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

Large Kilns PM 122 71 12% Sources with Data 1 3 0.0060

Equivalent PM 122 71 12% Sources with Data 1 3 0.0020

Small Kilns PM 103 21 12% Sources with Data 1 3 0.0093

Equivalent PM 103 21 12% Sources with Data 1 3 0.0023
UPL Unit of Measure Data Distribution

0.018 lb/ton lognormal

0.0059 gr/dscf @ 7% O2 lognormal

0.029 lb/ton lognormal

0.0072 gr/dscf @ 7% O2 lognormal
Runs

514 PM 

lb/ton

514 PM 

conc

599 PM 

Small 

lb/ton

599 PM 

Small 

conc

Example 

5

Example 

6

Example 

7

Example 

8

Example 

9

1 2.99E-03 1.02E-03 8.11E-03 2.03E-03

2 9.53E-03 3.18E-03 1.53E-02 3.77E-03

3 5.61E-03 1.78E-03 4.53E-03 1.12E-03

Average 6.04E-03 2.00E-03 9.30E-03 2.31E-03 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Standard Deviation 3.29E-03 1.09E-03 5.47E-03 1.35E-03 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1 8.74E-11 8.93E-13 2.01E-12 6.30E-15 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

2 1.48E-10 1.98E-12 1.26E-09 4.61E-12 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

3 3.43E-14 2.10E-15 5.19E-10 1.96E-12 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Testing Normality Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data

Sample Size 3.00 3.00 3.00 3.00 0.00 0.00 0.00 0.00 0.00

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

SE Kurtosis 2.83 2.83 2.83 2.83 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Result Kurtosisnormal normal normal normal #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Skewness 0.58 0.85 0.93 0.90 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

SE Skewness 1.22 1.22 1.22 1.22 0.00 0.00 0.00 0.00 0.00

Result Skewnessnormal normal normal normal #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

0.47198482 0.69088383 0.76234674 0.73496458 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

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

1 2 3 4 5 6 7 8 9
Example 

10

Example 

11

Example 

12

Example 

13

Example 

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! #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 Raw Data Raw Data Raw Data Raw Data Raw Data

0.00 0.00 0.00 0.00 0.00 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!

#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 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!

#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 15 16 17 18 19
Example 

20 Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example 7 Example 8

-5.81E+00 -6.88E+00 -4.81E+00 -6.20E+00 #NUM! #NUM! #NUM! #NUM!

-4.65E+00 -5.75E+00 -4.18E+00 -5.58E+00 #NUM! #NUM! #NUM! #NUM!

-5.18E+00 -6.33E+00 -5.40E+00 -6.79E+00 #NUM! #NUM! #NUM! #NUM!

#DIV/0! -5.22E+00 -6.32E+00 -4.80E+00 -6.19E+00 #NUM! #NUM! #NUM! #NUM!

#DIV/0! 5.81E-01 5.67E-01 6.08E-01 6.06E-01 #NUM! #NUM! #NUM! #NUM!

#DIV/0! 1.27E-01 1.00E-01 7.46E-08 1.39E-08 #NUM! #NUM! #NUM! #NUM!

#DIV/0! 1.01E-01 1.06E-01 1.44E-01 1.40E-01 #NUM! #NUM! #NUM! #NUM!

#DIV/0! 1.31E-06 5.57E-09 1.29E-01 1.30E-01 #NUM! #NUM! #NUM! #NUM!

Raw Data Log Data Log Data Log Data Log Data Log Data Log Data Log Data Log Data

0.00 3.00 3.00 3.00 3.00 0.00 0.00 0.00 0.00

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

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

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

#DIV/0! -0.26 0.07 0.12 0.08 #NUM! #NUM! #NUM! #NUM!

0.00 1.22 1.22 1.22 1.22 0.00 0.00 0.00 0.00

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

#DIV/0! 0.21330941 0.05598517 0.09985236 0.06584724 #NUM! #NUM! #NUM! #NUM!

#DIV/0!

20 1 2 3 4 5 6 7 8
Example 9

Example 

10

Example 

11

Example 

12

Example 

13

Example 

14

Example 

15

Example 

16

Example 

17

Example 

18

#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!

#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 Log Data Log Data Log Data Log Data Log Data

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

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

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

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

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

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

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

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

9 10 11 12 13 14 15 16 17 18
Example 

19

Example 

20

#NUM! #NUM!

#NUM! #NUM!

#NUM! #NUM!

#NUM! #NUM!

#NUM! #NUM!

#NUM! #NUM!

#NUM! #NUM!

#NUM! #NUM!

Log Data Log Data

0.00 0.00

#NUM! #NUM!

#DIV/0! #DIV/0!

#NUM! #NUM!

#NUM! #NUM!

0.00 0.00

#NUM! #NUM!

#NUM! #NUM!

19 20
Raw Data

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-Large-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-Large-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 599 1 8.11E-03 2 1.53E-02 3 4.53E-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 9.30E-03

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

Skewness 0.93 0.12

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 599 1 -4.81E+00 2 -4.18E+00 3 -5.40E+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 -4.80E+00

0.369349906

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-Small-lbton 12/1/2014
13.71806635

0.598873984

24.90643227

7. Calculate

1.330173276

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

3.484 z value from "z-stat"

a- Calculate 0.009916872

b- Calculate 9.83443E-05

c- Calculate 0.446793757

d-calculate 0.157221474

e- Calculate 0.016461

f- Calculate UPL 2.90E-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-Small-lbton 12/1/2014
∆z z

0.101 24.90643 1.330173

-5 =f(a) 412.9166

1 -4.899 375.2538

2 -4.798 340.0945

3 -4.697 307.3309

4 -4.596 276.8569

5 -4.495 248.5693

6 -4.394 222.3667

7 -4.293 198.1502

8 -4.192 175.8232

9 -4.091 155.2912

10 -3.99 136.4622

11 -3.889 119.2462

12 -3.788 103.5559

13 -3.687 89.30591

14 -3.586 76.41324

15 -3.485 64.79719

16 -3.384 54.37934

17 -3.283 45.08354

18 -3.182 36.83593

19 -3.081 29.56492

20 -2.98 23.20119

21 -2.879 17.67773

22 -2.778 12.92977

23 -2.677 8.894862

24 -2.576 5.512804

25 -2.475 2.725688

26 -2.374 0.477885

27 -2.273 -1.28396

28 -2.172 -2.61091

29 -2.071 -3.55177

30 -1.97 -4.15304

31 -1.869 -4.45896

32 -1.768 -4.51148

33 -1.667 -4.35027

34 -1.566 -4.01273

35 -1.465 -3.53397

36 -1.364 -2.94683

37 -1.263 -2.28186

38 -1.162 -1.56734

39 -1.061 -0.82926

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -0.09134

41 -0.859 0.624982

42 -0.758 1.300552

43 -0.657 1.91849

44 -0.556 2.464199

45 -0.455 2.925361

46 -0.354 3.291937

47 -0.253 3.556168

48 -0.152 3.712574

49 -0.051 3.757956

50 0.05 3.691392

51 0.151 3.514242

52 0.252 3.230144

53 0.353 2.845018

54 0.454 2.367062

55 0.555 1.806751

56 0.656 1.176845

57 0.757 0.492379

58 0.858 -0.22933

59 0.959 -0.96869

60 1.06 -1.70381

61 1.161 -2.41056

62 1.262 -3.06248

63 1.363 -3.63088

64 1.464 -4.08474

65 1.565 -4.39081

66 1.666 -4.51352

67 1.767 -4.41505

68 1.868 -4.05528

69 1.969 -3.39182

70 2.07 -2.38

71 2.171 -0.97287

72 2.272 0.878804

73 2.373 3.226532

74 2.474 6.124104

75 2.575 9.627587

76 2.676 13.79533

77 2.777 18.68797

78 2.878 24.3684

79 2.979 30.90182

80 3.08 38.35569

81 3.181 46.79976

82 3.282 56.30606

83 3.383 66.94889

84 3.484 78.80484

85 3.585 91.95278

86 3.686 106.4738

87 3.787 122.4515

88 3.888 139.9713
89 3.989 159.1214

90 4.09 179.9921

91 4.191 202.6757

92 4.292 227.2673

93 4.393 253.8638

94 4.494 282.5646

95 4.595 313.4715

96 4.696 346.6883

97 4.797 382.3214

98 4.898 420.479

99 4.999 461.2722
distribution of Z

Normal 

distribution

1.48672E-06 0.000613891

2.45094E-06 0.000919724

3.9995E-06 0.001360209

6.46025E-06 0.001985433

1.03291E-05 0.00285968

1.63472E-05 0.004063424

2.56093E-05 0.005694647

3.97118E-05 0.007868896

6.09553E-05 0.010717347

9.26132E-05 0.014382012

0.000139285 0.019007128

0.000207351 0.024725792

0.000305546 0.031641088

0.000445674 0.039801316

0.000643469 0.049169559

0.000919619 0.059588731

0.001300942 0.070744376

0.001821704 0.082128862

0.002525035 0.093011996

0.003464389 0.102424363

0.004704958 0.109160612

0.006324913 0.11181007

0.008416337 0.108821326

0.011085658 0.0986054

0.014453386 0.079678686

0.018652949 0.050842128

0.023828414 0.011387237

0.030130931 -0.038686852

0.03771375 -0.098467268

0.046725789 -0.165959148

0.057303789 -0.237984837

0.069563239 -0.310179458

0.083588399 -0.377106983

0.099421884 -0.432511795

0.117054396 -0.469707425

0.136415346 -0.482087746

0.157365126 -0.463728123

0.179689839 -0.410027034

0.203099244 -0.318325173

0.227228529 -0.188431409

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -0.022985307

0.275855243 0.172404637

0.299326439 0.389289469

0.321498297 0.616791231

0.341807853 0.842282666

0.359712192 1.052288161

0.37471238 1.233529721

0.386376486 1.37401981

0.394360216 1.464091615

0.398423793 1.49725891

0.398443914 1.470812517

0.394419966 1.386087066

0.386474058 1.248367034

0.374844865 1.066440548

0.359875719 0.851847973

0.34199778 0.617904949

0.32170943 0.378602133

0.299553264 0.14749381

0.276092166 -0.0633161

0.25188591 -0.243998237

0.227469632 -0.38756574

0.203335281 -0.490151319

0.17991684 -0.55099227

0.157579839 -0.572152916

0.136615273 -0.558038323

0.117237788 -0.514768897

0.099587708 -0.449491561

0.083736272 -0.369700079

0.069693339 -0.282626253

0.05741676 -0.194747569

0.046822635 -0.111438064

0.037795734 -0.036770395

0.030199481 0.02653942

0.023885038 0.077065847

0.018699163 0.11451561

0.014490659 0.139510086

0.011115369 0.153340208

0.008439746 0.157721699

0.006343145 0.1545723

0.004718997 0.145825584

0.003475077 0.133288992

0.002533081 0.118547577

0.001827693 0.102910217

0.001305351 0.087391828

0.000922829 0.072723399

0.00064578 0.059381302

0.00044732 0.047627868

0.000306705 0.037556517

0.000208159 0.029136228
0.000139842 0.022251823

9.29928E-05 0.016737965

6.12113E-05 0.012406041

3.98826E-05 0.009064008

2.5722E-05 0.00652989

1.64209E-05 0.004639961

1.03767E-05 0.003252789

6.49066E-06 0.002250236

4.01874E-06 0.001536449

2.46298E-06 0.001035629

1.49417E-06 0.000689219
Absolute values

0.000613891 3.10015E-05

0.000919724 9.28922E-05

0.001360209 0.000137381

0.001985433 0.000200529

0.00285968 0.000288828

0.004063424 0.000410406

0.005694647 0.000575159

0.007868896 0.000794758

0.010717347 0.001082452

0.014382012 0.001452583

0.019007128 0.00191972

0.024725792 0.002497305

0.031641088 0.00319575

0.039801316 0.004019933

0.049169559 0.004966125

0.059588731 0.006018462

0.070744376 0.007145182

0.082128862 0.008295015

0.093011996 0.009394212

0.102424363 0.010344861

0.109160612 0.011025222

0.11181007 0.011292817

0.108821326 0.010990954

0.0986054 0.009959145

0.079678686 0.008047547

0.050842128 0.005135055

0.011387237 0.001150111

0.038686852 0.003907372

0.098467268 0.009945194

0.165959148 0.016761874

0.237984837 0.024036469

0.310179458 0.031328125

0.377106983 0.038087805

0.432511795 0.043683691

0.469707425 0.04744045

0.482087746 0.048690862

0.463728123 0.04683654

0.410027034 0.04141273

0.318325173 0.032150842

0.188431409 0.019031572

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.022985307 0.002321516

0.172404637 0.017412868

0.389289469 0.039318236

0.616791231 0.062295914

0.842282666 0.085070549

1.052288161 0.106281104

1.233529721 0.124586502

1.37401981 0.138776001

1.464091615 0.147873253

1.49725891 0.15122315

1.470812517 0.148552064

1.386087066 0.139994794

1.248367034 0.12608507

1.066440548 0.107710495

0.851847973 0.086036645

0.617904949 0.0624084

0.378602133 0.038238815

0.14749381 0.014896875

0.0633161 0.006394926

0.243998237 0.024643822

0.38756574 0.03914414

0.490151319 0.049505283

0.55099227 0.055650219

0.572152916 0.057787444

0.558038323 0.056361871

0.514768897 0.051991659

0.449491561 0.045398648

0.369700079 0.037339708

0.282626253 0.028545252

0.194747569 0.019669504

0.111438064 0.011255244

0.036770395 0.00371381

0.02653942 0.002680481

0.077065847 0.007783651

0.11451561 0.011566077

0.139510086 0.014090519

0.153340208 0.015487361

0.157721699 0.015929892

0.1545723 0.015611802

0.145825584 0.014728384

0.133288992 0.013462188

0.118547577 0.011973305

0.102910217 0.010393932

0.087391828 0.008826575

0.072723399 0.007345063

0.059381302 0.005997511

0.047627868 0.004810415

0.037556517 0.003793208

0.029136228 0.002942759
0.022251823 0.002247434

0.016737965 0.001690534

0.012406041 0.00125301

0.009064008 0.000915465

0.00652989 0.000659519

0.004639961 0.000468636

0.003252789 0.000328532

0.002250236 0.000227274

0.001536449 0.000155181

0.001035629 0.000104599

0.000689219 3.48056E-05
Cumulative normalization

3.10015E-05 1.11285E-05

0.000123894 4.44738E-05

0.000261275 9.37891E-05

0.000461804 0.000165772

0.000750631 0.000269452

0.001161037 0.000416774

0.001736196 0.000623238

0.002530955 0.00090853

0.003613407 0.001297095

0.00506599 0.001818525

0.00698571 0.002507642

0.009483015 0.003404093

0.012678765 0.004551262

0.016698698 0.005994287

0.021664823 0.007776964

0.027683285 0.009937395

0.034828467 0.012502282

0.043123482 0.015479921

0.052517694 0.018852136

0.062862554 0.022565603

0.073887776 0.026523298

0.085180593 0.03057705

0.096171547 0.034522444

0.106130693 0.038097451

0.11417824 0.040986258

0.119313295 0.042829575

0.120463406 0.043242427

0.124370778 0.044645046

0.134315972 0.048215046

0.151077846 0.054232011

0.175114314 0.062860318

0.20644244 0.074106091

0.244530245 0.087778369

0.288213936 0.103459387

0.335654386 0.120488959

0.384345248 0.137967388

0.431181789 0.154780176

0.472594519 0.169645993

0.504745362 0.181187095

0.523776934 0.188018808

( f x ) ( f b)



 




0.52609845 0.188852156

0.543511318 0.195102807

0.582829555 0.209216769

0.645125469 0.231578967

0.730196018 0.262116515

0.836477123 0.30026796

0.961063625 0.344990444

1.099839625 0.394806494

1.247712878 0.447888161

1.398936028 0.502172332

1.547488093 0.555497669

1.687482886 0.605751227

1.813567957 0.651011648

1.921278452 0.689676197

2.007315097 0.720560542

2.069723497 0.742963119

2.107962313 0.756689604

2.122859187 0.762037095

2.129254113 0.764332665

2.153897935 0.773178992

2.193042075 0.787230459

2.242547358 0.805001239

2.298197577 0.824977849

2.355985022 0.845721654

2.412346892 0.865953724

2.464338551 0.884617031

2.509737199 0.900913662

2.547076907 0.914317398

2.575622158 0.92456421

2.595291663 0.93162492

2.606546907 0.935665185

2.610260717 0.936998321

2.612941198 0.937960527

2.620724849 0.940754603

2.632290926 0.944906447

2.646381444 0.949964482

2.661868805 0.955523939

2.677798697 0.96124225

2.693410499 0.966846377

2.708138883 0.972133385

2.721601071 0.976965871

2.733574377 0.981263897

2.743968309 0.984994979

2.752794883 0.98816343

2.760139947 0.990800068

2.766137458 0.992952979

2.770947873 0.994679761

2.774741081 0.996041399

2.77768384 0.997097753
2.779931274 0.997904508

2.781621809 0.998511355

2.782874819 0.998961145

2.783790284 0.999289767

2.784449803 0.999526512

2.784918439 0.999694737

2.78524697 0.99981267

2.785474244 0.999894253

2.785629425 0.999949958

2.785734024 0.999987506

2.78576883 1

139.2201487
Raw Data

Runs 599 1 2.03E-03 2 3.77E-03 3 1.12E-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 2.31E-03

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

Skewness 0.90 0.08

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 599 1 -6.20E+00 2 -5.58E+00 3 -6.79E+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 -6.19E+00

0.367033652

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-Small-conc 12/1/2014
13.60669361

0.589934377

25.06475796

7. Calculate

1.323894336

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

3.484 z value from "z-stat"

a- Calculate 0.002459994

b- Calculate 6.05157E-06

c- Calculate 0.443446493

d-calculate 0.156022976

e- Calculate 0.004068

f- Calculate UPL 7.18E-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-Small-conc 12/1/2014
∆z z

0.101 25.06476 1.323894

-5 =f(a) 416.185

1 -4.899 378.2311

2 -4.798 342.7997

3 -4.697 309.782

4 -4.596 279.0713

5 -4.495 250.5634

6 -4.394 224.1564

7 -4.293 199.7506

8 -4.192 177.2485

9 -4.091 156.5552

10 -3.99 137.5777

11 -3.889 120.2257

12 -3.788 104.411

13 -3.687 90.04759

14 -3.586 77.05193

15 -3.485 65.34269

16 -3.384 54.84089

17 -3.283 45.46979

18 -3.182 37.15498

19 -3.081 29.82435

20 -2.98 23.40806

21 -2.879 17.83859

22 -2.778 13.0507

23 -2.677 8.981463

24 -2.576 5.570227

25 -2.475 2.758651

26 -2.374 0.490686

27 -2.273 -1.28742

28 -2.172 -2.62713

29 -2.071 -3.57759

30 -1.97 -4.18568

31 -1.869 -4.49597

32 -1.768 -4.55072

33 -1.667 -4.38991

34 -1.566 -4.05123

35 -1.465 -3.57005

36 -1.364 -2.97947

37 -1.263 -2.31028

38 -1.162 -1.59097

39 -1.061 -0.84775

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -0.10451

41 -0.859 0.617125

42 -0.758 1.297854

43 -0.657 1.920659

44 -0.556 2.470819

45 -0.455 2.935911

46 -0.354 3.305807

47 -0.253 3.572675

48 -0.152 3.730981

49 -0.051 3.777483

50 0.05 3.71124

51 0.151 3.533604

52 0.252 3.248223

53 0.353 2.861042

54 0.454 2.380302

55 0.555 1.816539

56 0.656 1.182588

57 0.757 0.493577

58 0.858 -0.23307

59 0.959 -0.97763

60 1.06 -1.71809

61 1.161 -2.43013

62 1.262 -3.08714

63 1.363 -3.66022

64 1.464 -4.11817

65 1.565 -4.42749

66 1.666 -4.55238

67 1.767 -4.45476

68 1.868 -4.09424

69 1.969 -3.42814

70 2.07 -2.41148

71 2.171 -0.99698

72 2.272 0.864917

73 2.373 3.226086

74 2.474 6.140687

75 2.575 9.665179

76 2.676 13.85832

77 2.777 18.78115

78 2.878 24.49702

79 2.979 31.07158

80 3.08 38.57276

81 3.181 47.07079

82 3.282 56.63822

83 3.383 67.34986

84 3.484 79.28284

85 3.585 92.51657

86 3.686 107.1328

87 3.787 123.2155

88 3.888 140.8509
89 3.989 160.1278

90 4.09 181.137

91 4.191 203.9716

92 4.292 228.7273

93 4.393 255.5017

94 4.494 284.3949

95 4.595 315.5094

96 4.696 348.9498

97 4.797 384.823

98 4.898 423.2384

99 4.999 464.3076
distribution of Z

Normal 

distribution

1.48672E-06 0.00061875

2.45094E-06 0.000927022

3.9995E-06 0.001371028

6.46025E-06 0.002001268

1.03291E-05 0.002882552

1.63472E-05 0.004096022

2.56093E-05 0.00574048

3.97118E-05 0.007932448

6.09553E-05 0.010804227

9.26132E-05 0.014499072

0.000139285 0.019162512

0.000207351 0.024928892

0.000305546 0.031902354

0.000445674 0.040131863

0.000643469 0.049580535

0.000919619 0.060090387

0.001300942 0.07134482

0.001821704 0.082832487

0.002525035 0.093817614

0.003464389 0.103323142

0.004704958 0.110133936

0.006324913 0.112827533

0.008416337 0.109839129

0.011085658 0.099565434

0.014453386 0.080508648

0.018652949 0.051456981

0.023828414 0.011692272

0.030130931 -0.038791199

0.03771375 -0.099078831

0.046725789 -0.167165893

0.057303789 -0.239855592

0.069563239 -0.312754163

0.083588399 -0.380387341

0.099421884 -0.436453338

0.117054396 -0.474214065

0.136415346 -0.487009838

0.157365126 -0.468864849

0.179689839 -0.41513362

0.203099244 -0.323124977

0.227228529 -0.192632427

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -0.026300091

0.275855243 0.170237143

0.299326439 0.388482034

0.321498297 0.61748847

0.341807853 0.844545238

0.359712192 1.056082845

0.37471238 1.238726673

0.386376486 1.380397662

0.394360216 1.471350295

0.398423793 1.505039262

0.398443914 1.478721087

0.394419966 1.393723865

0.386474058 1.255353764

0.374844865 1.072446741

0.359875719 0.856612727

0.34199778 0.621252443

0.32170943 0.380449756

0.299553264 0.147852541

0.276092166 -0.064348664

0.25188591 -0.246251116

0.227469632 -0.390812433

0.203335281 -0.494130352

0.17991684 -0.555428408

0.157579839 -0.576777136

0.136615273 -0.562604983

0.117237788 -0.519068967

0.099587708 -0.45336132

0.083736272 -0.373025154

0.069693339 -0.285341364

0.05741676 -0.19683266

0.046822635 -0.112911738

0.037795734 -0.037681672

0.030199481 0.026120048

0.023885038 0.077055193

0.018699163 0.114825714

0.014490659 0.140054815

0.011115369 0.154040288

0.008439746 0.158508112

0.006343145 0.155388153

0.004718997 0.146626671

0.003475077 0.134043312

0.002533081 0.119234119

0.001827693 0.103517299

0.001305351 0.087915229

0.000922829 0.073164505

0.00064578 0.059745389

0.00044732 0.047922624

0.000306705 0.037790843

0.000208159 0.029319328
0.000139842 0.022392556

9.29928E-05 0.016844434

6.12113E-05 0.012485365

3.98826E-05 0.009122237

2.5722E-05 0.006572019

1.64209E-05 0.004670016

1.03767E-05 0.003273935

6.49066E-06 0.002264914

4.01874E-06 0.001546502

2.46298E-06 0.001042426

1.49417E-06 0.000693755
Absolute values

0.00061875 3.12469E-05

0.000927022 9.36292E-05

0.001371028 0.000138474

0.002001268 0.000202128

0.002882552 0.000291138

0.004096022 0.000413698

0.00574048 0.000579788

0.007932448 0.000801177

0.010804227 0.001091227

0.014499072 0.001464406

0.019162512 0.001935414

0.024928892 0.002517818

0.031902354 0.003222138

0.040131863 0.004053318

0.049580535 0.005007634

0.060090387 0.006069129

0.07134482 0.007205827

0.082832487 0.008366081

0.093817614 0.009475579

0.103323142 0.010435637

0.110133936 0.011123528

0.112827533 0.011395581

0.109839129 0.011093752

0.099565434 0.010056109

0.080508648 0.008131373

0.051456981 0.005197155

0.011692272 0.001180919

0.038791199 0.003917911

0.099078831 0.010006962

0.167165893 0.016883755

0.239855592 0.024225415

0.312754163 0.03158817

0.380387341 0.038419121

0.436453338 0.044081787

0.474214065 0.047895621

0.487009838 0.049187994

0.468864849 0.04735535

0.41513362 0.041928496

0.323124977 0.032635623

0.192632427 0.019455875

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.026300091 0.002656309

0.170237143 0.017193951

0.388482034 0.039236685

0.61748847 0.062366335

0.844545238 0.085299069

1.056082845 0.106664367

1.238726673 0.125111394

1.380397662 0.139420164

1.471350295 0.14860638

1.505039262 0.152008965

1.478721087 0.14935083

1.393723865 0.14076611

1.255353764 0.12679073

1.072446741 0.108317121

0.856612727 0.086517885

0.621252443 0.062746497

0.380449756 0.038425425

0.147852541 0.014933107

0.064348664 0.006499215

0.246251116 0.024871363

0.390812433 0.039472056

0.494130352 0.049907166

0.555428408 0.056098269

0.576777136 0.058254491

0.562604983 0.056823103

0.519068967 0.052425966

0.45336132 0.045789493

0.373025154 0.037675541

0.285341364 0.028819478

0.19683266 0.019880099

0.112911738 0.011404086

0.037681672 0.003805849

0.026120048 0.002638125

0.077055193 0.007782575

0.114825714 0.011597397

0.140054815 0.014145536

0.154040288 0.015558069

0.158508112 0.016009319

0.155388153 0.015694203

0.146626671 0.014809294

0.134043312 0.013538375

0.119234119 0.012042646

0.103517299 0.010455247

0.087915229 0.008879438

0.073164505 0.007389615

0.059745389 0.006034284

0.047922624 0.004840185

0.037790843 0.003816875

0.029319328 0.002961252
0.022392556 0.002261648

0.016844434 0.001701288

0.012485365 0.001261022

0.009122237 0.000921346

0.006572019 0.000663774

0.004670016 0.000471672

0.003273935 0.000330667

0.002264914 0.000228756

0.001546502 0.000156197

0.001042426 0.000105285

0.000693755 3.50346E-05
Cumulative normalization

3.12469E-05 1.11452E-05

0.000124876 4.4541E-05

0.00026335 9.3932E-05

0.000465478 0.000166027

0.000756616 0.000269871

0.001170314 0.000417429

0.001750102 0.000624229

0.00255128 0.000909994

0.003642507 0.001299214

0.005106913 0.001821541

0.007042327 0.002511867

0.009560145 0.003409926

0.012782283 0.004559202

0.016835601 0.006004945

0.021843235 0.007791075

0.027912364 0.009955821

0.035118191 0.012526005

0.043484272 0.01551003

0.052959851 0.018889793

0.063395488 0.022611991

0.074519016 0.026579546

0.085914597 0.030644137

0.097008349 0.034601072

0.107064457 0.038187899

0.115195831 0.041088209

0.120392986 0.042941937

0.121573905 0.043363149

0.125491816 0.044760595

0.135498778 0.048329892

0.152382533 0.054352013

0.176607948 0.062992768

0.208196119 0.07425968

0.24661524 0.087963066

0.290697027 0.103686219

0.338592648 0.120769695

0.387780642 0.138314137

0.435135991 0.155204909

0.477064487 0.170160023

0.50970011 0.181800542

0.529155985 0.18874009

( f x ) ( f b)



 




0.531812294 0.189687546

0.549006245 0.195820309

0.588242931 0.209815304

0.650609266 0.232060215

0.735908335 0.262484805

0.842572703 0.30053

0.967684096 0.345154905

1.10710426 0.394883483

1.25571064 0.447888612

1.407719606 0.502107381

1.557070435 0.555378042

1.697836546 0.605586694

1.824627276 0.650810587

1.932944397 0.689445288

2.019462282 0.720304607

2.082208779 0.742685115

2.120634204 0.756390749

2.135567311 0.76171711

2.142066526 0.764035259

2.166937889 0.772906411

2.206409944 0.786985359

2.25631711 0.804786316

2.312415379 0.824795524

2.37066987 0.845573817

2.427492973 0.86584156

2.479918939 0.884540926

2.525708432 0.900873186

2.563383973 0.91431135

2.59220345 0.924590721

2.612083549 0.931681583

2.623487635 0.935749208

2.627293483 0.937106683

2.629931608 0.938047653

2.637714183 0.940823553

2.64931158 0.944960128

2.663457116 0.950005578

2.679015185 0.955554851

2.695024505 0.961265078

2.710718708 0.966862908

2.725528002 0.972145107

2.739066376 0.976973993

2.751109022 0.981269381

2.76156427 0.984998573

2.770443708 0.988165704

2.777833323 0.990801442

2.783867607 0.992953759

2.788707792 0.994680162

2.792524667 0.996041571

2.795485919 0.997097794
2.797747567 0.997904482

2.799448855 0.9985113

2.800709877 0.998961083

2.801631223 0.99928971

2.802294997 0.999526466

2.802766669 0.999694702

2.803097336 0.999812645

2.803326093 0.999894238

2.803482289 0.999949951

2.803587574 0.999987504

2.803622609 1

140.1275143
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

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

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 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

27

28

29

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 

b- denominator first term

a- Numerator 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 24.12133 1.363077

-5 =f(a) 396.6768

1 -4.899 360.4593

2 -4.798 326.6518

3 -4.697 295.1499

4 -4.596 265.852

5 -4.495 238.6581

6 -4.394 213.4709

7 -4.293 190.195

8 -4.192 168.7373

9 -4.091 149.0069

10 -3.99 130.9151

11 -3.889 114.3752

12 -3.788 99.30312

13 -3.687 85.61659

14 -3.586 73.23572

15 -3.485 62.08277

16 -3.384 52.08221

17 -3.283 43.16072

18 -3.182 35.24714

19 -3.081 28.27255

20 -2.98 22.1702

21 -2.879 16.87554

22 -2.778 12.32624

23 -2.677 8.46214

24 -2.576 5.225296

25 -2.475 2.559955

26 -2.374 0.412563

27 -2.273 -1.26824

28 -2.172 -2.5316

29 -2.071 -3.42449

30 -1.97 -3.99166

31 -1.869 -4.27569

32 -1.768 -4.31694

33 -1.667 -4.15357

34 -1.566 -3.82156

35 -1.465 -3.35468

36 -1.364 -2.78451

37 -1.263 -2.14041

38 -1.162 -1.44958

39 -1.061 -0.737

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -0.02543

41 -0.859 0.664518

42 -0.758 1.314469

43 -0.657 1.908233

44 -0.556 2.431816

45 -0.455 2.873425

46 -0.354 3.223463

47 -0.253 3.474533

48 -0.152 3.621435

49 -0.051 3.661166

50 0.05 3.592923

51 0.151 3.418097

52 0.252 3.140281

53 0.353 2.765264

54 0.454 2.301034

55 0.555 1.757774

56 0.656 1.147869

57 0.757 0.4859

58 0.858 -0.21136

59 0.959 -0.92492

60 1.06 -1.63362

61 1.161 -2.31408

62 1.262 -2.94074

63 1.363 -3.48582

64 1.464 -3.91936

65 1.565 -4.20919

66 1.666 -4.32095

67 1.767 -4.21808

68 1.868 -3.86183

69 1.969 -3.21124

70 2.07 -2.22315

71 2.171 -0.85222

72 2.272 0.949112

73 2.373 3.230583

74 2.474 6.044142

75 2.575 9.443932

76 2.676 13.48629

77 2.777 18.22976

78 2.878 23.73508

79 2.979 30.06518

80 3.08 37.2852

81 3.181 45.46245

82 3.282 54.66648

83 3.383 64.96901

84 3.484 76.44396

85 3.585 89.16746

86 3.686 103.2178

87 3.787 118.6756

88 3.888 135.6234
89 3.989 154.1462

90 4.09 174.3312

91 4.191 196.2676

92 4.292 220.047

93 4.393 245.7629

94 4.494 273.5115

95 4.595 303.3907

96 4.696 335.5008

97 4.797 369.9444

98 4.898 406.8261

99 4.999 446.2528
distribution of Z

Normal 

distribution

1.48672E-06 0.000589747

2.45094E-06 0.000883464

3.9995E-06 0.001306445

6.46025E-06 0.001906741

1.03291E-05 0.002746009

1.63472E-05 0.003901404

2.56093E-05 0.005466833

3.97118E-05 0.007552981

6.09553E-05 0.010285427

9.26132E-05 0.013800006

0.000139285 0.018234501

0.000207351 0.023715784

0.000305546 0.030341661

0.000445674 0.038157079

0.000643469 0.047124921

0.000919619 0.057092498

0.001300942 0.067755949

0.001821704 0.078626046

0.002525035 0.089000254

0.003464389 0.097947106

0.004704958 0.104309835

0.006324913 0.10673633

0.008416337 0.103741783

0.011085658 0.093808392

0.014453386 0.075523218

0.018652949 0.047750703

0.023828414 0.009830716

0.030130931 -0.03821313

0.03771375 -0.095476112

0.046725789 -0.160011912

0.057303789 -0.228737477

0.069563239 -0.297430965

0.083588399 -0.360845903

0.099421884 -0.412955775

0.117054396 -0.447330438

0.136415346 -0.457629943

0.157365126 -0.43818421

0.179689839 -0.384610712

0.203099244 -0.29440934

0.227228529 -0.167466401

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -0.006400143

0.275855243 0.183310648

0.299326439 0.393455457

0.321498297 0.613493584

0.341807853 0.831213719

0.359712192 1.033605832

0.37471238 1.207871534

0.386376486 1.342478021

0.394360216 1.42815002

0.398423793 1.45869584

0.398443914 1.431578111

0.394419966 1.348165671

0.386474058 1.213637172

0.374844865 1.036545118

0.359875719 0.828086167

0.34199778 0.601154969

0.32170943 0.369280442

0.299553264 0.145552829

0.276092166 -0.058353815

0.25188591 -0.23297474

0.227469632 -0.371599393

0.203335281 -0.470535074

0.17991684 -0.529088588

0.157579839 -0.549294713

0.136615273 -0.535443759

0.117237788 -0.493475585

0.099587708 -0.43031327

0.083736272 -0.353206452

0.069693339 -0.26914386

0.05741676 -0.184378871

0.046822635 -0.104093763

0.037795734 -0.032210166

0.030199481 0.028662687

0.023885038 0.077162607

0.018699163 0.113020402

0.014490659 0.1368488

0.011115369 0.149905125

0.008439746 0.153854586

0.006343145 0.150555076

0.004718997 0.141877499

0.003475077 0.129568938

0.002533081 0.115160063

0.001827693 0.09991357

0.001305351 0.084807386

0.000922829 0.070544709

0.00064578 0.057582596

0.00044732 0.046171384

0.000306705 0.036398428

0.000208159 0.028231168
0.000139842 0.021556082

9.29928E-05 0.016211547

6.12113E-05 0.012013793

3.98826E-05 0.008776044

2.5722E-05 0.00632152

1.64209E-05 0.004491301

1.03767E-05 0.003148183

6.49066E-06 0.002177621

4.01874E-06 0.001486709

2.46298E-06 0.001002002

1.49417E-06 0.000666778
Absolute values

0.000589747 2.97822E-05

0.000883464 8.92299E-05

0.001306445 0.000131951

0.001906741 0.000192581

0.002746009 0.000277347

0.003901404 0.000394042

0.005466833 0.00055215

0.007552981 0.000762851

0.010285427 0.001038828

0.013800006 0.001393801

0.018234501 0.001841685

0.023715784 0.002395294

0.030341661 0.003064508

0.038157079 0.003853865

0.047124921 0.004759617

0.057092498 0.005766342

0.067755949 0.006843351

0.078626046 0.007941231

0.089000254 0.008989026

0.097947106 0.009892658

0.104309835 0.010535293

0.10673633 0.010780369

0.103741783 0.01047792

0.093808392 0.009474648

0.075523218 0.007627845

0.047750703 0.004822821

0.009830716 0.000992902

0.03821313 0.003859526

0.095476112 0.009643087

0.160011912 0.016161203

0.228737477 0.023102485

0.297430965 0.030040527

0.360845903 0.036445436

0.412955775 0.041708533

0.447330438 0.045180374

0.457629943 0.046220624

0.43818421 0.044256605

0.384610712 0.038845682

0.29440934 0.029735343

0.167466401 0.016914107

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.006400143 0.000646414

0.183310648 0.018514375

0.393455457 0.039739001

0.613493584 0.061962852

0.831213719 0.083952586

1.033605832 0.104394189

1.207871534 0.121995025

1.342478021 0.13559028

1.42815002 0.144243152

1.45869584 0.14732828

1.431578111 0.144589389

1.348165671 0.136164733

1.213637172 0.122577354

1.036545118 0.104691057

0.828086167 0.083636703

0.601154969 0.060716652

0.369280442 0.037297325

0.145552829 0.014700836

0.058353815 0.005893735

0.23297474 0.023530449

0.371599393 0.037531539

0.470535074 0.047524042

0.529088588 0.053437947

0.549294713 0.055478766

0.535443759 0.05407982

0.493475585 0.049841034

0.43031327 0.04346164

0.353206452 0.035673852

0.26914386 0.02718353

0.184378871 0.018622266

0.104093763 0.01051347

0.032210166 0.003253227

0.028662687 0.002894931

0.077162607 0.007793423

0.113020402 0.011415061

0.1368488 0.013821729

0.149905125 0.015140418

0.153854586 0.015539313

0.150555076 0.015206063

0.141877499 0.014329627

0.129568938 0.013086463

0.115160063 0.011631166

0.09991357 0.010091271

0.084807386 0.008565546

0.070544709 0.007125016

0.057582596 0.005815842

0.046171384 0.00466331

0.036398428 0.003676241

0.028231168 0.002851348
0.021556082 0.002177164

0.016211547 0.001637366

0.012013793 0.001213393

0.008776044 0.00088638

0.00632152 0.000638474

0.004491301 0.000453621

0.003148183 0.000317966

0.002177621 0.00021994

0.001486709 0.000150158

0.001002002 0.000101202

0.000666778 3.36723E-05
Cumulative normalization

2.97822E-05 1.10416E-05

0.000119012 4.41231E-05

0.000250963 9.30431E-05

0.000443544 0.000164441

0.000720891 0.000267266

0.001114933 0.000413355

0.001667083 0.000618061

0.002429934 0.000900884

0.003468762 0.001286024

0.004862562 0.001802767

0.006704247 0.002485561

0.009099541 0.003373603

0.012164049 0.004509751

0.016017914 0.00593855

0.020777531 0.00770315

0.026543873 0.009840989

0.033387224 0.012378122

0.041328455 0.015322287

0.05031748 0.018654917

0.060210138 0.022322563

0.070745431 0.026228463

0.081525801 0.030225223

0.092003721 0.034109851

0.101478368 0.037622523

0.109106214 0.040450502

0.113929034 0.042238536

0.114921937 0.042606648

0.118781463 0.044037546

0.12842455 0.047612665

0.144585753 0.053604338

0.167688239 0.062169452

0.197728766 0.073306805

0.234174202 0.086818741

0.275882735 0.102281941

0.32106311 0.119032305

0.367283734 0.136168336

0.411540339 0.152576218

0.450386021 0.166978032

0.480121364 0.17800224

0.497035471 0.184273048

( f x ) ( f b)



 




0.497681885 0.184512702

0.516196261 0.191376801

0.555935262 0.206109807

0.617898114 0.229082179

0.701850699 0.260207118

0.806244889 0.298910664

0.928239914 0.34413962

1.063830194 0.394408938

1.208073346 0.447886258

1.355401625 0.502507372

1.499991015 0.556113058

1.636155747 0.606595351

1.758733102 0.652040202

1.863424159 0.69085381

1.947060862 0.721861638

2.007777513 0.744371988

2.045074838 0.758199757

2.059775674 0.763650007

2.065669409 0.765835076

2.089199858 0.774558855

2.126731397 0.788473457

2.174255439 0.806092723

2.227693386 0.825904535

2.283172152 0.846472969

2.337251972 0.866522752

2.387093006 0.885001029

2.430554646 0.901114183

2.466228498 0.914340059

2.493412028 0.924418196

2.512034294 0.931322295

2.522547764 0.935220103

2.525800991 0.936426219

2.528695922 0.937499498

2.536489345 0.940388865

2.547904406 0.944620933

2.561726135 0.949745259

2.576866552 0.955358482

2.592405866 0.961119594

2.607611928 0.966757154

2.621941556 0.972069781

2.635028018 0.976921512

2.646659185 0.9812337

2.656750455 0.984974981

2.665316001 0.988150608

2.672441017 0.990792166

2.678256859 0.992948357

2.682920169 0.994677252

2.68659641 0.996040197

2.689447758 0.997097318
2.691624922 0.997904489

2.693262289 0.998511533

2.694475682 0.998961392

2.695362062 0.999290012

2.696000536 0.999526723

2.696454157 0.9996949

2.696772123 0.999812785

2.696992063 0.999894326

2.697142221 0.999949996

2.697243423 0.999987516

2.697277095 1

134.7207183
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

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!

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

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

N

i

i

df n



 

      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 
#NUM!

Conclusion

interval of differences for t-values used

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

Current Confidence Level is

#NUM!

new t-statistic from tab Recalculate t-statistic
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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

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

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

Calculations Calculations



 

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!

f.1.) Calculate 

Calculations

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

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

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

e.1.) Calculate 

Calculations

 

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

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

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

e.1.) Calculate 

Calculations

 

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

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

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

e.1.) Calculate 

Calculations

 

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

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

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

e.1.) Calculate 

Calculations

 

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

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

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

e.1.) Calculate 

Calculations

 

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

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

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

e.1.) Calculate 

Calculations

 

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

    

 