Subcategory Pollutant

No. of 

Sources 

Industryw

ide

No. of 

Sources 

with Data

Determination 

of No. of 

Sources in 

MACT Floor

No. of 

Sources 

in MACT 

Floor (N)

No. Data 

Points in 

MACT 

Floor (n)

Average 

of Top 

Sources

UPL Unit of Measure Data 

Distribution

All Kilns PM 225 31 12% Sources 

Industrywide 1 3 0.0060 0.018 lb/ton lognormal

Equivalent PM 225 31 12% Sources 

Industrywide 1 3 0.0020 0.0059 gr/dscf @ 7% O2 lognormal

Large Kilns Mercury 122 19 12% Sources with 

Data 1 3 1.5E-05 2.0E-05 lb/ton normal

Equivalent Mercury 122 19 12% Sources with 

Data 1 3 10 13 μg/dscm @ 7% O2 lognormal

Equivalent Mercury 122 19 12% Sources with 

Data 1 3 0.00018 0.00024 lb/hr normal

Small Kilns Mercury 103 2

12% Sources with 

Data 1 3 1.8E-04 2.0E-04 lb/ton lognormal

Equivalent Mercury 103 2

12% Sources with 

Data 1 3 62 70 μg/dscm @ 7% O2 normal

Equivalent Mercury 103 2

12% Sources with 

Data 1 3 0.0010 0.0011 lb/hr lognormal
Runs

514 PM lb/ton 514 PM conc 568 Hg Large lb/ton 568 Hg Large conc

1 2.99E-03 1.02E-03 1.37E-05 1.17E+01

2 9.53E-03 3.18E-03 1.50E-05 9.98E+00

3 5.61E-03 1.78E-03 1.56E-05 9.71E+00

Average 6.04E-03 2.00E-03 1.48E-05 1.05E+01

Standard Deviation 3.29E-03 1.09E-03 9.40E-07 1.06E+00

1 8.74E-11 8.93E-13 1.13E-24 2.18E+00

2 1.48E-10 1.98E-12 2.34E-27 4.99E-02

3 3.43E-14 2.10E-15 4.31E-25 3.04E-01

Testing Normality Raw Data Raw Data Raw Data Raw Data

Sample Size 3.00 3.00 3.00 3.00

Kurtosis -2.00 -2.00 -2.00 -2.00

SE Kurtosis 2.83 2.83 2.83 2.83

Result Kurtosis normal normal normal normal

Skewness 0.58 0.85 -1.00 1.61

SE Skewness 1.22 1.22 1.22 1.22

Result Skewness normal normal normal normal

lb/ton 0.471984818 0.690883834 0.812869883 1.312133767

Lognormal Lognormal Normal Lognormal

1 2 3 4
568 Hg Large lb/hr 531 Hg Small lb/ton 531 Hg Small conc 531 Hg Small lb/hr Example 9

1.63E-04 1.84E-04 6.03E+01 1.04E-03

1.78E-04 1.69E-04 6.21E+01 9.55E-04

1.85E-04 1.76E-04 6.32E+01 9.96E-04

1.75E-04 1.76E-04 6.19E+01 9.97E-04 #DIV/0!

1.12E-05 7.42E-06 1.48E+00 4.20E-05 #DIV/0!

2.24E-20 3.16E-21 6.01E+00 3.23E-18 #DIV/0!

4.66E-23 2.91E-21 1.60E-03 2.98E-18 #DIV/0!

8.59E-21 5.41E-28 3.48E+00 5.53E-25 #DIV/0!

Raw Data Raw Data Raw Data Raw Data Raw Data

3.00 3.00 3.00 3.00 0.00

-2.00 -2.00 -2.00 -2.00 #DIV/0!

2.83 2.83 2.83 2.83 #DIV/0!

normal normal normal normal #DIV/0!

-1.00 0.09 -0.60 0.09 #DIV/0!

1.22 1.22 1.22 1.22 0.00

normal normal normal normal #DIV/0!

0.812869883 0.075456359 0.488415014 0.075456359 #DIV/0!

Normal Lognormal Normal Lognormal #DIV/0!

5 6 7 8 9
Example 10 Example 11 Example 12 Example 13 Example 14

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Raw Data Raw Data Raw Data Raw Data Raw Data

0.00 0.00 0.00 0.00 0.00

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

0.00 0.00 0.00 0.00 0.00

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

10 11 12 13 14
Example 15 Example 16 Example 17 Example 18 Example 19

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Raw Data Raw Data Raw Data Raw Data Raw Data

0.00 0.00 0.00 0.00 0.00

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

0.00 0.00 0.00 0.00 0.00

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

15 16 17 18 19
Example 20 Example 21 Example 1 Example 2

-5.81E+00 -6.88E+00

-4.65E+00 -5.75E+00

-5.18E+00 -6.33E+00

#DIV/0! #DIV/0! -5.22E+00 -6.32E+00

#DIV/0! #DIV/0! 5.81E-01 5.67E-01

#DIV/0! #DIV/0! 1.27E-01 1.00E-01

#DIV/0! #DIV/0! 1.01E-01 1.06E-01

#DIV/0! #DIV/0! 1.31E-06 5.57E-09

Raw Data Raw Data Log Data Log Data

0.00 0.00 3.00 3.00

#DIV/0! #DIV/0! -2.00 -2.00

#DIV/0! #DIV/0! 2.83 2.83

#DIV/0! #DIV/0! normal normal

#DIV/0! #DIV/0! -0.26 0.07

0.00 0.00 1.22 1.22

#DIV/0! #DIV/0! normal normal

#DIV/0! #DIV/0! 0.213309407 0.055985172

#DIV/0! #DIV/0!

20 21
Example 3 Example 4 Example 5 Example 6 Example 7

-1.12E+01 2.46E+00 -8.72E+00 -8.60E+00 4.10E+00

-1.11E+01 2.30E+00 -8.63E+00 -8.69E+00 4.13E+00

-1.11E+01 2.27E+00 -8.59E+00 -8.64E+00 4.15E+00

-1.11E+01 2.34E+00 -8.65E+00 -8.64E+00 4.12E+00

6.44E-02 9.91E-02 6.44E-02 4.21E-02 2.39E-02

2.53E-05 1.65E-04 2.53E-05 3.19E-06 4.19E-07

6.81E-08 3.40E-06 6.81E-08 3.11E-06 1.36E-10

9.00E-06 2.45E-05 9.00E-06 5.67E-15 2.35E-07

Log Data Log Data Log Data Log Data Log Data

3.00 3.00 3.00 3.00 3.00

-2.00 -2.00 -2.00 -2.00 -2.00

2.83 2.83 2.83 2.83 2.83

normal normal normal normal normal

-1.06 1.58 -1.06 0.03 -0.63

1.22 1.22 1.22 1.22 1.22

normal normal normal normal normal

0.863705636 1.293580098 0.863705636 0.023935519 0.513997038
Example 8 Example 9 Example 10 Example 11 Example 12

-6.87E+00 #NUM! #NUM! #NUM! #NUM!

-6.95E+00 #NUM! #NUM! #NUM! #NUM!

-6.91E+00 #NUM! #NUM! #NUM! #NUM!

-6.91E+00 #NUM! #NUM! #NUM! #NUM!

4.21E-02 #NUM! #NUM! #NUM! #NUM!

3.19E-06 #NUM! #NUM! #NUM! #NUM!

3.11E-06 #NUM! #NUM! #NUM! #NUM!

5.67E-15 #NUM! #NUM! #NUM! #NUM!

Log Data Log Data Log Data Log Data Log Data

3.00 0.00 0.00 0.00 0.00

-2.00 #NUM! #NUM! #NUM! #NUM!

2.83 #DIV/0! #DIV/0! #DIV/0! #DIV/0!

normal #NUM! #NUM! #NUM! #NUM!

0.03 #NUM! #NUM! #NUM! #NUM!

1.22 0.00 0.00 0.00 0.00

normal #NUM! #NUM! #NUM! #NUM!

0.023935519 #NUM! #NUM! #NUM! #NUM!
Example 13 Example 14 Example 15 Example 16 Example 17

#NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM!

Log Data Log Data Log Data Log Data Log Data

0.00 0.00 0.00 0.00 0.00

#NUM! #NUM! #NUM! #NUM! #NUM!

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM!

0.00 0.00 0.00 0.00 0.00

#NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM!
Example 18 Example 19 Example 20 Example 21

#NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM!

Log Data Log Data Log Data Log Data

0.00 0.00 0.00 0.00

#NUM! #NUM! #NUM! #NUM!

#DIV/0! #DIV/0! #DIV/0! #DIV/0!

#NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM!

0.00 0.00 0.00 0.00

#NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM!
Raw Data lb/ton

Runs 514 1 2.99E-03 2 9.53E-03 3 5.61E-03 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 3.00 3.00

Kurtosis #DIV/0! #DIV/0!

SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 6.04E-03

Result Kurtosis #DIV/0! #DIV/0!

Skewness 0.58 -0.26

SE Skewness 1.22 1.22 Result Raw Data Result Log Data

Result Skewness normal normal #DIV/0! #DIV/0!

1. Take LN( ) of Raw Data

Runs 514 1 -5.81E+00 2 -4.65E+00 3 -5.18E+00 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

3

3. Calculate -5.22E+00

0.337445542

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

0900006481934216 PM-lbton 12/1/2014
12.2520905

0.483277475

27.35208269

7. Calculate

1.24411698

9. substitue the values of in cell D7 and F7 respectively

3.484 z value from "z-stat"

a- Calculate 0.006424721

b- Calculate 4.1277E-05

c- Calculate 0.401363291

d-calculate 0.140949221

e- Calculate 0.010103

f- Calculate UPL 1.82E-02

11. In column B, go down the the row number from Step 10. above, and copy the z value.

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat"

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is.

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

0900006481934216 PM-lbton 12/1/2014
∆z z

0.101 27.35208 1.244117

-5 =f(a) 463.2035

1 -4.899 421.0576

2 -4.798 381.7066

3 -4.697 345.03

4 -4.596 310.91

5 -4.495 279.2314

6 -4.394 249.8814

7 -4.293 222.7498

8 -4.192 197.7287

9 -4.091 174.7132

10 -3.99 153.6005

11 -3.889 134.2906

12 -3.788 116.6859

13 -3.687 100.6914

14 -3.586 86.21459

15 -3.485 73.16549

16 -3.384 61.4567

17 -3.283 51.00332

18 -3.182 41.723

19 -3.081 33.53592

20 -2.98 26.3648

21 -2.879 20.13487

22 -2.778 14.77392

23 -2.677 10.21226

24 -2.576 6.382752

25 -2.475 3.220766

26 -2.374 0.664222

27 -2.273 -1.34643

28 -2.172 -2.8682

29 -2.071 -3.95557

30 -1.97 -4.66049

31 -1.869 -5.03237

32 -1.768 -5.11808

33 -1.667 -4.96198

34 -1.566 -4.60588

35 -1.465 -4.08904

36 -1.364 -3.44822

37 -1.263 -2.71761

38 -1.162 -1.9289

39 -1.061 -1.11122

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -0.29118

41 -0.859 0.507151

42 -0.758 1.262234

43 -0.657 1.955065

44 -0.556 2.569176

45 -0.455 3.090631

46 -0.354 3.508029

47 -0.253 3.812502

48 -0.152 3.997719

49 -0.051 4.05988

50 0.05 3.99772

51 0.151 3.812508

52 0.252 3.508048

53 0.353 3.090678

54 0.454 2.569268

55 0.555 1.955224

56 0.656 1.262487

57 0.757 0.50753

58 0.858 -0.29064

59 0.959 -1.11048

60 1.06 -1.92791

61 1.161 -2.71633

62 1.262 -3.44659

63 1.363 -4.08701

64 1.464 -4.60338

65 1.565 -4.95895

66 1.666 -5.11444

67 1.767 -5.02804

68 1.868 -4.6554

69 1.969 -3.94963

70 2.07 -2.86133

71 2.171 -1.33853

72 2.272 0.673248

73 2.373 3.23102

74 2.474 6.394343

75 2.575 10.2253

76 2.676 14.78852

77 2.777 20.15115

78 2.878 26.38289

79 2.979 33.55595

80 3.08 41.7451

81 3.181 51.02763

82 3.282 61.48336

83 3.383 73.19465

84 3.484 86.2464

85 3.585 100.726

86 3.686 116.7235

87 3.787 134.3314

88 3.888 153.6446
89 3.989 174.7607

90 4.09 197.7799

91 4.191 222.8047

92 4.292 249.9404

93 4.393 279.2947

94 4.494 310.9777

95 4.595 345.1022

96 4.696 381.7836

97 4.797 421.1397

98 4.898 463.2908

99 4.999 508.3599
distribution of Z

Normal 

distribution

1.48672E-06 0.000688654

2.45094E-06 0.001031987

3.9995E-06 0.001526636

6.46025E-06 0.002228979

1.03291E-05 0.003211417

1.63472E-05 0.004564666

2.56093E-05 0.006399279

3.97118E-05 0.008845786

6.09553E-05 0.012052605

9.26132E-05 0.016180747

0.000139285 0.021394248

0.000207351 0.027845257

0.000305546 0.035652911

0.000445674 0.04487554

0.000643469 0.055476427

0.000919619 0.067284381

0.001300942 0.079951609

0.001821704 0.092912943

0.002525035 0.10535202

0.003464389 0.116181478

0.004704958 0.124045247

0.006324913 0.127351286

0.008416337 0.124342289

0.011085658 0.113209674

0.014453386 0.092252383

0.018652949 0.06007678

0.023828414 0.015827368

0.030130931 -0.040569091

0.03771375 -0.108170447

0.046725789 -0.184827021

0.057303789 -0.267063539

0.069563239 -0.350067665

0.083588399 -0.427812432

0.099421884 -0.493329872

0.117054396 -0.539138381

0.136415346 -0.557808155

0.157365126 -0.542629187

0.179689839 -0.488327241

0.203099244 -0.391758021

0.227228529 -0.252500893

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -0.073273625

0.275855243 0.139900292

0.299326439 0.377819919

0.321498297 0.628550081

0.341807853 0.878164461

0.359712192 1.11173754

0.37471238 1.314501734

0.386376486 1.473061259

0.394360216 1.576541345

0.398423793 1.617552685

0.398443914 1.592867053

0.394419966 1.503729262

0.386474058 1.355769593

0.374844865 1.158524603

0.359875719 0.924617077

0.34199778 0.668682393

0.32170943 0.406154008

0.299553264 0.152032198

0.276092166 -0.080243366

0.25188591 -0.279714154

0.227469632 -0.438541779

0.203335281 -0.552326028

0.17991684 -0.620099454

0.157579839 -0.64403015

0.136615273 -0.628891587

0.117237788 -0.581376068

0.099587708 -0.509335428

0.083736272 -0.421029404

0.069693339 -0.32445035

0.05741676 -0.226775201

0.046822635 -0.133974896

0.037795734 -0.050590672

0.030199481 0.02033173

0.023885038 0.077173047

0.018699163 0.119568861

0.014490659 0.148171378

0.011115369 0.164379871

0.008439746 0.170070626

0.006343145 0.167350506

0.004718997 0.158350443

0.003475077 0.145067465

0.002533081 0.129257107

0.001827693 0.112372732

0.001305351 0.09554474

0.000922829 0.079590688

0.00064578 0.065046903

0.00044732 0.052212756

0.000306705 0.041200145

0.000208159 0.031982427
0.000139842 0.024438844

9.29928E-05 0.018392104

6.12113E-05 0.013638163

3.98826E-05 0.009968273

2.5722E-05 0.007184023

1.64209E-05 0.005106528

1.03767E-05 0.00358101

6.49066E-06 0.002478027

4.01874E-06 0.001692449

2.46298E-06 0.001141074

1.49417E-06 0.000759577
Absolute values

0.000688654 3.4777E-05

0.001031987 0.000104231

0.001526636 0.00015419

0.002228979 0.000225127

0.003211417 0.000324353

0.004564666 0.000461031

0.006399279 0.000646327

0.008845786 0.000893424

0.012052605 0.001217313

0.016180747 0.001634255

0.021394248 0.002160819

0.027845257 0.002812371

0.035652911 0.003600944

0.04487554 0.004532429

0.055476427 0.005603119

0.067284381 0.006795723

0.079951609 0.008075112

0.092912943 0.009384207

0.10535202 0.010640554

0.116181478 0.011734329

0.124045247 0.01252857

0.127351286 0.01286248

0.124342289 0.012558571

0.113209674 0.011434177

0.092252383 0.009317491

0.06007678 0.006067755

0.015827368 0.001598564

0.040569091 0.004097478

0.108170447 0.010925215

0.184827021 0.018667529

0.267063539 0.026973417

0.350067665 0.035356834

0.427812432 0.043209056

0.493329872 0.049826317

0.539138381 0.054452976

0.557808155 0.056338624

0.542629187 0.054805548

0.488327241 0.049321051

0.391758021 0.03956756

0.252500893 0.02550259

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.073273625 0.007400636

0.139900292 0.014129929

0.377819919 0.038159812

0.628550081 0.063483558

0.878164461 0.088694611

1.11173754 0.112285492

1.314501734 0.132764675

1.473061259 0.148779187

1.576541345 0.159230676

1.617552685 0.163372821

1.592867053 0.160879572

1.503729262 0.151876655

1.355769593 0.136932729

1.158524603 0.117010985

0.924617077 0.093386325

0.668682393 0.067536922

0.406154008 0.041021555

0.152032198 0.015355252

0.080243366 0.00810458

0.279714154 0.02825113

0.438541779 0.04429272

0.552326028 0.055784929

0.620099454 0.062630045

0.64403015 0.065047045

0.628891587 0.06351805

0.581376068 0.058718983

0.509335428 0.051442878

0.421029404 0.04252397

0.32445035 0.032769485

0.226775201 0.022904295

0.133974896 0.013531464

0.050590672 0.005109658

0.02033173 0.002053505

0.077173047 0.007794478

0.119568861 0.012076455

0.148171378 0.014965309

0.164379871 0.016602367

0.170070626 0.017177133

0.167350506 0.016902401

0.158350443 0.015993395

0.145067465 0.014651814

0.129257107 0.013054968

0.112372732 0.011349646

0.09554474 0.009650019

0.079590688 0.00803866

0.065046903 0.006569737

0.052212756 0.005273488

0.041200145 0.004161215

0.031982427 0.003230225
0.024438844 0.002468323

0.018392104 0.001857603

0.013638163 0.001377455

0.009968273 0.001006796

0.007184023 0.000725586

0.005106528 0.000515759

0.00358101 0.000361682

0.002478027 0.000250281

0.001692449 0.000170937

0.001141074 0.000115248

0.000759577 3.83586E-05
Cumulative normalization

3.4777E-05 1.13583E-05

0.000139008 4.54005E-05

0.000293198 9.57597E-05

0.000518325 0.000169287

0.000842678 0.000275222

0.001303709 0.000425797

0.001950036 0.00063689

0.002843461 0.000928686

0.004060774 0.001326266

0.005695029 0.001860021

0.007855848 0.002565753

0.010668219 0.003484285

0.014269163 0.004660369

0.018801593 0.006140679

0.024404712 0.007970682

0.031200435 0.010190193

0.039275547 0.012827559

0.048659754 0.01589248

0.059300308 0.01936773

0.071034638 0.023200212

0.083563208 0.027292095

0.096425687 0.031493035

0.108984259 0.035594717

0.120418436 0.039329168

0.129735926 0.042372299

0.135803681 0.044354054

0.137402245 0.044876151

0.141499723 0.046214405

0.152424939 0.049782626

0.171092468 0.05587952

0.198065885 0.064689152

0.233422719 0.076236842

0.276631775 0.090349102

0.326458092 0.106622587

0.380911069 0.124407158

0.437249692 0.142807589

0.49205524 0.160707312

0.541376291 0.176815775

0.580943851 0.189738706

0.606446442 0.19806796

( f x ) ( f b)



 




0.613847078 0.200485039

0.627977007 0.205099933

0.666136819 0.217563088

0.729620377 0.238297085

0.818314988 0.267265118

0.930600479 0.303938032

1.063365155 0.347299534

1.212144342 0.395891443

1.371375017 0.447896851

1.534747839 0.501255102

1.695627411 0.553799048

1.847504066 0.603402602

1.984436795 0.648125408

2.10144778 0.686341688

2.194834105 0.716842054

2.262371027 0.738899897

2.303392582 0.75229771

2.318747834 0.7573128

2.326852414 0.759959789

2.355103543 0.769186727

2.399396263 0.783652916

2.455181192 0.801872508

2.517811236 0.822327745

2.582858282 0.843572383

2.646376332 0.864317646

2.705095315 0.883495513

2.756538193 0.900296973

2.799062163 0.914185481

2.831831648 0.924888132

2.854735943 0.932368772

2.868267408 0.936788205

2.873377066 0.938457041

2.87543057 0.939127724

2.883225048 0.941673433

2.895301503 0.945617654

2.910266812 0.950505387

2.926869179 0.95592779

2.944046312 0.961537914

2.960948714 0.967058309

2.976942108 0.972281819

2.991593922 0.977067163

3.00464889 0.981330971

3.015998536 0.985037813

3.025648555 0.98818955

3.033687214 0.99081501

3.040256952 0.992960714

3.04553044 0.994683057

3.049691655 0.996042127

3.05292188 0.997097132
3.055390203 0.997903297

3.057247806 0.998509998

3.05862526 0.99895988

3.059632056 0.999288704

3.060357642 0.999525684

3.060873401 0.999694133

3.061235083 0.99981226

3.061485364 0.999894002

3.061656301 0.999949831

3.06177155 0.999987472

3.061809909 1

153.2361334
Raw Data ranked by lb/ton, use concentration data

Runs 514 1 1.02E-03 2 3.18E-03 3 1.78E-03 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 3.00 3.00

Kurtosis #DIV/0! #DIV/0!

SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 2.00E-03

Result Kurtosis #DIV/0! #DIV/0!

Skewness 0.85 0.07

SE Skewness 1.22 1.22 Result Raw Data Result Log Data

Result Skewness normal normal #DIV/0! #DIV/0!

1. Take LN( ) of Raw Data

Runs 514 1 -6.88E+00 2 -5.75E+00 3 -6.33E+00 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

3

3. Calculate -6.32E+00

0.321232234

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

0900006481934216 PM-conc 12/1/2014
11.56063048

0.430526856

28.85228652

7. Calculate

1.200673476

9. substitue the values of in cell D7 and F7 respectively

3.484 z value from "z-stat"

a- Calculate 0.002111346

b- Calculate 4.45778E-06

c- Calculate 0.378825754

d-calculate 0.132874948

e- Calculate 0.003224

f- Calculate UPL 5.86E-03

11. In column B, go down the the row number from Step 10. above, and copy the z value.

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat"

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is.

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

0900006481934216 PM-conc 12/1/2014
∆z z

0.101 28.85229 1.200673

-5 =f(a) 493.879

1 -4.899 448.9942

2 -4.798 407.0825

3 -4.697 368.0158

4 -4.596 331.6691

5 -4.495 297.9197

6 -4.394 266.6478

7 -4.293 237.7364

8 -4.192 211.0708

9 -4.091 186.5395

10 -3.99 164.0332

11 -3.889 143.4457

12 -3.788 124.6732

13 -3.687 107.6147

14 -3.586 92.17184

15 -3.485 78.24911

16 -3.384 65.75353

17 -3.283 54.59484

18 -3.182 44.6855

19 -3.081 35.94065

20 -2.98 28.27811

21 -2.879 21.61841

22 -2.778 15.88474

23 -2.677 11.00301

24 -2.576 6.901806

25 -2.475 3.512406

26 -2.374 0.768779

27 -2.273 -1.39242

28 -2.172 -3.03183

29 -2.071 -4.20742

30 -1.97 -4.97447

31 -1.869 -5.38554

32 -1.768 -5.49053

33 -1.667 -5.33665

34 -1.566 -4.96839

35 -1.465 -4.42759

36 -1.364 -3.75336

37 -1.263 -2.98215

38 -1.162 -2.1477

39 -1.061 -1.28108

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -0.41066

41 -0.859 0.437898

42 -0.758 1.241592

43 -0.657 1.980129

44 -0.556 2.6359

45 -0.455 3.193989

46 -0.354 3.642168

47 -0.253 3.970899

48 -0.152 4.173336

49 -0.051 4.245323

50 0.05 4.185393

51 0.151 3.994769

52 0.252 3.677366

53 0.353 3.239788

54 0.454 2.691329

55 0.555 2.043973

56 0.656 1.312396

57 0.757 0.513962

58 0.858 -0.33127

59 0.959 -1.20057

60 1.06 -2.06848

61 1.161 -2.90689

62 1.262 -3.68497

63 1.363 -4.36923

64 1.464 -4.92347

65 1.565 -5.3088

66 1.666 -5.48364

67 1.767 -5.40372

68 1.868 -5.0221

69 1.969 -4.28911

70 2.07 -3.15244

71 2.171 -1.55704

72 2.272 0.554806

73 2.373 3.243498

74 2.474 6.572137

75 2.575 10.60651

76 2.676 15.4151

77 2.777 21.06908

78 2.878 27.64229

79 2.979 35.2113

80 3.08 43.85534

81 3.181 53.65635

82 3.282 64.69893

83 3.383 77.0704

84 3.484 90.86077

85 3.585 106.1627

86 3.686 123.0716

87 3.787 141.6856

88 3.888 162.1053
89 3.989 184.4343

90 4.09 208.7787

91 4.191 235.2473

92 4.292 263.9516

93 4.393 295.0059

94 4.494 328.527

95 4.595 364.6346

96 4.696 403.451

97 4.797 445.101

98 4.898 489.7124

99 4.999 537.4155
distribution of Z

Normal 

distribution

1.48672E-06 0.00073426

2.45094E-06 0.001100458

3.9995E-06 0.001628127

6.46025E-06 0.002377473

1.03291E-05 0.003425839

1.63472E-05 0.004870167

2.56093E-05 0.006828654

3.97118E-05 0.009440931

6.09553E-05 0.012865876

9.26132E-05 0.017276016

0.000139285 0.022847365

0.000207351 0.029743563

0.000305546 0.038093374

0.000445674 0.047961043

0.000643469 0.059309735

0.000919619 0.071959376

0.001300942 0.085541534

0.001821704 0.099455631

0.002525035 0.112832439

0.003464389 0.124512395

0.004704958 0.133047326

0.006324913 0.136734543

0.008416337 0.133691335

0.011085658 0.121975616

0.014453386 0.099754472

0.018652949 0.065516732

0.023828414 0.018318787

0.030130931 -0.041954788

0.03771375 -0.114341673

0.046725789 -0.196595176

0.057303789 -0.285055794

0.069563239 -0.374635636

0.083588399 -0.458944983

0.099421884 -0.530579646

0.117054396 -0.581572166

0.136415346 -0.603990604

0.157365126 -0.590647504

0.179689839 -0.535861365

0.203099244 -0.436196588

0.227228529 -0.291098433

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -0.103339333

0.275855243 0.12079643

0.299326439 0.371641392

0.321498297 0.636608094

0.341807853 0.900971442

0.359712192 1.148916769

0.37471238 1.364765255

0.386376486 1.534261981

0.394360216 1.645797856

0.398423793 1.691437805

0.398443914 1.667644387

0.394419966 1.575616834

0.386474058 1.421206737

0.374844865 1.214417959

0.359875719 0.968543939

0.34199778 0.699034335

0.32170943 0.422210168

0.299553264 0.153958978

0.276092166 -0.091462079

0.25188591 -0.302405574

0.227469632 -0.470516011

0.203335281 -0.591072493

0.17991684 -0.662988673

0.157579839 -0.688502973

0.136615273 -0.672621059

0.117237788 -0.622391438

0.099587708 -0.546102677

0.083736272 -0.45248751

0.069693339 -0.35000673

0.05741676 -0.246267083

0.046822635 -0.147605411

0.037795734 -0.058849316

0.030199481 0.016754864

0.023885038 0.077471072

0.018699163 0.122893462

0.014490659 0.153695357

0.011115369 0.171344555

0.008439746 0.177817661

0.006343145 0.175339083

0.004718997 0.166162026

0.003475077 0.152400712

0.002533081 0.135915856

0.001827693 0.118249807

0.001305351 0.100603957

0.000922829 0.083848961

0.00064578 0.068557805

0.00044732 0.055052392

0.000306705 0.043455725

0.000208159 0.033743607
0.000139842 0.025791619

9.29928E-05 0.019414913

6.12113E-05 0.014399788

3.98826E-05 0.010527075

2.5722E-05 0.007588148

1.64209E-05 0.005394705

1.03767E-05 0.003783692

6.49066E-06 0.002618663

4.01874E-06 0.001788744

2.46298E-06 0.001206149

1.49417E-06 0.000802991
Absolute values

0.00073426 3.70801E-05

0.001100458 0.000111146

0.001628127 0.000164441

0.002377473 0.000240125

0.003425839 0.00034601

0.004870167 0.000491887

0.006828654 0.000689694

0.009440931 0.000953534

0.012865876 0.001299454

0.017276016 0.001744878

0.022847365 0.002307584

0.029743563 0.0030041

0.038093374 0.003847431

0.047961043 0.004844065

0.059309735 0.005990283

0.071959376 0.007267897

0.085541534 0.008639695

0.099455631 0.010045019

0.112832439 0.011396076

0.124512395 0.012575752

0.133047326 0.01343778

0.136734543 0.013810189

0.133691335 0.013502825

0.121975616 0.012319537

0.099754472 0.010075202

0.065516732 0.00661719

0.018318787 0.001850198

0.041954788 0.004237434

0.114341673 0.011548509

0.196595176 0.019856113

0.285055794 0.028790635

0.374635636 0.037838199

0.458944983 0.046353443

0.530579646 0.053588544

0.581572166 0.058738789

0.603990604 0.061003051

0.590647504 0.059655398

0.535861365 0.054121998

0.436196588 0.044055855

0.291098433 0.029400942

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.103339333 0.010437273

0.12079643 0.012200439

0.371641392 0.037535781

0.636608094 0.064297417

0.900971442 0.090998116

1.148916769 0.116040594

1.364765255 0.137841291

1.534261981 0.15496046

1.645797856 0.166225583

1.691437805 0.170835218

1.667644387 0.168432083

1.575616834 0.1591373

1.421206737 0.14354188

1.214417959 0.122656214

0.968543939 0.097822938

0.699034335 0.070602468

0.422210168 0.042643227

0.153958978 0.015549857

0.091462079 0.00923767

0.302405574 0.030542963

0.470516011 0.047522117

0.591072493 0.059698322

0.662988673 0.066961856

0.688502973 0.0695388

0.672621059 0.067934727

0.622391438 0.062861535

0.546102677 0.05515637

0.45248751 0.045701238

0.35000673 0.03535068

0.246267083 0.024872975

0.147605411 0.014908147

0.058849316 0.005943781

0.016754864 0.001692241

0.077471072 0.007824578

0.122893462 0.01241224

0.153695357 0.015523231

0.171344555 0.0173058

0.177817661 0.017959584

0.175339083 0.017709247

0.166162026 0.016782365

0.152400712 0.015392472

0.135915856 0.013727501

0.118249807 0.011943231

0.100603957 0.010161

0.083848961 0.008468745

0.068557805 0.006924338

0.055052392 0.005560292

0.043455725 0.004389028

0.033743607 0.003408104
0.025791619 0.002604953

0.019414913 0.001960906

0.014399788 0.001454379

0.010527075 0.001063235

0.007588148 0.000766403

0.005394705 0.000544865

0.003783692 0.000382153

0.002618663 0.000264485

0.001788744 0.000180663

0.001206149 0.000121821

0.000802991 4.0551E-05
Cumulative normalization

3.70801E-05 1.14751E-05

0.000148226 4.58713E-05

0.000312667 9.67604E-05

0.000552792 0.000171071

0.000898802 0.00027815

0.001390689 0.000430373

0.002080383 0.000643811

0.003033917 0.000938899

0.00433337 0.001341038

0.006078248 0.001881021

0.008385832 0.002595143

0.011389931 0.003524815

0.015237362 0.00471547

0.020081428 0.006214552

0.026071711 0.008068351

0.033339608 0.010317529

0.041979303 0.012991235

0.052024321 0.016099844

0.063420398 0.01962656

0.07599615 0.023518348

0.08943393 0.027676906

0.103244119 0.031950713

0.116746943 0.0361294

0.129066481 0.039941898

0.139141682 0.043059847

0.145758872 0.045107653

0.14760907 0.045680229

0.151846503 0.046991578

0.163395012 0.050565467

0.183251125 0.056710292

0.21204176 0.065620061

0.249879959 0.077329759

0.296233403 0.09167465

0.349821947 0.10825857

0.408560736 0.126436323

0.469563787 0.145314793

0.529219185 0.163776208

0.583341183 0.180525214

0.627397038 0.194159076

0.65679798 0.203257715

( f x ) ( f b)



 




0.667235252 0.206487713

0.679435692 0.210263355

0.716971472 0.221879464

0.78126889 0.241777433

0.872267006 0.269938405

0.988307599 0.305849212

1.12614889 0.34850663

1.28110935 0.396461877

1.447334933 0.447903314

1.618170152 0.500771285

1.786602235 0.552895563

1.945739535 0.602143406

2.089281415 0.646564972

2.211937629 0.6845231

2.309760567 0.714796133

2.380363035 0.736645312

2.423006262 0.74984201

2.438556119 0.754654187

2.447793789 0.757512947

2.478336752 0.766965006

2.525858869 0.781671563

2.585557191 0.800146261

2.652519047 0.820868788

2.722057847 0.842388796

2.789992574 0.863412395

2.852854109 0.882866005

2.908010479 0.899935117

2.953711718 0.914078171

2.989062398 0.925018062

3.013935373 0.932715443

3.02884352 0.937329031

3.0347873 0.939168439

3.036479542 0.939692133

3.04430412 0.942113587

3.05671636 0.945954773

3.072239591 0.950758711

3.089545391 0.956114296

3.107504975 0.961672206

3.125214222 0.967152645

3.141996587 0.972346243

3.157389059 0.977109715

3.17111656 0.981357932

3.18305979 0.985053976

3.19322079 0.988198476

3.201689535 0.990819279

3.208613873 0.992962137

3.214174165 0.994682867

3.218563193 0.99604113

3.221971298 0.997095828
3.224576251 0.997901976

3.226537157 0.998508813

3.227991536 0.998958896

3.229054771 0.999287933

3.229821174 0.99952511

3.230366039 0.999693728

3.230748192 0.999811992

3.231012676 0.999893842

3.23119334 0.999949751

3.231315161 0.999987451

3.231355712 1

161.833731
Raw Data

Runs 568

1 1.37E-05

2 1.50E-05

3 1.56E-05

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -1.00

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

ni = number test runs = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 1.48E-05 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1.48E-05

Mean =

Pooled Variance = 8.83E-13

m= number future runs = 3

Term1 6.67E-01

Term2 5.89E-13

Squared root Term2 7.67E-07

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 6.964556734 99

UPL POOLED VARIANCE = 2.01E-05

Use for large sample sets and small sample sets that are Normal.

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

0900006481934216 Hg-Large-lbton 12/1/2014
Raw Data

Runs 568 1 1.17E+01 2 9.98E+00 3 9.71E+00 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 3.00 3.00

Kurtosis #DIV/0! #DIV/0!

SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 1.05E+01

Result Kurtosis #DIV/0! #DIV/0! variance 1.13

Skewness 1.61 1.58

SE Skewness 1.22 1.22 Result Raw Data Result Log Data

Result Skewness normal normal #DIV/0! #DIV/0!

1. Take LN( ) of Raw Data

Runs 568 1 2.46E+00 2 2.30E+00 3 2.27E+00 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

3

3. Calculate 2.34E+00

0.009813596

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

0900006481934216 Hg-Large-conc 12/1/2014
3.159255142

0.000291772

10829.83433

7. Calculate

0.172570439

9. substitue the values of in cell D7 and F7 respectively

3.484 z value from "z-stat"

a- Calculate 10.46892065

b- Calculate 109.5982996

c- Calculate 0.009861907

d-calculate 0.003295275

e- Calculate 2.548128

f- Calculate UPL 1.34E+01

11. In column B, go down the the row number from Step 10. above, and copy the z value.

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat"

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is.

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

0900006481934216 Hg-Large-conc 12/1/2014
∆z z

0.101 10829.83 0.17257

-5 =f(a) 215632.3

1 -4.899 196238.2

2 -4.798 178114.4

3 -4.697 161206.6

4 -4.596 145462.1

5 -4.495 130828.8

6 -4.394 117256.2

7 -4.293 104694.7

8 -4.192 93095.75

9 -4.091 82412.08

10 -3.99 72597.49

11 -3.889 63606.92

12 -3.788 55396.4

13 -3.687 47923.14

14 -3.586 41145.42

15 -3.485 35022.7

16 -3.384 29515.53

17 -3.283 24585.6

18 -3.182 20195.73

19 -3.081 16309.85

20 -2.98 12893.05

21 -2.879 9911.502

22 -2.778 7332.541

23 -2.677 5124.613

24 -2.576 3257.292

25 -2.475 1701.281

26 -2.374 428.4088

27 -2.273 -588.37

28 -2.172 -1374.97

29 -2.071 -1956.2

30 -1.97 -2355.7

31 -1.869 -2596.02

32 -1.768 -2698.57

33 -1.667 -2683.64

34 -1.566 -2570.39

35 -1.465 -2376.84

36 -1.364 -2119.9

37 -1.263 -1815.35

38 -1.162 -1477.85

39 -1.061 -1120.91

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -756.935

41 -0.859 -397.199

42 -0.758 -51.8453

43 -0.657 270.1075

44 -0.556 560.7674

45 -0.455 813.3691

46 -0.354 1022.274

47 -0.253 1182.97

48 -0.152 1292.072

49 -0.051 1347.322

50 0.05 1347.586

51 0.151 1292.86

52 0.252 1184.265

53 0.353 1024.05

54 0.454 815.5871

55 0.555 563.3792

56 0.656 273.0538

57 0.757 -48.6347

58 0.858 -393.805

59 0.959 -753.45

60 1.06 -1117.43

61 1.161 -1474.5

62 1.262 -1812.25

63 1.363 -2117.18

64 1.464 -2374.65

65 1.565 -2568.89

66 1.666 -2682.99

67 1.767 -2698.96

68 1.868 -2597.62

69 1.969 -2358.72

70 2.07 -1960.85

71 2.171 -1381.48

72 2.272 -596.962

73 2.373 417.4893

74 2.474 1687.781

75 2.575 3240.945

76 2.676 5105.145

77 2.777 7309.666

78 2.878 9884.922

79 2.979 12862.45

80 3.08 16274.93

81 3.181 20156.15

82 3.282 24541.02

83 3.383 29465.6

84 3.484 34967.05

85 3.585 41083.69

86 3.686 47854.93

87 3.787 55321.34

88 3.888 63524.58
89 3.989 72507.48

90 4.09 82313.95

91 4.191 92989.07

92 4.292 104579

93 4.393 117131.1

94 4.494 130693.8

95 4.595 145316.6

96 4.696 161050.3

97 4.797 177946.6

98 4.898 196058.6

99 4.999 215440.2
distribution of Z

Normal 

distribution

1.48672E-06 0.320584728

2.45094E-06 0.480968059

3.9995E-06 0.712368881

6.46025E-06 1.041434637

1.03291E-05 1.502490538

1.63472E-05 2.138691351

2.56093E-05 3.002845525

3.97118E-05 4.157611109

6.09553E-05 5.674674747

9.26132E-05 7.632445337

0.000139285 10.11174029

0.000207351 13.18893815

0.000305546 16.92614391

0.000445674 21.35809064

0.000643469 26.4758084

0.000919619 32.20754271

0.001300942 38.3979977

0.001821704 44.78768437

0.002525035 50.99491248

0.003464389 56.50367799

0.004704958 60.66123887

0.006324913 62.68938565

0.008416337 61.71313917

0.011085658 56.80970624

0.014453386 47.07890339

0.018652949 31.73391561

0.023828414 10.20830309

0.030130931 -17.72815034

0.03771375 -51.85545301

0.046725789 -91.4047783

0.057303789 -134.990404

0.069563239 -180.5875352

0.083588399 -225.5693993

0.099421884 -266.8128203

0.117054396 -300.8751322

0.136415346 -324.2372193

0.157365126 -333.5984632

0.179689839 -326.2005536

0.203099244 -300.1497382

0.227228529 -254.7024581

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -190.4784967

0.275855243 -109.5694599

0.299326439 -15.51867645

0.321498297 86.83909555

0.341807853 191.6746972

0.359712192 292.5787914

0.37471238 383.0587505

0.386376486 457.0718861

0.394360216 509.5419173

0.398423793 536.8049788

0.398443914 536.9374517

0.394419966 509.9298876

0.386474058 457.6878688

0.374844865 383.8597112

0.359875719 293.5099852

0.34199778 192.6744437

0.32170943 87.84398392

0.299553264 -14.56868599

0.276092166 -108.7265328

0.25188591 -189.783421

0.227469632 -254.1824092

0.203335281 -299.8174229

0.17991684 -326.0546888

0.157579839 -333.6254108

0.136615273 -324.4134915

0.117237788 -301.1704294

0.099587708 -267.1932449

0.083736272 -226.0005213

0.069693339 -181.0370431

0.05741676 -135.4301614

0.046822635 -91.81218949

0.037795734 -52.21413206

0.030199481 -18.02795645

0.023885038 9.971748348

0.018699163 31.56008415

0.014490659 46.96343636

0.011115369 56.74556679

0.008439746 61.6917223

0.006343145 62.70149656

0.004718997 60.69788418

0.003475077 56.55664587

0.002533081 51.05714379

0.001827693 44.85345646

0.001305351 38.4629548

0.000922829 32.26861206

0.00064578 26.53104138

0.00044732 21.4064641

0.000306705 16.96735009

0.000208159 13.2231846
0.000139842 10.13957335

9.29928E-05 7.654604526

6.12113E-05 5.691980245

3.98826E-05 4.17088268

2.5722E-05 3.012849008

1.64209E-05 2.146107647

1.03767E-05 1.507901753

6.49066E-06 1.045322392

4.01874E-06 0.715120523

2.46298E-06 0.482887348

1.49417E-06 0.321904473
Absolute values

0.320584728 0.016189529

0.480968059 0.048577774

0.712368881 0.071949257

1.041434637 0.105184898

1.502490538 0.151751544

2.138691351 0.216007826

3.002845525 0.303287398

4.157611109 0.419918722

5.674674747 0.573142149

7.632445337 0.770876979

10.11174029 1.021285769

13.18893815 1.332082753

16.92614391 1.709540535

21.35809064 2.157167155

26.4758084 2.674056648

32.20754271 3.252961814

38.3979977 3.878197768

44.78768437 4.523556122

50.99491248 5.150486161

56.50367799 5.706871477

60.66123887 6.126785126

62.68938565 6.331627951

61.71313917 6.233027056

56.80970624 5.737780331

47.07890339 4.754969242

31.73391561 3.205125477

10.20830309 1.031038612

17.72815034 1.790543185

51.85545301 5.237400755

91.4047783 9.231882608

134.990404 13.6340308

180.5875352 18.23934105

225.5693993 22.78250933

266.8128203 26.94809485

300.8751322 30.38838836

324.2372193 32.74795915

333.5984632 33.69344479

326.2005536 32.94625591

300.1497382 30.31512356

254.7024581 25.72494826

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
190.4784967 19.23832816

109.5694599 11.06651545

15.51867645 1.567386321

86.83909555 8.77074865

191.6746972 19.35914441

292.5787914 29.55045794

383.0587505 38.6889338

457.0718861 46.16426049

509.5419173 51.46373365

536.8049788 54.21730286

536.9374517 54.23068262

509.9298876 51.50291865

457.6878688 46.22647475

383.8597112 38.76983083

293.5099852 29.6445085

192.6744437 19.46011881

87.84398392 8.872242376

14.56868599 1.471437285

108.7265328 10.98137981

189.783421 19.16812552

254.1824092 25.67242333

299.8174229 30.28155971

326.0546888 32.93152357

333.6254108 33.69616649

324.4134915 32.76576264

301.1704294 30.41821337

267.1932449 26.98651774

226.0005213 22.82605265

181.0370431 18.28474136

135.4301614 13.6784463

91.81218949 9.273031139

52.21413206 5.273627338

18.02795645 1.820823602

9.971748348 1.007146583

31.56008415 3.187568499

46.96343636 4.743307072

56.74556679 5.731302246

61.6917223 6.230863952

62.70149656 6.332851153

60.69788418 6.130486302

56.55664587 5.712221233

51.05714379 5.156771522

44.85345646 4.530199103

38.4629548 3.884758435

32.26861206 3.259129818

26.53104138 2.67963518

21.4064641 2.162052874

16.96735009 1.713702359

13.2231846 1.335541645
10.13957335 1.024096908

7.654604526 0.773115057

5.691980245 0.574890005

4.17088268 0.421259151

3.012849008 0.30429775

2.146107647 0.216756872

1.507901753 0.152298077

1.045322392 0.105577562

0.715120523 0.072227173

0.482887348 0.048771622

0.321904473 0.016256176
Cumulative normalization

0.016189529 1.28182E-05

0.064767303 5.128E-05

0.13671656 0.000108246

0.241901458 0.000191527

0.393653002 0.000311677

0.609660829 0.000482703

0.912948227 0.000722833

1.332866949 0.001055307

1.906009098 0.001509096

2.676886077 0.002119443

3.698171846 0.002928053

5.0302546 0.003982739

6.739795135 0.00533628

8.896962289 0.007044232

11.57101894 0.009161435

14.82398075 0.01173699

18.70217852 0.01480758

23.22573464 0.018389137

28.3762208 0.02246707

34.08309228 0.026985525

40.2098774 0.03183645

46.54150536 0.03684956

52.77453241 0.041784602

58.51231274 0.04632753

63.26728198 0.05009231

66.47240746 0.05262999

67.50344607 0.053446322

69.29398926 0.054863997

74.53139001 0.059010745

83.76327262 0.066320152

97.39730342 0.077114991

115.6366445 0.091556116

138.4191538 0.109594326

165.3672487 0.130930667

195.755637 0.154990884

228.5035962 0.180919308

262.1970409 0.207596327

295.1432969 0.233681754

325.4584204 0.257683963

351.1833687 0.27805187

( f x ) ( f b)



 




370.4216968 0.293283949

381.4882123 0.30204594

383.0555986 0.303286929

391.8263473 0.31023123

411.1854917 0.325558967

440.7359496 0.348955747

479.4248834 0.379587979

525.5891439 0.416138853

577.0528776 0.456885621

631.2701804 0.499812547

685.500863 0.542750066

737.0037817 0.583527859

783.2302564 0.620127992

822.0000873 0.650824274

851.6445958 0.67429552

871.1047146 0.689703204

879.976957 0.696727863

881.4483942 0.697892884

892.429774 0.706587468

911.5978996 0.721763964

937.2703229 0.742090283

967.5518826 0.766065918

1000.483406 0.792139681

1034.179573 0.818818854

1066.945335 0.844761374

1097.363549 0.868845206

1124.350066 0.890211969

1147.176119 0.908284654

1165.46086 0.922761725

1179.139307 0.93359173

1188.412338 0.940933717

1193.685965 0.945109148

1195.506789 0.946550797

1196.513935 0.947348213

1199.701504 0.949871992

1204.444811 0.953627539

1210.176113 0.958165337

1216.406977 0.963098667

1222.739828 0.968112745

1228.870315 0.972966601

1234.582536 0.977489291

1239.739307 0.981572201

1244.269506 0.985159018

1248.154265 0.988234802

1251.413395 0.990815241

1254.09303 0.99293686

1256.255083 0.994648681

1257.968785 0.996005517

1259.304327 0.997062941
1260.328424 0.997873777

1261.101539 0.998485896

1261.676429 0.998941069

1262.097688 0.999274604

1262.401986 0.999515534

1262.618743 0.999687153

1262.771041 0.999807736

1262.876618 0.999891327

1262.948845 0.999948514

1262.997617 0.999987129

1263.013873 1

63776.35047
Raw Data rank by lb/ton use lb/hr

Runs 568

1 1.63E-04

2 1.78E-04

3 1.85E-04

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -1.00

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

ni = number test runs = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 1.75E-04 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1.75E-04

Mean =

Pooled Variance = 1.25E-10

m= number future runs = 3

Term1 6.67E-01

Term2 8.31E-11

Squared root Term2 9.12E-06

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 6.964556734 99

UPL POOLED VARIANCE = 2.39E-04

Use for large sample sets and small sample sets that are Normal.

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

0900006481934216 Hg-Large-lbhr 12/1/2014
Raw Data rank by lb/ton use lb/ton

Runs 531 1 1.84E-04 2 1.69E-04 3 1.76E-04 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 3.00 3.00

Kurtosis #DIV/0! #DIV/0!

SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 1.76E-04

Result Kurtosis #DIV/0! #DIV/0!

Skewness 0.09 0.03

SE Skewness 1.22 1.22 Result Raw Data Result Log Data

Result Skewness normal normal #DIV/0! #DIV/0!

1. Take LN( ) of Raw Data

Runs 531 1 -8.60E+00 2 -8.69E+00 3 -8.64E+00 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

3

3. Calculate -8.64E+00

0.001773919

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

0900006481934216 Hg-Small-lbton 12/1/2014
3.028455212

9.45713E-06

320231.8612

7. Calculate

0.07302593

9. substitue the values of in cell D7 and F7 respectively

3.484 z value from "z-stat "

a- Calculate 0.000176319

b- Calculate 3.10885E-08

c- Calculate 0.001775493

d-calculate 0.000592093

e- Calculate 1.82E-05

f- Calculate UPL 2.0E-04

11. In column B, go down the the row number from Step 10. above, and copy the z value.

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat "

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is.

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

0900006481934216 Hg-Small-lbton 12/1/2014
∆z z

0.101 320231.9 0.073026

-5 =f(a) 6377891

1 -4.899 5804260

2 -4.798 5268199

3 -4.697 4768108

4 -4.596 4302421

5 -4.495 3869604

6 -4.394 3468158

7 -4.293 3096616

8 -4.192 2753546

9 -4.091 2437548

10 -3.99 2147255

11 -3.889 1881334

12 -3.788 1638485

13 -3.687 1417443

14 -3.586 1216973

15 -3.485 1035876

16 -3.384 872986

17 -3.283 727169.3

18 -3.182 597326.2

19 -3.081 482390.2

20 -2.98 381328

21 -2.879 293140

22 -2.778 216859.4

23 -2.677 151553.1

24 -2.576 96321.29

25 -2.475 50297.27

26 -2.374 12647.86

27 -2.273 -17426.9

28 -2.172 -40693.5

29 -2.071 -57885.4

30 -1.97 -69702.4

31 -1.869 -76811.1

32 -1.768 -79845

33 -1.667 -79403.9

34 -1.566 -76054.5

35 -1.465 -70330.2

36 -1.364 -62730.9

37 -1.263 -53723.4

38 -1.162 -43741

39 -1.061 -33183.7

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -22418.4

41 -0.859 -11778.3

42 -0.758 -1563.57

43 -0.657 7959.066

44 -0.556 16556.17

45 -0.455 24027.64

46 -0.354 30206.71

47 -0.253 34959.9

48 -0.152 38187.09

49 -0.051 39821.47

50 0.05 39829.55

51 0.151 38211.15

52 0.252 34999.45

53 0.353 30260.93

54 0.454 24095.38

55 0.555 16635.93

56 0.656 8049.039

57 0.757 -1465.52

58 0.858 -11674.7

59 0.959 -22311.9

60 1.06 -33077.6

61 1.161 -43638.7

62 1.262 -53628.7

63 1.363 -62647.9

64 1.464 -70263.3

65 1.565 -76008.6

66 1.666 -79384.1

67 1.767 -79856.7

68 1.868 -76860.1

69 1.969 -69794.7

70 2.07 -58027.5

71 2.171 -40892.3

72 2.272 -17689.3

73 2.373 12314.41

74 2.474 49884.99

75 2.575 95822.1

76 2.676 150958.6

77 2.777 216160.8

78 2.878 292328.3

79 2.979 380393.8

80 3.08 481323.7

81 3.181 596117.4

82 3.282 725807.8

83 3.383 871461.1

84 3.484 1034177

85 3.585 1215088

86 3.686 1415360

87 3.787 1636193

88 3.888 1878820
89 3.989 2144506

90 4.09 2434551

91 4.191 2750289

92 4.292 3093084

93 4.393 3464337

94 4.494 3865480

95 4.595 4297979

96 4.696 4763334

97 4.797 5263077

98 4.898 5798774

99 4.999 6372025
distribution of Z

Normal 

distribution

1.48672E-06 9.482135231

2.45094E-06 14.22589281

3.9995E-06 21.07017796

6.46025E-06 30.80315723

1.03291E-05 44.44008603

1.63472E-05 63.25737464

2.56093E-05 88.8169625

3.97118E-05 122.9721103

6.09553E-05 167.8431131

9.26132E-05 225.7490785

0.000139285 299.0803286

0.000207351 390.0958602

0.000305546 500.6324415

0.000445674 631.7171745

0.000643469 783.0844331

0.000919619 952.6114436

0.001300942 1135.704304

0.001821704 1324.687107

0.002525035 1508.269244

0.003464389 1671.187086

0.004704958 1794.132213

0.006324913 1854.084705

0.008416337 1825.161903

0.011085658 1680.066356

0.014453386 1392.168772

0.018652949 938.1924676

0.023828414 301.3784363

0.030130931 -525.0880094

0.03771375 -1534.705674

0.046725789 -2704.740072

0.057303789 -3994.210046

0.069563239 -5343.231447

0.083588399 -6674.1139

0.099421884 -7894.483006

0.117054396 -8902.512616

0.136415346 -9594.112976

0.157365126 -9871.654295

0.179689839 -9653.544101

0.203099244 -8883.758627

0.227228529 -7540.291364

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -5641.457864

0.275855243 -3249.104849

0.299326439 -468.0164235

0.321498297 2558.8262

0.341807853 5659.029172

0.359712192 8643.036797

0.37471238 11318.82738

0.386376486 13507.68414

0.394360216 15059.47049

0.398423793 15865.82151

0.398443914 15869.84028

0.394419966 15071.24222

0.386474058 13526.38094

0.374844865 11343.15265

0.359875719 8671.340423

0.34199778 5689.45084

0.32170943 2589.451717

0.299553264 -439.0018051

0.276092166 -3223.280932

0.25188591 -5620.06389

0.227469632 -7524.158318

0.203335281 -8873.282863

0.17991684 -9648.701009

0.157579839 -9872.046658

0.136615273 -9599.045599

0.117237788 -8911.083046

0.099587708 -7905.677738

0.083736272 -6686.900326

0.069693339 -5356.637602

0.05741676 -4007.386645

0.046822635 -2717.001999

0.037795734 -1545.55264

0.030199481 -534.2062408

0.023885038 294.1300802

0.018699163 932.807623

0.014490659 1388.525396

0.011115369 1677.960975

0.008439746 1824.342699

0.006343145 1854.280771

0.004718997 1795.0773

0.003475077 1672.637104

0.002533081 1510.013548

0.001827693 1326.554202

0.001305351 1137.562967

0.000922829 954.3684071

0.00064578 784.6798087

0.00044732 633.1186513

0.000306705 501.829117

0.000208159 391.0923349
0.000139842 299.8914797

9.29928E-05 226.395735

6.12113E-05 168.3487008

3.98826E-05 123.3602238

2.5722E-05 89.10975263

1.64209E-05 63.47460261

1.03767E-05 44.59868829

6.49066E-06 30.91717381

4.01874E-06 21.15091801

2.46298E-06 14.28223616

1.49417E-06 9.520894664
Absolute values

9.482135231 0.478847829

14.22589281 1.436815174

21.07017796 2.128087974

30.80315723 3.11111888

44.44008603 4.488448689

63.25737464 6.388994839

88.8169625 8.970513213

122.9721103 12.42018314

167.8431131 16.95215442

225.7490785 22.80065693

299.0803286 30.20711319

390.0958602 39.39968188

500.6324415 50.56387659

631.7171745 63.80343462

783.0844331 79.09152774

952.6114436 96.21375581

1135.704304 114.7061347

1324.687107 133.7933978

1508.269244 152.3351937

1671.187086 168.7898957

1794.132213 181.2073535

1854.084705 187.2625552

1825.161903 184.3413522

1680.066356 169.6867019

1392.168772 140.609046

938.1924676 94.75743923

301.3784363 30.43922207

525.0880094 53.03388895

1534.705674 155.005273

2704.740072 273.1787473

3994.210046 403.4152146

5343.231447 539.6663761

6674.1139 674.0855039

7894.483006 797.3427836

8902.512616 899.1537742

9594.112976 969.0054106

9871.654295 997.0370838

9653.544101 975.0079542

8883.758627 897.2596213

7540.291364 761.5694277

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
5641.457864 569.7872442

3249.104849 328.1595898

468.0164235 47.26965877

2558.8262 258.4414462

5659.029172 571.5619464

8643.036797 872.9467164

11318.82738 1143.201565

13507.68414 1364.276098

15059.47049 1521.006519

15865.82151 1602.447973

15869.84028 1602.853868

15071.24222 1522.195464

13526.38094 1366.164475

11343.15265 1145.658418

8671.340423 875.8053828

5689.45084 574.6345348

2589.451717 261.5346234

439.0018051 44.33918231

3223.280932 325.5513742

5620.06389 567.6264529

7524.158318 759.9399901

8873.282863 896.2015692

9648.701009 974.5188019

9872.046658 997.0767124

9599.045599 969.5036055

8911.083046 900.0193876

7905.677738 798.4734515

6686.900326 675.3769329

5356.637602 541.0203978

4007.386645 404.7460512

2717.001999 274.4172019

1545.55264 156.1008167

534.2062408 53.95483033

294.1300802 29.7071381

932.807623 94.21356992

1388.525396 140.241065

1677.960975 169.4740585

1824.342699 184.2586126

1854.280771 187.2823578

1795.0773 181.3028073

1672.637104 168.9363475

1510.013548 152.5113683

1326.554202 133.9819744

1137.562967 114.8938597

954.3684071 96.39120912

784.6798087 79.25266068

633.1186513 63.94498378

501.829117 50.68474082

391.0923349 39.50032582
299.8914797 30.28903945

226.395735 22.86596924

168.3487008 17.00321878

123.3602238 12.4593826

89.10975263 9.000085016

63.47460261 6.410934864

44.59868829 4.504467517

30.91717381 3.122634555

21.15091801 2.136242719

14.28223616 1.442505852

9.520894664 0.480805181
Cumulative normalization

0.478847829 1.28191E-05

1.915663003 5.12837E-05

4.043750977 0.000108254

7.154869857 0.000191541

11.64331855 0.0003117

18.03231338 0.000482738

27.0028266 0.000722885

39.42300974 0.001055382

56.37516416 0.001509204

79.17582108 0.002119593

109.3829343 0.002928259

148.7826162 0.003983017

199.3464927 0.005336648

263.1499274 0.007044712

342.2414551 0.009162049

438.4552109 0.01173776

553.1613456 0.014808525

686.9547434 0.01839027

839.2899371 0.022468393

1008.079833 0.026987019

1189.287186 0.03183807

1376.549741 0.036851222

1560.891094 0.041786172

1730.577796 0.046328807

1871.186842 0.050093012

1965.944281 0.052629736

1996.383503 0.053444616

2049.417392 0.054864371

2204.422665 0.059013973

2477.601412 0.066327164

2881.016627 0.077126878

3420.683003 0.091574133

4094.768507 0.109619885

4892.11129 0.130965322

5791.265065 0.155036313

6760.270475 0.180977282

7757.307559 0.207668678

8732.315513 0.233770339

9629.575134 0.257790622

10391.14456 0.278178381

( f x ) ( f b)



 




10960.93181 0.293431993

11289.0914 0.302217061

11336.36105 0.303482503

11594.8025 0.310401166

12166.36445 0.325702288

13039.31116 0.349071697

14182.51273 0.379676022

15546.78883 0.41619867

17067.79535 0.456917103

18670.24332 0.499815783

20273.09719 0.542725329

21795.29265 0.583475591

23161.45713 0.620048793

24307.11554 0.650718889

25182.92093 0.674164826

25757.55546 0.689548204

26019.09009 0.696549673

26063.42927 0.697736665

26388.98064 0.706451908

26956.6071 0.721647675

27716.54709 0.741991813

28612.74865 0.765983771

29587.26746 0.792072337

30584.34417 0.818764794

31553.84777 0.8447191

32453.86716 0.868813264

33252.34061 0.89018897

33927.71755 0.908269294

34468.73794 0.922752797

34873.484 0.933588139

35147.9012 0.940934484

35304.00201 0.945113414

35357.95684 0.946557824

35387.66398 0.947353105

35481.87755 0.94987527

35622.11862 0.953629624

35791.59268 0.958166566

35975.85129 0.963099301

36163.13365 0.968112983

36344.43645 0.972966589

36513.3728 0.977489136

36665.88417 0.981571975

36799.86614 0.985158768

36914.76 0.988234559

37011.15121 0.990815021

37090.40387 0.992936671

37154.34886 0.994648524

37205.0336 0.996005391

37244.53392 0.997062843
37274.82296 0.997873703

37297.68893 0.998485841

37314.69215 0.998941029

37327.15153 0.999274576

37336.15162 0.999515515

37342.56255 0.99968714

37347.06702 0.999807728

37350.18966 0.999891323

37352.3259 0.999948512

37353.7684 0.999987129

37354.24921 1

1886224.019
Raw Data rank by lb/ton using concentration data

Runs 531

1 6.03E+01

2 6.21E+01

3 6.32E+01

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -0.60

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

ni = number test runs = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 6.19E+01 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

6.19E+01

Mean =

Pooled Variance = 2.18E+00

m= number future runs = 3

Term1 6.67E-01

Term2 1.45E+00

Squared root Term2 1.21E+00

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 6.964556734 99

UPL POOLED VARIANCE = 70

Use for large sample sets and small sample sets that are Normal.

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

0900006481934216 Hg-Small-conc 12/1/2014
Raw Data rank by lb/ton using lb/hr data

Runs 531 1 1.04E-03 2 9.55E-04 3 9.96E-04 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 3.00 3.00

Kurtosis #DIV/0! #DIV/0!

SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 9.97E-04

Result Kurtosis #DIV/0! #DIV/0!

Skewness 0.09 0.03

SE Skewness 1.22 1.22 Result Raw Data Result Log Data

Result Skewness normal normal #DIV/0! #DIV/0!

1. Take LN( ) of Raw Data

Runs 531 1 -6.87E+00 2 -6.95E+00 3 -6.91E+00 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

3

3. Calculate -6.91E+00

0.001773919

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

0900006481934216 Hg-Small-lbhr 12/1/2014
3.028455212

9.45713E-06

320231.8612

7. Calculate

0.07302593

9. substitue the values of in cell D7 and F7 respectively

3.484 z value from "z-stat "

a- Calculate 0.000997086

b- Calculate 9.9418E-07

c- Calculate 0.001775493

d-calculate 0.000592093

e- Calculate 0.000103

f- Calculate UPL 0.0011

11. In column B, go down the the row number from Step 10. above, and copy the z value.

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat "

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is.

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

0900006481934216 Hg-Small-lbhr 12/1/2014
∆z z

0.101 320231.9 0.073026

-5 =f(a) 6377891

1 -4.899 5804260

2 -4.798 5268199

3 -4.697 4768108

4 -4.596 4302421

5 -4.495 3869604

6 -4.394 3468158

7 -4.293 3096616

8 -4.192 2753546

9 -4.091 2437548

10 -3.99 2147255

11 -3.889 1881334

12 -3.788 1638485

13 -3.687 1417443

14 -3.586 1216973

15 -3.485 1035876

16 -3.384 872986

17 -3.283 727169.3

18 -3.182 597326.2

19 -3.081 482390.2

20 -2.98 381328

21 -2.879 293140

22 -2.778 216859.4

23 -2.677 151553.1

24 -2.576 96321.29

25 -2.475 50297.27

26 -2.374 12647.86

27 -2.273 -17426.9

28 -2.172 -40693.5

29 -2.071 -57885.4

30 -1.97 -69702.4

31 -1.869 -76811.1

32 -1.768 -79845

33 -1.667 -79403.9

34 -1.566 -76054.5

35 -1.465 -70330.2

36 -1.364 -62730.9

37 -1.263 -53723.4

38 -1.162 -43741

39 -1.061 -33183.7

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -22418.4

41 -0.859 -11778.3

42 -0.758 -1563.57

43 -0.657 7959.066

44 -0.556 16556.17

45 -0.455 24027.64

46 -0.354 30206.71

47 -0.253 34959.9

48 -0.152 38187.09

49 -0.051 39821.47

50 0.05 39829.55

51 0.151 38211.15

52 0.252 34999.45

53 0.353 30260.93

54 0.454 24095.38

55 0.555 16635.93

56 0.656 8049.039

57 0.757 -1465.52

58 0.858 -11674.7

59 0.959 -22311.9

60 1.06 -33077.6

61 1.161 -43638.7

62 1.262 -53628.7

63 1.363 -62647.9

64 1.464 -70263.3

65 1.565 -76008.6

66 1.666 -79384.1

67 1.767 -79856.7

68 1.868 -76860.1

69 1.969 -69794.7

70 2.07 -58027.5

71 2.171 -40892.3

72 2.272 -17689.3

73 2.373 12314.41

74 2.474 49884.99

75 2.575 95822.1

76 2.676 150958.6

77 2.777 216160.8

78 2.878 292328.3

79 2.979 380393.8

80 3.08 481323.7

81 3.181 596117.4

82 3.282 725807.8

83 3.383 871461.1

84 3.484 1034177

85 3.585 1215088

86 3.686 1415360

87 3.787 1636193

88 3.888 1878820
89 3.989 2144506

90 4.09 2434551

91 4.191 2750289

92 4.292 3093084

93 4.393 3464337

94 4.494 3865480

95 4.595 4297979

96 4.696 4763334

97 4.797 5263077

98 4.898 5798774

99 4.999 6372025
distribution of Z

Normal 

distribution

1.48672E-06 9.482135231

2.45094E-06 14.22589281

3.9995E-06 21.07017796

6.46025E-06 30.80315723

1.03291E-05 44.44008603

1.63472E-05 63.25737464

2.56093E-05 88.8169625

3.97118E-05 122.9721103

6.09553E-05 167.8431131

9.26132E-05 225.7490785

0.000139285 299.0803286

0.000207351 390.0958602

0.000305546 500.6324415

0.000445674 631.7171745

0.000643469 783.0844331

0.000919619 952.6114436

0.001300942 1135.704304

0.001821704 1324.687107

0.002525035 1508.269244

0.003464389 1671.187086

0.004704958 1794.132213

0.006324913 1854.084705

0.008416337 1825.161903

0.011085658 1680.066356

0.014453386 1392.168772

0.018652949 938.1924676

0.023828414 301.3784363

0.030130931 -525.0880094

0.03771375 -1534.705674

0.046725789 -2704.740072

0.057303789 -3994.210046

0.069563239 -5343.231447

0.083588399 -6674.1139

0.099421884 -7894.483006

0.117054396 -8902.512616

0.136415346 -9594.112976

0.157365126 -9871.654295

0.179689839 -9653.544101

0.203099244 -8883.758627

0.227228529 -7540.291364

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -5641.457864

0.275855243 -3249.104849

0.299326439 -468.0164235

0.321498297 2558.8262

0.341807853 5659.029172

0.359712192 8643.036797

0.37471238 11318.82738

0.386376486 13507.68414

0.394360216 15059.47049

0.398423793 15865.82151

0.398443914 15869.84028

0.394419966 15071.24222

0.386474058 13526.38094

0.374844865 11343.15265

0.359875719 8671.340423

0.34199778 5689.45084

0.32170943 2589.451717

0.299553264 -439.0018051

0.276092166 -3223.280932

0.25188591 -5620.06389

0.227469632 -7524.158318

0.203335281 -8873.282863

0.17991684 -9648.701009

0.157579839 -9872.046658

0.136615273 -9599.045599

0.117237788 -8911.083046

0.099587708 -7905.677738

0.083736272 -6686.900326

0.069693339 -5356.637602

0.05741676 -4007.386645

0.046822635 -2717.001999

0.037795734 -1545.55264

0.030199481 -534.2062408

0.023885038 294.1300802

0.018699163 932.807623

0.014490659 1388.525396

0.011115369 1677.960975

0.008439746 1824.342699

0.006343145 1854.280771

0.004718997 1795.0773

0.003475077 1672.637104

0.002533081 1510.013548

0.001827693 1326.554202

0.001305351 1137.562967

0.000922829 954.3684071

0.00064578 784.6798087

0.00044732 633.1186513

0.000306705 501.829117

0.000208159 391.0923349
0.000139842 299.8914797

9.29928E-05 226.395735

6.12113E-05 168.3487008

3.98826E-05 123.3602238

2.5722E-05 89.10975263

1.64209E-05 63.47460261

1.03767E-05 44.59868829

6.49066E-06 30.91717381

4.01874E-06 21.15091801

2.46298E-06 14.28223616

1.49417E-06 9.520894664
Absolute values

9.482135231 0.478847829

14.22589281 1.436815174

21.07017796 2.128087974

30.80315723 3.11111888

44.44008603 4.488448689

63.25737464 6.388994839

88.8169625 8.970513213

122.9721103 12.42018314

167.8431131 16.95215442

225.7490785 22.80065693

299.0803286 30.20711319

390.0958602 39.39968188

500.6324415 50.56387659

631.7171745 63.80343462

783.0844331 79.09152774

952.6114436 96.21375581

1135.704304 114.7061347

1324.687107 133.7933978

1508.269244 152.3351937

1671.187086 168.7898957

1794.132213 181.2073535

1854.084705 187.2625552

1825.161903 184.3413522

1680.066356 169.6867019

1392.168772 140.609046

938.1924676 94.75743923

301.3784363 30.43922207

525.0880094 53.03388895

1534.705674 155.005273

2704.740072 273.1787473

3994.210046 403.4152146

5343.231447 539.6663761

6674.1139 674.0855039

7894.483006 797.3427836

8902.512616 899.1537742

9594.112976 969.0054106

9871.654295 997.0370838

9653.544101 975.0079542

8883.758627 897.2596213

7540.291364 761.5694277

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
5641.457864 569.7872442

3249.104849 328.1595898

468.0164235 47.26965877

2558.8262 258.4414462

5659.029172 571.5619464

8643.036797 872.9467164

11318.82738 1143.201565

13507.68414 1364.276098

15059.47049 1521.006519

15865.82151 1602.447973

15869.84028 1602.853868

15071.24222 1522.195464

13526.38094 1366.164475

11343.15265 1145.658418

8671.340423 875.8053828

5689.45084 574.6345348

2589.451717 261.5346234

439.0018051 44.33918231

3223.280932 325.5513742

5620.06389 567.6264529

7524.158318 759.9399901

8873.282863 896.2015692

9648.701009 974.5188019

9872.046658 997.0767124

9599.045599 969.5036055

8911.083046 900.0193876

7905.677738 798.4734515

6686.900326 675.3769329

5356.637602 541.0203978

4007.386645 404.7460512

2717.001999 274.4172019

1545.55264 156.1008167

534.2062408 53.95483033

294.1300802 29.7071381

932.807623 94.21356992

1388.525396 140.241065

1677.960975 169.4740585

1824.342699 184.2586126

1854.280771 187.2823578

1795.0773 181.3028073

1672.637104 168.9363475

1510.013548 152.5113683

1326.554202 133.9819744

1137.562967 114.8938597

954.3684071 96.39120912

784.6798087 79.25266068

633.1186513 63.94498378

501.829117 50.68474082

391.0923349 39.50032582
299.8914797 30.28903945

226.395735 22.86596924

168.3487008 17.00321878

123.3602238 12.4593826

89.10975263 9.000085016

63.47460261 6.410934864

44.59868829 4.504467517

30.91717381 3.122634555

21.15091801 2.136242719

14.28223616 1.442505852

9.520894664 0.480805181
Cumulative normalization

0.478847829 1.28191E-05

1.915663003 5.12837E-05

4.043750977 0.000108254

7.154869857 0.000191541

11.64331855 0.0003117

18.03231338 0.000482738

27.0028266 0.000722885

39.42300974 0.001055382

56.37516416 0.001509204

79.17582108 0.002119593

109.3829343 0.002928259

148.7826162 0.003983017

199.3464927 0.005336648

263.1499274 0.007044712

342.2414551 0.009162049

438.4552109 0.01173776

553.1613456 0.014808525

686.9547434 0.01839027

839.2899371 0.022468393

1008.079833 0.026987019

1189.287186 0.03183807

1376.549741 0.036851222

1560.891094 0.041786172

1730.577796 0.046328807

1871.186842 0.050093012

1965.944281 0.052629736

1996.383503 0.053444616

2049.417392 0.054864371

2204.422665 0.059013973

2477.601412 0.066327164

2881.016627 0.077126878

3420.683003 0.091574133

4094.768507 0.109619885

4892.11129 0.130965322

5791.265065 0.155036313

6760.270475 0.180977282

7757.307559 0.207668678

8732.315513 0.233770339

9629.575134 0.257790622

10391.14456 0.278178381

( f x ) ( f b)



 




10960.93181 0.293431993

11289.0914 0.302217061

11336.36105 0.303482503

11594.8025 0.310401166

12166.36445 0.325702288

13039.31116 0.349071697

14182.51273 0.379676022

15546.78883 0.41619867

17067.79535 0.456917103

18670.24332 0.499815783

20273.09719 0.542725329

21795.29265 0.583475591

23161.45713 0.620048793

24307.11554 0.650718889

25182.92093 0.674164826

25757.55546 0.689548204

26019.09009 0.696549673

26063.42927 0.697736665

26388.98064 0.706451908

26956.6071 0.721647675

27716.54709 0.741991813

28612.74865 0.765983771

29587.26746 0.792072337

30584.34417 0.818764794

31553.84777 0.8447191

32453.86716 0.868813264

33252.34061 0.89018897

33927.71755 0.908269294

34468.73794 0.922752797

34873.484 0.933588139

35147.9012 0.940934484

35304.00201 0.945113414

35357.95684 0.946557824

35387.66398 0.947353105

35481.87755 0.94987527

35622.11862 0.953629624

35791.59268 0.958166566

35975.85129 0.963099301

36163.13365 0.968112983

36344.43645 0.972966589

36513.3728 0.977489136

36665.88417 0.981571975

36799.86614 0.985158768

36914.76 0.988234559

37011.15121 0.990815021

37090.40387 0.992936671

37154.34886 0.994648524

37205.0336 0.996005391

37244.53392 0.997062843
37274.82296 0.997873703

37297.68893 0.998485841

37314.69215 0.998941029

37327.15153 0.999274576

37336.15162 0.999515515

37342.56255 0.99968714

37347.06702 0.999807728

37350.18966 0.999891323

37352.3259 0.999948512

37353.7684 0.999987129

37354.24921 1

1886224.019
Raw Data

Runs

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Testing Normality Raw Data

Sample Size 0.00

Kurtosis #DIV/0!

SE Kurtosis 0.00

Result Kurtosis #DIV/0!

Skewness #DIV/0!

SE Skewness 0.00 Result Raw Data

Result Skewness #DIV/0! #DIV/0!

ni = number test runs = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 0

n =Total # test runs = 0

-1

This step is for QC only:

means #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

#DIV/0!

Mean =

Pooled Variance = #DIV/0!

m= number future runs = 3

Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0!

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = #NUM!

UPL POOLED VARIANCE = #DIV/0!

Use for large sample sets and small sample sets that are Normal.

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

0900006481934216 UPL Pooled Template 12/1/2014
Raw Data

Runs 123456789

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Testing Normality Raw Data Logtransformed Data

Sample Size 0.00 0.00

Kurtosis #DIV/0! #DIV/0!

SE Kurtosis 0.00 0.00 Average of Raw Data #DIV/0!

Result Kurtosis #DIV/0! #DIV/0!

Skewness #DIV/0! #DIV/0!

SE Skewness 0.00 0.00 Result Raw Data Result Log Data

Result Skewness #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1. Take LN( ) of Raw Data

Runs 123456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

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30

31

32

33

34

35

0

3. Calculate #DIV/0!

#DIV/0!

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

0900006481934216 Lognormal Template 12/1/2014
#DIV/0!

#DIV/0!

#DIV/0!

7. Calculate

#DIV/0!

9. substitue the values of in cell D7 and F7 respectively

3.484 z value from "z-stat "

a- Calculate #DIV/0!

b- Calculate #DIV/0!

c- Calculate #DIV/0!

d-calculate #DIV/0!

e- Calculate #DIV/0!

f- Calculate UPL #DIV/0!

11. In column B, go down the the row number from Step 10. above, and copy the z value.

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

8. Go to tab distribution for 99 percentile "z-stat "

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is.

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

0900006481934216 Lognormal Template 12/1/2014
∆z z

0.101 14.78551 3.387135

-5 =f(a) 173.6307

1 -4.899 156.5385

2 -4.798 140.6598

3 -4.697 125.9391

4 -4.596 112.3223

5 -4.495 99.75638

6 -4.394 88.18975

7 -4.293 77.57188

8 -4.192 67.85352

9 -4.091 58.98665

10 -3.99 50.92447

11 -3.889 43.62141

12 -3.788 37.03312

13 -3.687 31.11647

14 -3.586 25.82959

15 -3.485 21.13179

16 -3.384 16.98365

17 -3.283 13.34694

18 -3.182 10.18468

19 -3.081 7.461103

20 -2.98 5.141681

21 -2.879 3.193102

22 -2.778 1.583284

23 -2.677 0.281372

24 -2.576 -0.74226

25 -2.475 -1.51602

26 -2.374 -2.06709

27 -2.273 -2.42141

28 -2.172 -2.6037

29 -2.071 -2.63747

30 -1.97 -2.54498

31 -1.869 -2.34728

32 -1.768 -2.06419

33 -1.667 -1.71429

34 -1.566 -1.31496

35 -1.465 -0.88233

36 -1.364 -0.43132

37 -1.263 0.024392

38 -1.162 0.472344

39 -1.061 0.901303

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 1.301261

41 -0.859 1.663439

42 -0.758 1.980282

43 -0.657 2.245462

44 -0.556 2.45388

45 -0.455 2.601659

46 -0.354 2.686152

47 -0.253 2.705937

48 -0.152 2.660818

49 -0.051 2.551826

50 0.05 2.381219

51 0.151 2.152479

52 0.252 1.870318

53 0.353 1.540671

54 0.454 1.170702

55 0.555 0.7688

56 0.656 0.344581

57 0.757 -0.09111

58 0.858 -0.52622

59 0.959 -0.94743

60 1.06 -1.34023

61 1.161 -1.68888

62 1.262 -1.97639

63 1.363 -2.18458

64 1.464 -2.294

65 1.565 -2.28401

66 1.666 -2.13273

67 1.767 -1.81705

68 1.868 -1.31264

69 1.969 -0.59394

70 2.07 0.365836

71 2.171 1.594699

72 2.272 3.121887

73 2.373 4.977865

74 2.474 7.194324

75 2.575 9.804181

76 2.676 12.84158

77 2.777 16.34189

78 2.878 20.34171

79 2.979 24.87887

80 3.08 29.9924

81 3.181 35.7226

82 3.282 42.11095

83 3.383 49.20019

84 3.484 57.03427

85 3.585 65.65838

86 3.686 75.11893
87 3.787 85.46354

88 3.888 96.74107

89 3.989 109.0016

90 4.09 122.2965

91 4.191 136.6783

92 4.292 152.2006

93 4.393 168.9187

94 4.494 186.8885

95 4.595 206.1677

96 4.696 226.8148

97 4.797 248.8898

98 4.898 272.4537

99 4.999 297.569
distribution of Z

Normal 

distribution

1.48672E-06 0.00025814

2.45094E-06 0.000383666

3.9995E-06 0.000562569

6.46025E-06 0.000813598

1.03291E-05 0.001160187

1.63472E-05 0.001630742

2.56093E-05 0.002258474

3.97118E-05 0.003080516

6.09553E-05 0.004136028

9.26132E-05 0.005462942

0.000139285 0.007093014

0.000207351 0.009044929

0.000305546 0.011315317

0.000445674 0.0138678

0.000643469 0.016620542

0.000919619 0.0194332

0.001300942 0.022094744

0.001821704 0.02431417

0.002525035 0.025716663

0.003464389 0.025848163

0.004704958 0.024191389

0.006324913 0.02019609

0.008416337 0.013325453

0.011085658 0.003119197

0.014453386 -0.010728213

0.018652949 -0.028278328

0.023828414 -0.049255451

0.030130931 -0.072959218

0.03771375 -0.098195275

0.046725789 -0.123237716

0.057303789 -0.145836905

0.069563239 -0.163284321

0.083588399 -0.172542023

0.099421884 -0.170438144

0.117054396 -0.153922026

0.136415346 -0.120363821

0.157365126 -0.067874755

0.179689839 0.004383042

0.203099244 0.095932777

0.227228529 0.204801723

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 0.327454986

0.275855243 0.458868267

0.299326439 0.592750629

0.321498297 0.72191232

0.341807853 0.838755395

0.359712192 0.93584856

0.37471238 1.006534541

0.386376486 1.045510485

0.394360216 1.049320814

0.398423793 1.016708275

0.398443914 0.948782077

0.394419966 0.848980757

0.386474058 0.722829327

0.374844865 0.577512693

0.359875719 0.421307289

0.34199778 0.262927915

0.32170943 0.110854823

0.299553264 -0.027293533

0.276092166 -0.145283984

0.25188591 -0.238643935

0.227469632 -0.304862135

0.203335281 -0.343408575

0.17991684 -0.355586415

0.157579839 -0.344245108

0.136615273 -0.31339536

0.117237788 -0.267772357

0.099587708 -0.212393663

0.083736272 -0.152153024

0.069693339 -0.091482269

0.05741676 -0.034102101

0.046822635 0.017129395

0.037795734 0.060272801

0.030199481 0.094279357

0.023885038 0.118896489

0.018699163 0.134527832

0.014490659 0.142069051

0.011115369 0.142738913

0.008439746 0.137921435

0.006343145 0.129030453

0.004718997 0.117403303

0.003475077 0.104225924

0.002533081 0.090488222

0.001827693 0.076965905

0.001305351 0.064223535

0.000922829 0.052632887

0.00064578 0.042400898

0.00044732 0.033602192
0.000306705 0.026212124

0.000208159 0.020137481

0.000139842 0.01524298

9.29928E-05 0.011372694

6.12113E-05 0.008366251

3.98826E-05 0.006070157

2.5722E-05 0.004344929

1.64209E-05 0.003068875

1.03767E-05 0.002139333

6.49066E-06 0.001472177

4.01874E-06 0.001000222

2.46298E-06 0.000671047

1.49417E-06 0.000444619
Absolute values

0.00025814 1.30361E-05

0.000383666 3.87503E-05

0.000562569 5.68195E-05

0.000813598 8.21734E-05

0.001160187 0.000117179

0.001630742 0.000164705

0.002258474 0.000228106

0.003080516 0.000311132

0.004136028 0.000417739

0.005462942 0.000551757

0.007093014 0.000716394

0.009044929 0.000913538

0.011315317 0.001142847

0.0138678 0.001400648

0.016620542 0.001678675

0.0194332 0.001962753

0.022094744 0.002231569

0.02431417 0.002455731

0.025716663 0.002597383

0.025848163 0.002610664

0.024191389 0.00244333

0.02019609 0.002039805

0.013325453 0.001345871

0.003119197 0.000315039

0.010728213 0.00108355

0.028278328 0.002856111

0.049255451 0.004974801

0.072959218 0.007368881

0.098195275 0.009917723

0.123237716 0.012447009

0.145836905 0.014729527

0.163284321 0.016491716

0.172542023 0.017426744

0.170438144 0.017214253

0.153922026 0.015546125

0.120363821 0.012156746

0.067874755 0.00685535

0.004383042 0.000442687

0.095932777 0.009689211

0.204801723 0.020684974

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.327454986 0.033072954

0.458868267 0.046345695

0.592750629 0.059867814

0.72191232 0.072913144

0.838755395 0.084714295

0.93584856 0.094520705

1.006534541 0.101659989

1.045510485 0.105596559

1.049320814 0.105981402

1.016708275 0.102687536

0.948782077 0.09582699

0.848980757 0.085747056

0.722829327 0.073005762

0.577512693 0.058328782

0.421307289 0.042552036

0.262927915 0.026555719

0.110854823 0.011196337

0.027293533 0.002756647

0.145283984 0.014673682

0.238643935 0.024103037

0.304862135 0.030791076

0.343408575 0.034684266

0.355586415 0.035914228

0.344245108 0.034768756

0.31339536 0.031652931

0.267772357 0.027045008

0.212393663 0.02145176

0.152153024 0.015367455

0.091482269 0.009239709

0.034102101 0.003444312

0.017129395 0.001730069

0.060272801 0.006087553

0.094279357 0.009522215

0.118896489 0.012008545

0.134527832 0.013587311

0.142069051 0.014348974

0.142738913 0.01441663

0.137921435 0.013930065

0.129030453 0.013032076

0.117403303 0.011857734

0.104225924 0.010526818

0.090488222 0.00913931

0.076965905 0.007773556

0.064223535 0.006486577

0.052632887 0.005315922

0.042400898 0.004282491

0.033602192 0.003393821
0.026212124 0.002647425

0.020137481 0.002033886

0.01524298 0.001539541

0.011372694 0.001148642

0.008366251 0.000844991

0.006070157 0.000613086

0.004344929 0.000438838

0.003068875 0.000309956

0.002139333 0.000216073

0.001472177 0.00014869

0.001000222 0.000101022

0.000671047 6.77757E-05

0.000444619 2.24533E-05
Cumulative normalization

1.30361E-05 7.04745E-06

5.17864E-05 2.79963E-05

0.000108606 5.87135E-05

0.000190779 0.000103137

0.000307958 0.000166486

0.000472663 0.000255527

0.000700769 0.000378843

0.001011901 0.000547045

0.00142964 0.000772879

0.001981397 0.001071165

0.002697791 0.001458456

0.003611329 0.001952325

0.004754176 0.00257016

0.006154824 0.003327366

0.007833499 0.004234876

0.009796252 0.005295962

0.012027821 0.006502373

0.014483552 0.007829968

0.017080935 0.009234142

0.0196916 0.010645496

0.02213493 0.011966388

0.024174735 0.013069129

0.025520606 0.013796721

0.025835645 0.013967035

0.026919195 0.014552814

0.029775306 0.016096859

0.034750106 0.018786291

0.042118987 0.02276999

0.05203671 0.02813162

0.064483719 0.034860611

0.079213247 0.042823556

0.095704963 0.05173916

0.113131707 0.06116025

0.13034596 0.070466465

0.145892085 0.078870872

0.15804883 0.085442943

0.164904181 0.089149021

0.165346868 0.089388343

0.175036079 0.094626437

0.195721052 0.105808962

( f x ) ( f b)



 




0.228794006 0.123688566

0.275139701 0.14874356

0.335007515 0.181108761

0.407920659 0.220526412

0.492634954 0.266323895

0.587155658 0.317422832

0.688815647 0.372381344

0.794412206 0.42946801

0.900393608 0.486762726

1.003081144 0.542276742

1.098908134 0.594081871

1.18465519 0.640437676

1.257660952 0.679905397

1.315989734 0.711438581

1.35854177 0.734442682

1.38509749 0.748798997

1.396293827 0.75485186

1.399050474 0.756342134

1.413724156 0.764274889

1.437827193 0.777305257

1.468618269 0.793951252

1.503302535 0.812701949

1.539216763 0.832117578

1.573985519 0.850913951

1.60563845 0.868025875

1.632683458 0.882646705

1.654135218 0.894243763

1.669502674 0.90255158

1.678742383 0.907546668

1.682186695 0.909408702

1.683916764 0.910343996

1.690004317 0.913634994

1.699526532 0.918782809

1.711535077 0.925274761

1.725122388 0.932620211

1.739471362 0.940377425

1.753887993 0.948171215

1.767818058 0.955701962

1.780850133 0.962747246

1.792707867 0.969157667

1.803234685 0.974848582

1.812373996 0.979789394

1.820147552 0.983991864

1.826634129 0.987498579

1.831950051 0.990372425

1.836232541 0.992687587

1.839626363 0.994522324
1.842273787 0.995953551

1.844307673 0.997053092

1.845847214 0.997885385

1.846995856 0.998506354

1.847840847 0.998963165

1.848453933 0.999294606

1.848892771 0.999531847

1.849202727 0.999699413

1.8494188 0.999816224

1.84956749 0.999896607

1.849668512 0.999951221

1.849736288 0.999987862

1.849758741 1

90.98161566
Raw Data

Runs

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

n for each source 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Sample size 0

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY

Testing Normality Raw Data Logtransformed

Sample Size 0.00 0.00

Kurtosis #DIV/0! #DIV/0! http://www.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics

SE Kurtosis 0.00 0.00 note: cannot use this equation when n=3 because the denominator in second component is not defined (the term n-3 in denominator of second compone

Result Kurtosis #DIV/0! #DIV/0!

Skewness #DIV/0! #DIV/0!

SE Skewness 0.00 0.00 Result Raw Data Result Log Data

Result Skewness #DIV/0! #DIV/0! #DIV/0! #DIV/0!

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n = 3 ONLY

Average #DIV/0!

Standard Deviation #DIV/0! http://www.spiderfinancial.com/support/documentation/numxl/reference-manual/statistical-tests/test_xkurt

1

2

3
4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 0.00

Kurtosis -3.00

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness #DIV/0!

SE Skewness 0.00 Result Raw Data

Result Skewness #DIV/0! #DIV/0!

### The kurtosis and skewness are pulled into "Calculations for Template skew", in cells H12 and H14 respectively

kurtosis= #DIV/0! depending on sample size

skewness= #DIV/0! same for n>3 and n=3

ni = number test runs = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 0

n =Total # test runs = 0

1

N

i df n

 

    

1

N

i

i

n





N

1

N

i

i

n




-1

This step is for QC only:

means #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Mean = #DIV/0!

Pooled Variance = #DIV/0!

m= number future runs = 3

Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0!

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = #NUM!

UPL based on t-statistic with df at 99% confidence level #DIV/0!

0

#NUM!

#NUM!

Adjustment for non-normality (skewness and kurtosis) for any sample size in particular for small samples 

a) n = number of test-runs =

b) Calculate u0

Current Confidence Level is

2

1

0

1

1

u

t

n







 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1 i

i

df n



      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 
Conclusion

interval of differences for t-values used

Adjusted UPL based on recalculated t-statistic #DIV/0!

new t-statistic from tab Recalculate t-statistic

#NUM!
Logtransformed

0 0 0 0 0 0 0 0 0 0 0 0 0 0

e term n-3 in denominator of second component is zero).
0 0 0 0 0 0 0 0 0 0 0 0 0 0
#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!






Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

c.1.7 #NUM! d.1.7

Calculations Calculations

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 ,

 

x

I a b 

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )



( )

w a

e w





a w  
Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

Result #DIV/0!

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



1 1 1 1  1 2   1 3  4 1   1

( ) , , ,

n n n n n n n n

P t I I I

             

        
4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

Corrected probability is::

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

#NUM! e.1.7

Calculations Calculations

d.1.) Calculate e.1.) Calculate 



 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
#NUM! Result3:

3 4 1 1

,

n

I

 

   

 
#NUM!

0

3 4 1 1

,

24 2 2

u

n

I

 

   

  

 

1 1

2 2

5 1

2 2



 








a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

#NUM! f.1.7 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
#NUM! Result4: #NUM!

a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.7 #NUM! h.1.7

h.1.) Calculate 

Calculations Calculations

g.1.) Calculate 0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
Result5: #NUM! Result6:

#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

3 1

2 6

w a

    

 
#NUM!
#NUM!

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM!

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM!

4) If you don't get a YES in row 11, play around with the cell G3, the interval for differences until you get one p-value closed to 0.01

5) Once you get a YES, use the t-statistic in the calculations of the UPL.

Go back to Template_nonnormal tab and paste the value in row 5 that has a Yes in row 11.

Copy the new t-value in cell E164 of Template_nonnormal n<3

1) Start with initial t-statistic from Template

2) Determine interval of differences for t-values

2) Candidate values for t-statistic

3) Corrected p-values

Acutal significance level with t-value

4) Is p-value closed enough to 0.01?
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

x=u0= #NUM! skewness= #DIV/0!

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
#NUM! u0= #NUM!

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.) Calculate h.1.) Calculate 

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 