13 11 8 13 11 9 6 9 7 5 6 4 2

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E.2.2 Critical values for the Wilcoxon Rank Sum Test test

Table E.4 Critical values for the Wilcoxon Rank Sum test (WRS) test
m is the number of reference area samples and n is the number of survey unit samples.

n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=2 α=0.001 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43
α=0.005 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 40 42
α=0.01 7 9 11 13 15 17 19 21 23 25 27 28 30 32 34 36 38 39 41
α=0.025 7 9 11 13 15 17 18 20 22 23 25 27 29 31 33 34 36 38 40
α=0.05 7 9 11 12 14 16 17 19 21 23 24 26 27 29 31 33 34 36 38
α=0.1 7 8 10 11 13 15 16 18 19 21 22 24 26 27 29 30 32 33 35
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=3 α=0.001 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 56 59 62 65
α=0.005 12 15 18 21 24 27 30 32 35 38 40 43 46 48 51 54 57 59 62
α=0.01 12 15 18 21 24 26 29 31 34 37 39 42 45 47 50 52 55 58 60
α=0.025 12 15 18 20 22 25 27 30 32 35 37 40 42 45 47 50 52 55 57
α=0.05 12 14 17 19 21 24 26 28 31 33 36 38 40 43 45 47 50 52 54
α=0.1 11 13 16 18 20 22 24 27 29 31 33 35 37 40 42 44 46 48 50
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=4 α=0.001 18 22 26 30 34 38 42 46 49 53 57 60 64 68 71 75 78 82 86
α=0.005 18 22 26 30 33 37 40 44 47 51 54 58 61 64 68 71 75 78 81
α=0.01 18 22 26 29 32 36 39 42 46 49 52 56 59 62 66 69 72 76 79
α=0.025 18 22 25 28 31 34 37 41 44 47 50 53 56 59 62 66 69 72 75
α=0.05 18 21 24 27 30 33 36 39 42 45 48 51 54 57 59 62 65 68 71
α=0.1 17 20 22 25 28 31 34 36 39 42 45 48 50 53 56 59 61 64 67
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=5 α=0.001 25 30 35 40 45 50 54 58 63 67 72 76 81 85 89 94 98 102 107
α=0.005 25 30 35 39 43 48 52 56 60 64 68 72 77 81 85 89 93 97 101
α=0.01 25 30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90 94 98
α=0.025 25 29 33 37 41 44 48 52 56 60 63 67 71 75 79 82 86 90 94
α=0.05 24 28 32 35 39 43 46 50 53 57 61 64 68 71 75 79 82 86 89
α=0.1 23 27 30 34 37 41 44 47 51 54 57 61 64 67 71 74 77 81 84
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=6 α=0.001 33 39 45 51 57 63 67 72 77 82 88 93 98 103 108 113 118 123 128
α=0.005 33 39 44 49 54 59 64 69 74 79 83 88 93 98 103 107 112 117 122
α=0.01 33 39 43 48 53 58 62 67 72 77 81 86 91 95 100 104 109 114 118
α=0.025 33 37 42 47 51 56 60 64 69 73 78 82 87 91 95 100 104 109 113
α=0.05 32 36 41 45 49 54 58 62 66 70 75 79 83 87 91 96 100 104 108
α=0.1 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 94 98 102
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=7 α=0.001 42 49 56 63 69 75 81 87 92 98 104 110 116 122 128 133 139 145 151
α=0.005 42 49 55 61 66 72 77 83 88 94 99 105 110 116 121 127 132 138 143
α=0.01 42 48 54 59 65 70 76 81 86 92 97 102 108 113 118 123 129 134 139
α=0.025 42 47 52 57 63 68 73 78 83 88 93 98 103 108 113 118 123 128 133
α=0.05 41 46 51 56 61 65 70 75 80 85 90 94 99 104 109 113 118 123 128
α=0.1 40 44 49 54 58 63 67 72 76 81 85 90 94 99 103 108 112 117 121
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=8 α=0.001 52 60 68 75 82 89 95 102 109 115 122 128 135 141 148 154 161 167 174
α=0.005 52 60 66 73 79 85 92 98 104 110 116 122 129 135 141 147 153 159 165
α=0.01 52 59 65 71 77 84 90 96 102 108 114 120 125 131 137 143 149 155 161
α=0.025 51 57 63 69 75 81 86 92 98 104 109 115 121 126 132 137 143 149 154
α=0.05 50 56 62 67 73 78 84 89 95 100 105 111 116 122 127 132 138 143 148
α=0.1 49 54 60 65 70 75 80 85 91 96 101 106 111 116 121 126 131 136 141
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=9 α=0.001 63 72 81 88 96 104 111 118 126 133 140 147 155 162 169 176 183 190 198
α=0.005 63 71 79 86 93 100 107 114 121 127 134 141 148 155 161 168 175 182 188
α=0.01 63 70 77 84 91 98 105 111 118 125 131 138 144 151 157 164 170 177 184
α=0.025 62 69 76 82 88 95 101 108 114 120 126 133 139 145 151 158 164 170 176
α=0.05 61 67 74 80 86 92 98 104 110 116 122 128 134 140 146 152 158 164 170
α=0.1 60 66 71 77 83 89 94 100 106 112 117 123 129 134 140 145 151 157 162
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n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=10 α=0.001 75 85 94 103 111 119 128 136 144 152 160 167 175 183 191 199 207 215 222
α=0.005 75 84 92 100 108 115 123 131 138 146 153 160 168 175 183 190 197 205 212
α=0.01 75 83 91 98 106 113 121 128 135 142 150 157 164 171 178 186 193 200 207
α=0.025 74 81 89 96 103 110 117 124 131 138 145 151 158 165 172 179 186 192 199
α=0.05 73 80 87 93 100 107 114 120 127 133 140 147 153 160 166 173 179 186 192
α=0.1 71 78 84 91 97 103 110 116 122 128 135 141 147 153 160 166 172 178 184
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=11 α=0.001 88 99 109 118 127 136 145 154 163 171 180 188 197 206 214 223 231 240 248
α=0.005 88 98 107 115 124 132 140 148 157 165 173 181 189 197 205 213 221 229 237
α=0.01 88 97 105 113 122 130 138 146 153 161 169 177 185 193 200 208 216 224 232
α=0.025 87 95 103 111 118 126 134 141 149 156 164 171 179 186 194 201 208 216 223
α=0.05 86 93 101 108 115 123 130 137 144 152 159 166 173 180 187 195 202 209 216
α=0.1 84 91 98 105 112 119 126 133 139 146 153 160 167 173 180 187 194 201 207
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=12 α=0.001 102 114 125 135 145 154 164 173 183 192 202 210 220 230 238 247 256 266 275
α=0.005 102 112 122 131 140 149 158 167 176 185 194 202 211 220 228 237 246 254 263
α=0.01 102 111 120 129 138 147 156 164 173 181 190 198 207 215 223 232 240 249 257
α=0.025 100 109 118 126 135 143 151 159 168 176 184 192 200 208 216 224 232 240 248
α=0.05 99 108 116 124 132 140 147 155 165 171 179 186 194 202 209 217 225 233 240
α=0.1 97 105 113 120 128 135 143 150 158 165 172 180 187 194 202 209 216 224 231
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=13 α=0.001 117 130 141 152 163 173 183 193 203 213 223 233 243 253 263 273 282 292 302
α=0.005 117 128 139 148 158 168 177 187 196 206 215 225 234 243 253 262 271 280 290
α=0.01 116 127 137 146 156 165 174 184 193 202 211 220 229 238 247 256 265 274 283
α=0.025 115 125 134 143 152 161 170 179 187 196 205 214 222 231 239 248 257 265 274
α=0.05 114 123 132 140 149 157 166 174 183 191 199 208 216 224 233 241 249 257 266
α=0.1 112 120 129 137 145 153 161 169 177 185 193 201 209 217 224 232 240 248 256
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=14 α=0.001 133 147 159 171 182 193 204 215 225 236 247 257 268 278 289 299 310 320 330
α=0.005 133 145 156 167 177 187 198 208 218 228 238 248 258 268 278 288 298 307 317
α=0.01 132 144 154 164 175 185 194 204 214 224 234 243 253 263 272 282 291 301 311
α=0.025 131 141 151 161 171 180 190 199 208 218 227 236 245 255 264 273 282 292 301
α=0.05 129 139 149 158 167 176 185 194 203 212 221 230 239 248 257 265 274 283 292
α=0.1 128 136 145 154 163 171 180 189 197 206 214 223 231 240 248 257 265 273 282
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=15 α=0.001 150 165 178 190 202 212 225 237 248 260 271 282 293 304 316 327 338 349 360
α=0.005 150 162 174 186 197 208 219 230 240 251 262 272 283 293 304 314 325 335 346
α=0.01 149 161 172 183 194 205 215 226 236 247 257 267 278 288 298 308 319 329 339
α=0.025 148 159 169 180 190 200 210 220 230 240 250 260 270 280 289 299 309 319 329
α=0.05 146 157 167 176 186 196 206 215 225 234 244 253 263 272 282 291 301 310 319
α=0.1 144 154 163 172 182 191 200 209 218 227 236 246 255 264 273 282 291 300 309
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=16 α=0.001 168 184 197 210 223 236 248 260 272 284 296 308 320 332 343 355 367 379 390
α=0.005 168 181 194 206 218 229 241 252 264 275 286 298 309 320 331 342 353 365 376
α=0.01 167 180 192 203 215 226 237 248 259 270 281 292 303 314 325 336 347 357 368
α=0.025 166 177 188 200 210 221 232 242 253 264 274 284 295 305 316 326 337 347 357
α=0.05 164 175 185 196 206 217 227 237 247 257 267 278 288 298 308 318 328 338 348
α=0.1 162 172 182 192 202 211 221 231 241 250 260 269 279 289 298 308 317 327 336
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=17 α=0.001 187 203 218 232 245 258 271 284 297 310 322 335 347 360 372 384 397 409 422
α=0.005 187 201 214 227 239 252 264 276 288 300 312 324 336 347 359 371 383 394 406
α=0.01 186 199 212 224 236 248 260 272 284 295 307 318 330 341 353 364 376 387 399
α=0.025 184 197 209 220 232 243 254 266 277 288 299 310 321 332 343 354 365 376 387
α=0.05 183 194 205 217 228 238 249 260 271 282 292 303 313 324 335 345 356 366 377
α=0.1 180 191 202 212 223 233 243 253 264 274 284 294 305 315 325 335 345 355 365
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=18 α=0.001 207 224 239 254 268 282 296 309 323 336 349 362 376 389 402 415 428 441 454
α=0.005 207 222 236 249 262 275 288 301 313 326 339 351 364 376 388 401 413 425 438
α=0.01 206 220 233 246 259 272 284 296 309 321 333 345 357 370 382 394 406 418 430
α=0.025 204 217 230 242 254 266 278 290 302 313 325 337 348 360 372 383 395 406 418
α=0.05 202 215 226 238 250 261 273 284 295 307 318 329 340 352 363 374 385 396 407
α=0.1 200 211 222 233 244 255 266 277 288 299 309 320 331 342 352 363 374 384 395
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=19 α=0.001 228 246 262 277 292 307 321 335 350 364 377 391 405 419 433 446 460 473 487
α=0.005 227 243 258 272 286 300 313 327 340 353 366 379 392 405 419 431 444 457 470
α=0.01 226 242 256 269 283 296 309 322 335 348 361 373 386 399 411 424 437 449 462
α=0.025 225 239 252 265 278 290 303 315 327 340 352 364 377 389 401 413 425 437 450
α=0.05 223 236 248 261 273 285 297 309 321 333 345 356 368 380 392 403 415 427 439
α=0.1 220 232 244 256 267 279 290 302 313 325 336 347 358 370 381 392 403 415 426
.
n = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m=20 α=0.001 250 269 286 302 317 333 348 363 377 392 407 421 435 450 464 479 493 507 521
α=0.005 249 266 281 296 311 325 339 353 367 381 395 409 422 436 450 463 477 490 504
α=0.01 248 264 279 293 307 321 335 349 362 376 389 402 416 429 442 456 469 482 495
α=0.025 247 261 275 289 302 315 329 341 354 367 380 393 406 419 431 444 457 470 482
α=0.05 245 258 271 284 297 310 322 335 347 360 372 385 397 409 422 434 446 459 471
α=0.1 242 254 267 279 291 303 315 327 339 351 363 375 387 399 410 422 434 446 458

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E.2.3 Rejecting null hypothesis based on Wilcoxon Rank Sum Test

Reject the null hypothesis if the test statistic (Wr) is greater than the table (critical) value. For n or m greater than 20, the table (critical) value can be calculated from:

m (n + m + 1)/2 + z √(n m (n + m + 1)/12) ……………………………………………………….. (E-3)

if there are few or no ties, and from

m (n + m + 1)/2 + z √( n m/12* ((n + m + 1)- Σgj=i tj(tj2 – 1)/((n + m)(n + m -1)) ) ) ………………….. (E-4)

if there are many ties, where g is the number of groups of tied measurements and tj is the number of tied measurements in the jth group. z is the (1- α) percentile of a standard normal distribution, which can be found in the following table:

α z
0.001 3.09
0.005 2.575
0.01 2.326
0.025 1.960
0.05 1.645
0.1 1.282

Other values can be found in Table E.12.

D.2.4 Power of the Wilcoxon Rank Sum Test

The power of the WRS test is computed from

Power = 1 – Φ ((Wc – 0.5 – 0.5 m (m + 1) – E(WMW))/√(Var(WMW)) ………………….. (E-5)

where Wc is the critical value found in Table E.4 for the appropriate vales of α, n and m. Values of Φ(z), the standard normal cumulative distribution function, are given in Table E.12.

WMW =Wr -0.5m(m+1) is the Mann-Whitney form of the WRS test statistic. Its mean is

E(WMW) = m n Pr ……………………………………………… (E-6)

and its variance is

Var(WMW) = m n Pr(1 – Pr) + m n (n + m – 2)(p2 – Pr2) ………………………………… (E-7)

Values of Pr and p2 as a function of Δ/σ are given in Table E.5.

The power calculated in Equation D-5 is an approximation, but the results are generally accurate enough to be used to determine if the sample design achieves the DQOs.
The retrospective power curve for the WRS test can be constructed using Equations E-5, E-6, and E-7, together with the actual number of concentration measurements obtained, N. The power as a function of Δ/σ is calculated. The values of Δ/σ are converted to dpm/100 cm2 using:

dpm/100 cm2 = DCGL – (Δ/σ)(observed standard deviation)

The results for this example are plotted in Figure E.2, showing the probability that the survey unit would have passed the release criterion using the WRS test versus dpm of residual radioactivity. This curve shows that the data quality objectives were easily achieved. The curve shows that a survey unit with less than 4,500 dpm/100 cm2 above background would almost always pass, and that one with more than 5,100 dpm/100 cm2 above background would almost always fail.
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Table E.5 Values of Pr and p2 for computing the mean and variance of WMW.

Δ/σ Prr p2 Δ/σ Prr p2
-6.0 1.11E-05 1.16E-07 0.7 0.689691 0.544073
-5.0 0.000204 6.14E-06 0.8 0.714196 0.574469
-4.0 0.002339 0.000174 0.9 0.737741 0.604402
-3.5 0.006664 0.000738 1 0.76025 0.633702
-3 0.016947 0.00269 1.1 0.781662 0.662216
-2.5 0.03855 0.008465 1.2 0.801928 0.6898
-2.0 0.07865 0.023066 1.3 0.821015 0.716331
-1.9 0.089555 0.027714 1.4 0.838901 0.741698
-1.8 0.101546 0.033114 1.5 0.855578 0.765812
-1.7 0.114666 0.039348 1.6 0.87105 0.788602
-1.6 0.12895 0.046501 1.7 0.885334 0.810016
-1.5 0.144422 0.054656 1.8 0.898454 0.830022
-1.4 0.161099 0.063897 1.9 0.910445 0.848605
-1.3 0.178985 0.074301 2 0.92135 0.865767
-1.2 0.198072 0.085944 2.1 0.931218 0.881527
-1.1 0.218338 0.098892 2.2 0.940103 0.895917
-1.0 0.23975 0.113202 2.3 0.948062 0.908982
-0.9 0.262259 0.12892 2.4 0.955157 0.920777
-0.8 0.285804 0.146077 2.5 0.96145 0.931365
-0.7 0.310309 0.164691 2.6 0.967004 0.940817
-0.6 0.335687 0.18476 2.7 0.971881 0.949208
-0.5 0.361837 0.206266 2.8 0.976143 0.956616
-0.4 0.388649 0.229172 2.9 0.979848 0.963118
-0.3 0.416002 0.253419 3 0.983053 0.968795
-0.2 0.443769 0.27893 3.1 0.985811 0.973725
-0.1 0.471814 0.305606 3.2 0.988174 0.977981
0.0 0.5 0.333333 3.3 0.990188 0.981636
0.1 0.528186 0.361978 3.4 0.991895 0.984758
0.2 0.556231 0.391392 3.5 0.993336 0.98741
0.3 0.583998 0.421415 4 0.997661 0.995497
0.4 0.611351 0.451875 5 0.999796 0.999599
0.5 0.638163 0.482593 6 0.999989 0.999978
0.6 0.664313 0.513387

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Figure D.2 Retrospective power curve for class 2 interior drywall survey unit
Figure E.2 Retrospective power curve for class 2 interior drywall survey unit.

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E.2.5 Spreadsheet Formulas for the Wilcoxon Rank Sum Test

The analysis for the WRS test is very well suited for calculation on a spreadsheet. This is how the analysis discussed above was done. This particular example was constructed using Excel 5.0TM. The formula sheet corresponding to Table 3.55 is given in Table E.6. The function in Column D of Table E.6 calculates the ranks of the data. The RANK function in Excel™ does not return tied ranks in the way needed for the WRS. The COUNTIF function is used to correct for this. Column E simply picks out the reference area ranks from Column D.

Table E.6 Spreadsheet formulas used in Table 3.55

A B C D E
1 Data Area Adjusted Data Ranks Reference Area Ranks
2 49 R “=IF(B2=”"R"",A2+160,A2)" =RANK+(COUNTIF -1) / 2 “=IF(B2=”"R"",D2,0)"
3 35 R “=IF(B3=”"R"",A3+160,A3)" =RANK+(COUNTIF -1) / 2 “=IF(B3=”"R"",D3,0)"
4 45 R “=IF(B4=”"R"",A4+160,A4)" =RANK+(COUNTIF -1) / 2 “=IF(B4=”"R"",D4,0)"
5 45 R “=IF(B5=”"R"",A5+160,A5)" =RANK+(COUNTIF -1) / 2 “=IF(B5=”"R"",D5,0)"
6 41 R “=IF(B6=”"R"",A6+160,A6)" =RANK+(COUNTIF -1) / 2 “=IF(B6=”"R"",D6,0)"
7 44 R “=IF(B7=”"R"",A7+160,A7)" =RANK+(COUNTIF -1) / 2 “=IF(B7=”"R"",D7,0)"
8 48 R “=IF(B8=”"R"",A8+160,A8)" =RANK+(COUNTIF -1) / 2 “=IF(B8=”"R"",D8,0)"
9 37 R “=IF(B9=”"R"",A9+160,A9)" =RANK+(COUNTIF -1) / 2 “=IF(B9=”"R"",D9,0)"
10 46 R “=IF(B10=”"R"",A10+160,A10)" =RANK+(COUNTIF -1) / 2 “=IF(B10=”"R"",D10,0)"
11 42 R “=IF(B11=”"R"",A11+160,A11)" =RANK+(COUNTIF -1) / 2 “=IF(B11=”"R"",D11,0)"
12 47 R “=IF(B12=”"R"",A12+160,A12)" =RANK+(COUNTIF -1) / 2 “=IF(B12=”"R"",D12,0)"
13 104 S “=IF(B13=”"R"",A13+160,A13)" =RANK+(COUNTIF -1) / 2 “=IF(B13=”"R"",D13,0)"
14 94 S “=IF(B14=”"R"",A14+160,A14)" =RANK+(COUNTIF -1) / 2 “=IF(B14=”"R"",D14,0)"
15 98 S “=IF(B15=”"R"",A15+160,A15)" =RANK+(COUNTIF -1) / 2 “=IF(B15=”"R"",D15,0)"
16 99 S “=IF(B16=”"R"",A16+160,A16)" =RANK+(COUNTIF -1) / 2 “=IF(B16=”"R"",D16,0)"
17 90 S “=IF(B17=”"R"",A17+160,A17)" =RANK+(COUNTIF -1) / 2 “=IF(B17=”"R"",D17,0)"
18 104 S “=IF(B18=”"R"",A18+160,A18)" =RANK+(COUNTIF -1) / 2 “=IF(B18=”"R"",D18,0)"
19 95 S “=IF(B19=”"R"",A19+160,A19)" =RANK+(COUNTIF -1) / 2 “=IF(B19=”"R"",D19,0)"
20 105 S “=IF(B20=”"R"",A20+160,A20)" =RANK+(COUNTIF -1) / 2 “=IF(B20=”"R"",D20,0)"
21 93 S “=IF(B21=”"R"",A21+160,A21)" =RANK+(COUNTIF -1) / 2 “=IF(B21=”"R"",D21,0)"
22 101 S “=IF(B22=”"R"",A22+160,A22)" =RANK+(COUNTIF -1) / 2 “=IF(B22=”"R"",D22,0)"
23 92 S “=IF(B23=”"R"",A23+160,A23)" =RANK+(COUNTIF -1) / 2 “=IF(B23=”"R"",D23,0)"
24 Sum= = SUM (D2:D23) = SUM (E2:E23)