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APPENDIX I STATISTICAL TABLES AND PROCEDURES I.1 ..., Lecture notes of Construction

St Charles School of Massage TherapyConstruction

For N greater than 50, the table (critical) value can be calculated from: N z. %. 2. 2. N z is the (1-α) percentile of a standard normal distribution, ...

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APPENDIX I
STATISTICAL TABLES AND PROCEDURES
I.1 Normal Distribution
Table I.1 Cumulative Normal Distribution Function �(z)
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.10 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5674 0.5714 0.5753
0.20 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.30 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.40 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.50 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.60 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.70 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.80 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.90 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.6315 0.8340 0.8365 0.8389
1.00 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.10 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.20 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.30 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.40 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.50 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.60 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.70 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.80 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.90 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.00 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.10 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.20 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.30 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.40 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.50 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.60 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.70 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.80 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.90 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.00 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.10 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.20 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.30 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.40 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
Negative values of z can be obtained from the relationship �(-z) = 1 - �(z).
August 2000 I-1 MARSSIM, Revision 1
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Download APPENDIX I STATISTICAL TABLES AND PROCEDURES I.1 ... and more Lecture notes Construction in PDF only on Docsity!

STATISTICAL TABLES AND PROCEDURES

I.1 Normal Distribution

I.2 Sample Sizes for Statistical Tests

Table I.2a Sample Sizes for Sign Test

(Number of measurements to be performed in each survey unit)

I.3 Critical Values for the SignTest

Table I.3 Critical Values for the Sign Test Statistic S+

N 4 5 6 7 8 9

Alpha 0.005 0.01 0.025 0.05 0.1 0.2 0.3 0.4 0. 4 4 4 4 3 3 3 2 2 5 5 5 4 4 3 3 3 2 6 6 5 5 5 4 4 3 3 7 6 6 6 5 5 4 4 3 7 7 7 6 6 5 5 4 4 8 8 7 7 6 6 5 5 4 9 9 8 8 7 6 6 5 5 10 9 9 8 8 7 6 6 5 10 10 9 9 8 7 7 6 6 11 11 10 9 9 8 7 7 6 12 11 11 10 9 9 8 7 7 12 12 11 11 10 9 9 8 7 13 13 12 11 11 10 9 9 8 14 13 12 12 11 10 10 9 8 14 14 13 12 12 11 10 10 9 15 14 14 13 12 11 11 10 9 16 15 14 14 13 12 11 11 10 16 16 15 14 13 12 12 11 10 17 16 16 15 14 13 12 12 11 18 17 16 15 15 14 13 12 11 18 18 17 16 15 14 13 13 12 19 18 17 17 16 15 14 13 12 19 19 18 17 16 15 14 14 13 20 19 19 18 17 16 15 14 13 21 20 19 18 17 16 15 15 14 21 21 20 19 18 17 16 15 14 22 21 20 19 19 17 16 16 15

MARSSIM, Revision 1 I-4^ August 2000

Table I.3 Critical Values for the Sign Test Statistic S+ (continued)

N

For N greater than 50, the table (critical) value can be calculated from:

N (^) % z 2 2

N

z is the (1-�) percentile of a standard normal distribution, which can be found on page I-10 or on

page 5-28 in Table 5.2.

Alpha 0.005 0.01 0.025 0.05 0.1 0.2 0.3 0.4 0. 23 22 21 20 19 18 17 16 15 23 23 22 21 20 18 17 17 16 24 23 22 21 20 19 18 17 16 24 24 23 22 21 19 19 18 17 25 24 23 22 21 20 19 18 17 26 25 24 23 22 21 20 19 18 26 26 24 23 22 21 20 19 18 27 26 25 24 23 22 21 20 19 27 27 26 25 23 22 21 20 19 28 27 26 25 24 23 22 21 20 29 28 27 26 25 23 22 21 20 29 28 27 26 25 24 23 22 21 30 29 28 27 26 24 23 22 21 30 30 28 27 26 25 24 23 22 31 30 29 28 27 25 24 23 22 32 31 30 29 27 26 25 24 23 32 31 30 29 28 26 25 24 23 33 32 31 30 28 27 26 25 24 33 33 31 30 29 27 26 25 24 34 33 32 31 30 28 27 26 25

August 2000 I-5^ MARSSIM, Revision 1

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 nm ( n % m %1) /12 (I.1)

if there are few or no ties, and from

nm g^ tj ( tj

m ( n % m %1)/2 % z

[( n % m %1) &

j ' 1 ( n % m )( n % m &1)^

j ]^ (I.2)

if there are many ties, where g is the number of groups of tied measurements and tj

tied measurements in the jth group. z is the (1-�) percentile of a standard normal distribution, which

can be found in the following table:

is the number of

� z

Other values can be found in Table I-1.

MARSSIM, Revision 1 I-10^ August 2000

August 2000 I-11 MARSSIM, Revision 1

I.5 Probability of Detecting an Elevated Area

Table I.5 isk that an Elevated Area with Length L/G and Shape S will not be Detected

and the Area (%) of the Elevated Area Relative to a Triangular Sample Grid Area of 0.866 G^2

Shape Parameter, S 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1. L/G Risk Area Risk Area Risk Area Risk Area Risk Area Risk Area Risk Area Risk Area Risk Area Risk Area 0.01 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 0.02 1.00^ <1%^ 1.00^ <1%^ 1.00^ <1%^ 1.00^ <1%^ 1.00^ <1%^ 1.00^ <1%^ 1.00^ <1%^ 1.00^ <1%^ 1.00^ <1%^ 1.00^ <1% 0.03 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 0.04 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 0.99 1% 0.99 1% 0.05 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 1.00 <1% 0.99 1% 0.99 1% 0.99 1% 0.99 1% 0.99 1% 0.06 1.00 <1% 1.00 <1% 1.00 <1% 0.99 <1% 0.99 1% 0.99 1% 0.99 1% 0.99 1% 0.99 1% 0.99 1% 0.07 1.00 <1% 1.00 <1% 0.99 1% 0.99 <1% 0.99 1% 0.99 1% 0.99 1% 0.99 1% 0.98 2% 0.98 2% 0.08 1.00 <1% 1.00 <1% 0.99 1% 0.99 <1% 0.99 1% 0.99 1% 0.98 2% 0.98 2% 0.98 2% 0.98 2% 0.09 1.00 <1% 0.99 1% 0.99 1% 0.99 1% 0.99 1% 0.98 2% 0.98 2% 0.98 2% 0.97 3% 0.97 3% 0.10 1.00 <1% 0.99 1% 0.99 1% 0.99 1% 0.98 2% 0.98 2% 0.97 3% 0.97 3% 0.97 3% 0.96 4% 0.11 1.00 <1% 0.99 1% 0.99 1% 0.98 2% 0.98 2% 0.97 3% 0.97 3% 0.96 4% 0.96 4% 0.96 4% 0.12 0.99^ 1%^ 0.99^ 1%^ 0.98^ 2%^ 0.98^ 2%^ 0.97^ 3%^ 0.97^ 3%^ 0.96^ 4%^ 0.96^ 4%^ 0.95^ 5%^ 0.95^ 5% 0.13 0.99^ 1%^ 0.99^ 1%^ 0.98^ 2%^ 0.98^ 2%^ 0.97^ 3%^ 0.96^ 4%^ 0.96^ 4%^ 0.95^ 5%^ 0.94^ 6%^ 0.94^ 6% 0.14 0.99 1% 0.99 1% 0.98 2% 0.97 3% 0.96 4% 0.96 4% 0.95 5% 0.94 6% 0.94 6% 0.93 7% 0.15 0.99 1% 0.98 2% 0.98 2% 0.97 3% 0.96 4% 0.95 5% 0.94 6% 0.93 7% 0.93 7% 0.92 8% 0.16 0.99 1% 0.98 2% 0.97 3% 0.96 4% 0.95 5% 0.94 6% 0.94 7% 0.93 7% 0.92 8% 0.91 9% 0.17 0.99 1% 0.98 2% 0.97 3% 0.96 4% 0.95 5% 0.94 6% 0.93 7% 0.92 8% 0.91 9% 0.90 10% 0.18 0.99 1% 0.98 2% 0.96 4% 0.95 5% 0.94 6% 0.93 7% 0.92 8% 0.91 9% 0.89 11% 0.88 12% 0.19 0.99 1% 0.97 3% 0.96 4% 0.95 5% 0.93 7% 0.92 8% 0.91 9% 0.90 10% 0.88 12% 0.87 13% 0.20 0.99 1% 0.97 3% 0.96 4% 0.94 6% 0.93 7% 0.91 9% 0.90 10% 0.88 12% 0.87 13% 0.85 15% 0.21 0.98 2% 0.97 3% 0.95 5% 0.94 6% 0.92 8% 0.90 10% 0.89 11% 0.87 13% 0.86 14% 0.84 16% 0.22 0.98^ 2%^ 0.96^ 4%^ 0.95^ 5%^ 0.93^ 7%^ 0.91^ 9%^ 0.89^ 11%^ 0.88^ 12%^ 0.86^ 14%^ 0.84^ 16%^ 0.82^ 18% 0.23 0.98^ 2%^ 0.96^ 4%^ 0.94^ 6%^ 0.92^ 8%^ 0.90^ 10%^ 0.88^ 12%^ 0.87^ 13%^ 0.85^ 15%^ 0.83^ 17%^ 0.81^ 19% 0.24 0.98 2% 0.96 4% 0.94 6% 0.92 8% 0.90 10% 0.87 13% 0.85 15% 0.83 17% 0.81 19% 0.79 21% 0.25 0.98 2% 0.95 5% 0.93 7% 0.91 9% 0.89 11% 0.86 14% 0.84 16% 0.82 18% 0.80 20% 0.77 23% 0.26 0.98 2% 0.95 5% 0.93 7% 0.90 10% 0.88 12% 0.85 15% 0.83 17% 0.80 20% 0.78 22% 0.75 25% 0.27 0.97 3% 0.95 5% 0.92 8% 0.89 11% 0.87 13% 0.84 16% 0.81 19% 0.79 21% 0.76 24% 0.74 26% 0.28 0.97 3% 0.94 6% 0.91 9% 0.89 11% 0.86 14% 0.83 17% 0.80 20% 0.77 23% 0.74 26% 0.72 28% 0.29 0.97 3% 0.94 6% 0.91 9% 0.88 12% 0.85 15% 0.82 18% 0.79 21% 0.76 24% 0.73 27% 0.69 31% 0.30 0.97 3% 0.93 7% 0.90 10% 0.87 13% 0.84 16% 0.80 20% 0.77 23% 0.74 26% 0.71 29% 0.67 33%

Guidance for using Table I.5 can be found in Gilbert 1987 and EPA 1989a.

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August 2000 I-13 MARSSIM, Revision 1

Table I.5 isk that an Elevated Area with Length L/G and Shape S will not be Detected

and the Area (%) of the Elevated Area Relative to a Triangular Sample Grid Area of 0.866G^2

(continued)

Shape Parameter, S 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1. L/G Risk Area Risk Area Risk Area Risk Area Risk Area Risk Area Risk Area Risk Area Risk Area Risk Area 0.66 0.84 16% 0.69 32% 0.55 47% 0.40 63% 0.27 79% 0.15 95% 0.05 111% 0.00 126% 0.00 142% 0.00 158% 0.67 0.84 16% 0.68 33% 0.53 49% 0.39 65% 0.25 81% 0.13 98% 0.03 114% 0.00 130% 0.00 147% 0.00 163% 0.68 0.84 17% 0.68 34% 0.52 50% 0.38 67% 0.24 84% 0.12 101% 0.02 117% 0.00 134% 0.00 151% 0.00 168% 0.69 0.83 17% 0.67 35% 0.51 52% 0.36 69% 0.22 86% 0.10 104% 0.01 121% 0.00 138% 0.00 155% 0.00 173% 0.70 0.83 18% 0.66 36% 0.50 53% 0.35 71% 0.21 89% 0.09 107% 0.01 124% 0.00 142% 0.00 160% 0.00 178% 0.71 0.82^ 18%^ 0.65^ 37%^ 0.49^ 55%^ 0.33^ 73%^ 0.20^ 91%^ 0.08^ 110%^ 0.00^ 128%^ 0.00^ 146%^ 0.00^ 165%^ 0.00^ 183% 0.72 0.82^ 19%^ 0.64^ 38%^ 0.48^ 56%^ 0.32^ 75%^ 0.18^ 94%^ 0.07^ 113%^ 0.00^ 132%^ 0.00^ 150%^ 0.00^ 169%^ 0.00^ 188% 0.73 0.81 19% 0.63 39% 0.46 58% 0.31 77% 0.17 97% 0.05 116% 0.00 135% 0.00 155% 0.00 174% 0.00 193% 0.74 0.81 20% 0.62 40% 0.45 60% 0.29 79% 0.15 99% 0.04 119% 0.00 139% 0.00 159% 0.00 179% 0.00 199% 0.75 0.80 20% 0.61 41% 0.44 61% 0.28 82% 0.14 102% 0.04 122% 0.00 143% 0.00 163% 0.00 184% 0.00 204% 0.76 0.80 21% 0.61 42% 0.43 63% 0.27 84% 0.13 105% 0.03 126% 0.00 147% 0.00 168% 0.00 189% 0.00 210% 0.77 0.79 22% 0.60 43% 0.42 65% 0.25 86% 0.12 108% 0.02 129% 0.00 151% 0.00 172% 0.00 194% 0.00 215% 0.78 0.79 22% 0.59 44% 0.40 66% 0.24 88% 0.10 110% 0.01 132% 0.00 154% 0.00 177% 0.00 199% 0.00 221% 0.79 0.78 23% 0.58 45% 0.39 68% 0.23 91% 0.09 113% 0.01 136% 0.00 158% 0.00 181% 0.00 204% 0.00 226% 0.80 0.78 23% 0.57 46% 0.38 70% 0.22 93% 0.08 116% 0.00 139% 0.00 163% 0.00 186% 0.00 209% 0.00 232% 0.81 0.77 24% 0.56 48% 0.37 71% 0.20 95% 0.07 119% 0.00 143% 0.00 167% 0.00 190% 0.00 214% 0.00 238% 0.82 0.77^ 24%^ 0.55^ 49%^ 0.36^ 73%^ 0.19^ 98%^ 0.06^ 122%^ 0.00^ 146%^ 0.00^ 171%^ 0.00^ 195%^ 0.00^ 220%^ 0.00^ 244% 0.83 0.76 25% 0.54 50% 0.35 75% 0.18 100% 0.05 125% 0.00 150% 0.00 175% 0.00 200% 0.00 225% 0.00 250% 0.84 0.76 26% 0.53 51% 0.33 77% 0.17 102% 0.05 128% 0.00 154% 0.00 179% 0.00 205% 0.00 230% 0.00 256% 0.85 0.75 26% 0.52 52% 0.32 79% 0.16 105% 0.04 131% 0.00 157% 0.00 183% 0.00 210% 0.00 236% 0.00 262% 0.86 0.74 27% 0.51 54% 0.31 80% 0.14 107% 0.03 134% 0.00 161% 0.00 188% 0.00 215% 0.00 241% 0.00 268% 0.87 0.74 27% 0.50 55% 0.30 82% 0.13 110% 0.02 137% 0.00 165% 0.00 192% 0.00 220% 0.00 247% 0.00 275% 0.88 0.73 28% 0.50 56% 0.29 84% 0.12 112% 0.02 140% 0.00 169% 0.00 197% 0.00 225% 0.00 253% 0.00 281% 0.89 0.73 29% 0.49 57% 0.28 86% 0.11 115% 0.01 144% 0.00 172% 0.00 201% 0.00 230% 0.00 259% 0.00 287% 0.90 0.72 29% 0.48 59% 0.27 88% 0.10 118% 0.01 147% 0.00 176% 0.00 206% 0.00 235% 0.00 264% 0.00 294% 0.91 0.72 30% 0.47 60% 0.26 90% 0.10 120% 0.01 150% 0.00 180% 0.00 210% 0.00 240% 0.00 270% 0.00 300% 0.92 0.71 31% 0.46 61% 0.25 92% 0.09 123% 0.00 154% 0.00 184% 0.00 215% 0.00 246% 0.00 276% 0.00 307% 0.93 0.71 31% 0.45 63% 0.24 94% 0.08 126% 0.00 157% 0.00 188% 0.00 220% 0.00 251% 0.00 282% 0.00 314% 0.94 0.70 32% 0.44 64% 0.23 96% 0.07 128% 0.00 160% 0.00 192% 0.00 224% 0.00 256% 0.00 288% 0.00 321% 0.95 0.69 33% 0.43 65% 0.22 98% 0.07 131% 0.00 164% 0.00 196% 0.00 229% 0.00 262% 0.00 295% 0.00 327% 0.96 0.69 33% 0.42 67% 0.21 100% 0.06 134% 0.00 167% 0.00 201% 0.00 234% 0.00 267% 0.00 301% 0.00 334% 0.97 0.68 34% 0.41 68% 0.20 102% 0.05 137% 0.00 171% 0.00 205% 0.00 239% 0.00 273% 0.00 307% 0.00 341% 0.98 0.68 35% 0.40 70% 0.19 105% 0.05 139% 0.00 174% 0.00 209% 0.00 244% 0.00 279% 0.00 314% 0.00 348% 0.99 0.67 36% 0.40 71% 0.18 107% 0.04 142% 0.00 178% 0.00 213% 0.00 249% 0.00 284% 0.00 320% 0.00 356% 1.00 0.67 36% 0.39 73% 0.17 109% 0.04 145% 0.00 181% 0.00 218% 0.00 254% 0.00 290% 0.00 326% 0.00 363%

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I.6 Random Numbers

Table I.6 1,000 Random Numbers Uniformly Distributed between Zero and One

 - z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0. Table I.1 Cumulative Normal Distribution Function � (z)
  • 0.00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.
  • 0.10 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5674 0.5714 0.
  • 0.20 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.
  • 0.30 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.
  • 0.40 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.
  • 0.50 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.
  • 0.60 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.
  • 0.70 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.
  • 0.80 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.
  • 0.90 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.6315 0.8340 0.8365 0.
  • 1.00 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.
  • 1.10 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.
  • 1.20 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.
  • 1.30 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.
  • 1.40 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.
  • 1.50 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.
  • 1.60 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.
  • 1.70 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.
  • 1.80 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.
  • 1.90 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.
  • 2.00 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.
  • 2.10 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.
  • 2.20 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.
  • 2.30 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.
  • 2.40 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.
  • 2.50 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.
  • 2.60 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.
  • 2.70 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.
  • 2.80 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.
  • 2.90 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.
  • 3.00 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.
  • 3.10 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.
  • 3.20 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.
  • 3.30 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.
  • 3.40 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.
  • August 2000 I-1 MARSSIM, Revision Negative values of z can be obtained from the relationship �(-z) = 1 - �(z). - 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. (�,�) or (�,�)
    • Δ/σΔ/σΔ/σΔ/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.
      • 0.1
      • 0.2
      • 0.3^468 398 341 282 195 333 281 227 150 234 185 117 143 83
      • 0.4^270 230 197 162 113 192 162 131 87 136 107 68 82 48
      • 0.5^178 152 130 107 75 126 107 87 58 89 71 45 54 33
      • 0.6^129 110 94 77 54 92 77 63 42 65 52 33 40 23
      • 0.7^99 83 72 59 41 70 59 48 33 50 40 26 30 18
      • 0.8^80 68 58 48 34 57 48 39 26 40 32 21 24 15
      • 0.9
      • 1.0
      • 1.1
      • 1.2
      • 1.3
      • 1.4
      • 1.5
      • 1.6
      • 1.7
      • 1.8
      • 1.9^30 26 22 18 14 22 18 15 10 16 12 9 10 6
      • 2.0^29 26 22 18 12 21 18 15 10 15 12 8 10 6
      • 2.5^28 23 21 17 12 20 17 14 10 15 11 8 9 5
      • 3.0^27 23 20 17 12 20 17 14 9 14 11 8 9 5
  • MARSSIM, Revision 1 I-2 August
  • August 2000 I-3 MARSSIM, Revision - 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. (�,�) or (�,�)
    • Δ/σΔ/σΔ/σΔ/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. - 0.1 - 0.2 - 0.3 - 0.4 - 0.5 - 0.6 - 0.7 - 0.8 - 0.9 - 1.0^64 55 47 39 27 46 39 32 21 32 26 16 20 12 - 1.1^55 47 40 33 23 39 33 27 18 28 22 14 17 10 - 1.2^48 41 35 29 20 34 29 24 16 24 19 12 15 9 - 1.3^43 36 31 26 18 30 26 21 14 22 17 11 13 8 - 1.4^38 32 28 23 16 27 23 19 13 19 15 10 12 7 - 1.5^35 30 25 21 15 25 21 17 11 18 14 9 11 7 - 1.6 - 1.7 - 1.8 - 1.9 - 2.0
      • 2.25
        • 2.5
      • 2.75
        • 3.0
        • 3.5
        • 4.0^18 15 13 11 8 13 11 9 6 9 7 5 6 4
    • n = m is the number of reference area samples and n is the number of survey unit samples.
  • m = 2 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 3 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 4 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 5 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 6 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
  • MARSSIM, Revision 1 I-6 August
    • n = Table I.4 Critical Values for the WRS Test (continued)
  • m = 7 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 8 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 9 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 10 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 11 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
  • August 2000 I-7 MARSSIM, Revision
    • n = Table I.4 Critical Values for the WRS Test (continued)
  • m = 12 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 13 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 14 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 15 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 16 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
  • MARSSIM, Revision 1 I-8 August
    • n = Table I.4 Critical Values for the WRS Test (continued)
  • m = 17 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 18 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 19 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
    • n =
  • m = 20 �=0.001
    • �=0.005
    • �=0.01
    • �=0.025
    • �=0.05
    • �=0.1
  • August 2000 I-9 MARSSIM, Revision
    • 0.163601 0.647423 0.555548 0.248859 0.259801 0.718368 0.305020 0.812482 0.601951 0. Table I.6 1,000 Random Numbers Uniformly Distributed between Zero and One
    • 0.934196 0.951102 0.979831 0.132364 0.157808 0.040605 0.997626 0.896462 0.360578 0.
    • 0.054552 0.965257 0.999181 0.172627 0.583713 0.852958 0.116336 0.748483 0.058602 0.
    • 0.972409 0.241889 0.799991 0.926726 0.585505 0.453993 0.877990 0.947022 0.910821 0.
    • 0.556401 0.621126 0.293328 0.984335 0.366531 0.912588 0.733824 0.092405 0.717362 0.
    • 0.625153 0.838711 0.196153 0.630553 0.867808 0.957094 0.830218 0.783518 0.141557 0.
    • 0.527330 0.124034 0.351792 0.161947 0.688925 0.140346 0.553577 0.890058 0.470457 0.
    • 0.826643 0.673286 0.550827 0.885295 0.690781 0.371540 0.108632 0.090765 0.618443 0.
    • 0.296068 0.891272 0.392367 0.649633 0.261410 0.523221 0.769081 0.358794 0.924341 0.
    • 0.848882 0.083603 0.274621 0.268003 0.272254 0.017727 0.309463 0.445986 0.244653 0.
    • 0.779276 0.484461 0.101393 0.995100 0.085164 0.611426 0.030270 0.494982 0.426236 0.
    • 0.095038 0.577943 0.186239 0.267852 0.786070 0.208937 0.184565 0.826397 0.256825 0.
    • 0.011672 0.844846 0.443407 0.915087 0.275906 0.883009 0.243728 0.865552 0.796671 0.
    • 0.215993 0.476035 0.354717 0.883172 0.840666 0.393867 0.374810 0.222167 0.114691 0.
    • 0.982374 0.101973 0.683995 0.730612 0.548200 0.084302 0.145212 0.337680 0.566173 0.
    • 0.860868 0.794380 0.819422 0.752871 0.158956 0.317468 0.062387 0.909843 0.779089 0.
    • 0.718917 0.696798 0.463655 0.762408 0.823097 0.843209 0.368678 0.996266 0.542048 0.
    • 0.800735 0.225556 0.398048 0.437067 0.642698 0.144068 0.104212 0.675095 0.318953 0.
    • 0.915538 0.711742 0.232159 0.242961 0.327863 0.156608 0.260175 0.385141 0.681475 0.
    • 0.975506 0.652654 0.928348 0.513444 0.744095 0.972031 0.527368 0.494287 0.602829 0.
    • 0.435196 0.272807 0.452254 0.793464 0.817291 0.828245 0.407518 0.441518 0.358966 0.
    • 0.692512 0.368151 0.821543 0.583707 0.802354 0.133831 0.569521 0.474516 0.437608 0.
    • 0.678823 0.930602 0.657348 0.025057 0.294093 0.499623 0.006423 0.290613 0.325204 0.
    • 0.642075 0.029842 0.289042 0.891009 0.813844 0.973093 0.952871 0.361623 0.709933 0.
    • 0.174285 0.863244 0.133649 0.773819 0.891664 0.246417 0.272407 0.517658 0.132225 0.
    • 0.951401 0.921291 0.210993 0.369411 0.196909 0.054389 0.364475 0.716718 0.096843 0.
    • 0.186824 0.005407 0.310843 0.998118 0.725887 0.143171 0.293721 0.841304 0.661969 0.
    • 0.105673 0.026338 0.878006 0.105936 0.612556 0.124601 0.922558 0.648985 0.896805 0.
    • 0.801080 0.619461 0.933720 0.275881 0.637352 0.644996 0.713379 0.302687 0.904515 0.
    • 0.101214 0.236405 0.945199 0.005975 0.893786 0.082317 0.648743 0.511871 0.298942 0.
    • 0.177754 0.930066 0.390527 0.575622 0.390428 0.600575 0.460949 0.191600 0.910079 0.
    • 0.846157 0.322467 0.156607 0.253388 0.739021 0.133498 0.293141 0.144834 0.626600 0.
    • 0.812147 0.306383 0.201517 0.306651 0.827112 0.277716 0.660224 0.268538 0.518416 0.
    • 0.691055 0.059046 0.104390 0.427038 0.148688 0.480788 0.026511 0.572705 0.745522 0.
    • 0.483819 0.797573 0.174899 0.892670 0.118990 0.813221 0.857964 0.279164 0.883509 0.
    • 0.165133 0.985134 0.214681 0.595309 0.741697 0.418602 0.301917 0.338913 0.680062 0.
    • 0.281668 0.476899 0.839512 0.057760 0.474156 0.898409 0.482638 0.198725 0.888281 0.
    • 0.554337 0.350955 0.942401 0.526759 0.509846 0.408165 0.800079 0.789263 0.564192 0.
  • MARSSIM, Revision 1 I-14 August
  • 0.873143 0.349662 0.238282 0.383195 0.568383 0.298471 0.490431 0.731405 0.339906 0. (continued)
  • 0.401675 0.061151 0.771468 0.795760 0.365952 0.221234 0.947374 0.375686 0.828215 0.
  • 0.574987 0.154831 0.808117 0.723544 0.134014 0.360957 0.166572 0.112314 0.242857 0.
  • 0.745415 0.929459 0.425406 0.118845 0.386382 0.867386 0.808757 0.009573 0.229879 0.
  • 0.613554 0.926550 0.857632 0.014438 0.004214 0.592513 0.280223 0.283447 0.943793 0.
  • 0.880368 0.303741 0.247850 0.341580 0.867155 0.542130 0.473418 0.650251 0.326222 0.
  • 0.567556 0.183534 0.696381 0.373333 0.716762 0.526636 0.306862 0.904790 0.151931 0.
  • 0.280015 0.237361 0.336240 0.424191 0.192603 0.770194 0.284572 0.992475 0.308979 0.
  • 0.502862 0.818555 0.238758 0.057148 0.461531 0.904929 0.521982 0.599127 0.239509 0.
  • 0.738375 0.794328 0.305231 0.887161 0.021104 0.469779 0.913966 0.266514 0.647901 0.
  • 0.366209 0.749763 0.634971 0.261038 0.869115 0.787951 0.678287 0.667142 0.216531 0.
  • 0.739267 0.554299 0.979969 0.489597 0.545130 0.931869 0.096443 0.374089 0.140070 0.
  • 0.375690 0.866922 0.256930 0.518074 0.217373 0.027043 0.801938 0.040364 0.624283 0.
  • 0.894101 0.178824 0.443631 0.110614 0.556232 0.969563 0.291364 0.695764 0.306903 0.
  • 0.668169 0.296926 0.324041 0.616290 0.799426 0.372555 0.070954 0.045748 0.505327 0.
  • 0.470107 0.135634 0.271284 0.494071 0.485610 0.382772 0.418470 0.004082 0.298068 0.
  • 0.047906 0.694949 0.309033 0.223989 0.008978 0.383695 0.479858 0.894958 0.597796 0.
  • 0.917713 0.072793 0.107402 0.007328 0.176598 0.576809 0.052969 0.421803 0.737514 0.
  • 0.839439 0.338565 0.254833 0.924413 0.871833 0.480599 0.172846 0.736102 0.471802 0.
  • 0.488244 0.260352 0.129716 0.153558 0.305933 0.777100 0.111924 0.412930 0.601453 0.
  • 0.488369 0.485094 0.322236 0.894264 0.781546 0.770237 0.707400 0.587451 0.571609 0.
  • 0.311380 0.270400 0.807264 0.348433 0.172763 0.914856 0.011893 0.014317 0.820797 0.
  • 0.028802 0.072165 0.944160 0.804761 0.770481 0.104256 0.112919 0.184068 0.940946 0.
  • 0.466082 0.603884 0.959713 0.547834 0.487552 0.455150 0.240324 0.428921 0.648821 0.
  • 0.720229 0.575779 0.939622 0.234554 0.767389 0.735335 0.941002 0.794021 0.291615 0.
  • 0.861579 0.778039 0.331677 0.608231 0.646094 0.498720 0.140520 0.259197 0.782477 0.
  • 0.849884 0.917789 0.816247 0.572502 0.753757 0.857324 0.988330 0.597085 0.186087 0.
  • 0.989999 0.994007 0.349735 0.954437 0.741124 0.791852 0.986074 0.444554 0.177531 0.
  • 0.337214 0.987184 0.344245 0.039033 0.549585 0.688526 0.225470 0.556251 0.157058 0.
  • 0.706330 0.082994 0.299909 0.613361 0.031334 0.941102 0.772731 0.198070 0.460602 0.
  • 0.417239 0.916556 0.707773 0.249767 0.169301 0.914420 0.732687 0.934912 0.985594 0.
  • 0.653326 0.529996 0.305465 0.181747 0.153359 0.353168 0.673377 0.448970 0.546347 0.
  • 0.099373 0.156385 0.067157 0.755573 0.689979 0.494021 0.996216 0.051811 0.049321 0.
  • 0.860299 0.210143 0.026232 0.838499 0.108975 0.455260 0.320633 0.150619 0.445073 0.
  • 0.067160 0.791992 0.363875 0.825052 0.047561 0.311194 0.447486 0.971659 0.876616 0.
  • 0.944317 0.348844 0.210015 0.769274 0.253032 0.239894 0.208165 0.600014 0.945046 0.
  • 0.917419 0.185575 0.743859 0.655124 0.185320 0.237660 0.271534 0.949825 0.441666 0.
  • 0.365705 0.800723 0.116707 0.386073 0.837800 0.244896 0.337304 0.869528 0.845737 0.
  • 0.911453 0.591254 0.920222 0.707522 0.782902 0.092884 0.426444 0.320336 0.226369 0.
  • August 2000 I-15 MARSSIM, Revision
    • 0.027171 0.058193 0.726183 0.057705 0.935493 0.688071 0.752543 0.932781 0.048914 0. (continued)
    • 0.768066 0.387888 0.655990 0.690208 0.746739 0.936409 0.685458 0.090931 0.242120 0.
    • 0.052305 0.899285 0.092643 0.058916 0.826653 0.772790 0.785028 0.967761 0.588503 0.
    • 0.623285 0.492051 0.644294 0.821341 0.600824 0.901289 0.774379 0.391874 0.810022 0.
    • 0.624284 0.308522 0.208541 0.297156 0.576129 0.373705 0.370345 0.372748 0.965550 0.
    • 0.853117 0.671602 0.018316 0.095780 0.871263 0.885420 0.919787 0.439594 0.460586 0.
    • 0.967796 0.933631 0.397054 0.682343 0.505977 0.406611 0.539543 0.066152 0.885414 0.
    • 0.759450 0.768853 0.115419 0.744466 0.607572 0.179839 0.413809 0.228607 0.362857 0.
    • 0.514703 0.108915 0.864053 0.076280 0.352557 0.674917 0.572689 0.588574 0.596215 0.
    • 0.826296 0.264540 0.255775 0.180449 0.405715 0.740170 0.423514 0.537793 0.877436 0.
    • 0.354198 0.792775 0.051583 0.806962 0.385851 0.655314 0.046701 0.860466 0.848112 0.
    • 0.744807 0.960789 0.123099 0.163569 0.621969 0.571558 0.482449 0.346358 0.795845 0.
    • 0.642312 0.356643 0.797708 0.505570 0.418534 0.634642 0.033111 0.393330 0.105093 0.
    • 0.824625 0.855876 0.770743 0.678619 0.927298 0.204828 0.831460 0.979875 0.566627 0.
    • 0.755877 0.679791 0.442388 0.899944 0.563383 0.197074 0.679568 0.244433 0.786084 0.
    • 0.625370 0.967123 0.321605 0.697578 0.122418 0.475395 0.068207 0.070374 0.353248 0.
    • 0.124012 0.133851 0.761154 0.501578 0.204221 0.866481 0.925783 0.329001 0.327832 0.
    • 0.825392 0.382001 0.847909 0.520741 0.404959 0.308849 0.418976 0.972838 0.452438 0.
    • 0.999194 0.297058 0.617183 0.570478 0.875712 0.581618 0.284410 0.405575 0.362205 0.
    • 0.536855 0.667083 0.636883 0.043774 0.113509 0.980045 0.237797 0.618925 0.670767 0.
    • 0.361632 0.797162 0.136063 0.487575 0.682796 0.952708 0.759989 0.058556 0.292400 0.
    • 0.923253 0.479871 0.022855 0.673915 0.733795 0.811955 0.417970 0.095675 0.831670 0.
    • 0.845432 0.202336 0.348421 0.050704 0.171916 0.600557 0.284838 0.606715 0.758190 0.
  • MARSSIM, Revision 1 I-16 August

I.7 Stem and Leaf Display

The construction of a stem and leaf display is a simple way to generate a crude histogram of the

data quickly. The “stems” of such a display are the most significant digits of the data. Consider the

sample data of Section 8.2.2.2:

Here the data span three decades, so one might consider using the stems 70, 80 and 90. However,

three is too few stems to be informative, just as three intervals would be too few for constructing a

histogram. Therefore, for this example, each decade is divided into two parts. This results in the six

stems 70, 75, 80, 85, 90, 95. The leaves are the least significant digits, so 90.7 has the stem 90 and

the leaf 0.7. 77.4 has the stem 75 and the leaf 7.4. Note that even though the stem is 75, the leaf is

not 2.4. The leaf is kept as 7.4 so that the data can be read directly from the display without any

calculations.

As shown in the top part of Figure I.1, simply arrange the leaves of the data into rows, one stem per

row. The result is a quick histogram of the data. In order to ensure this, the same number of digits

should be used for each leaf, so that each occupies the same amount of horizontal space.

If the stems are arranged in increasing order, as shown in the bottom half of Figure I.1, it is easy to

pick out the minimum (74.2), the maximum (92.4), and the median (between 84.1 and 84.4).

A stem and leaf display (or histogram) with two peaks may indicate that residual radioactivity is

distributed over only a portion of the survey unit. Further information on the construction and

interpretation of data plots is given in EPA QA/G-9 (EPA 1996a).

August 2000 I-17^ MARSSIM, Revision 1

Table I.7 Data for Quantile Plot

Data: 74.2 75.5 76.3 77.4 77.6 78.2 79.1 80.5 83.5 84.

Rank: 1 2 3 4 5 6 7 8 9 10

Percent: 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.

Data: 84.4 86.4 86.4 86.5 87.6 88.5 90.1 90.3 90.7 92.

Rank: 11 12.5 12.5 14 15 16 17 18 19 20

Percent: 52.5 60.0 60.0 67.5 72.5 77.5 82.5 87.5 92.5 97.

A useful aid to interpreting the quantile plot is the addition of boxes containing the middle 50%

and middle 75% of the data. These are shown as the dashed lines in Figure I.2. The 50% box has

its upper right corner at the 75th percentile and its lower left corner at the 25th percentile. These

points are also called the Quartiles. These are ~78 and ~88, respectively, as indicated by the

dashed lines. They bracket the middle half of the data values. The 75% box has its upper right

corner at the 87.5th percentile and its lower left corner at the 12.5th percentile. A sharp increase

within the 50% box can indicate two or more modes in the data. Outside the 75% box, sharp

increases can indicate outliers. The median (50th percentile) is indicated by the heavy solid line

at the value ~84, and can be used as an aid to judging the symmetry of the data distribution.

There are no especially unusual features in the example Quantile plot shown in Figure I.2, other

than the possibility of slight asymmetry around the median.

Another Quantile plot, for the example data of Section 8.3.3, is shown in Figure I.3.

August 2000 I-19^ MARSSIM, Revision 1

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Figure I.2 Example of a Quantile Plot

MARSSIM, Revision 1 I-20^ August 2000