Subcategory Pollutant

No. of 

Sources 

Industrywide

No. of 

Sources 

with Data

Determination of No. of 

Sources in MACT Floor

Floor tile roller kilns Hg 27 7 best performing 5 sources

PM 27 10 best performing 5 sources

D/F TEQ 27 6 best performing 5 sources

Floor tile press dryers DF/TEQ 5 4 best performing 5 sources

Floor tile spray dryers DF/TEQ 11 5 best performing 5 sources

Wall tile roller kilns Hg 14 1 best performing 5 sources

PM 14 1 best performing 5 sources

D/F TEQ 14 1 best performing 5 sources

Wall tile spray dryers D/F TEQ 5 1 best performing 5 sources

Tile glaze lines Hg 24 3 best performing 5 sources

PM 24 3 best performing 5 sources

First-fire sanitaryware tunnel kilns Hg 11 5 best performing 5 sources

PM 11 6 best performing 5 sources

D/F TEQ 11 5 best performing 5 sources

First-fire sanitaryware manual glaze application PM 14 3 best performing 5 sources

First-fire sanitaryware spray machine glaze application PM 4 3 best performing 5 sources

First-fire sanitaryware robot glaze application PM 10 4 best performing 5 sources
No. of 

Sources in 

MACT Floor 

(N)

No. Data 

Points in 

MACT Floor 

(n)

Average of 

Top 

Sources

UPL Unit of Measure Data Distribution

1 3 3.5E-05 3.9E-05 lb/ton Lognormal

1 3 0.020 0.03 lb/ton Normal

1 3 1.1 1.5 ng/dscm @ 7% O2 Lognormal

1 3 0.0070 0.012 ng/dscm @ 7% O2 Normal

1 3 0.010 0.015 ng/dscm @ 7% O2 Normal

1 6 5.0E-05 2.0E-04 lb/ton Normal

1 9 0.071 0.20 lb/ton Normal

1 6 0.065 0.17 ng/dscm @ 7% O2 Normal

1 3 0.053 0.12 ng/dscm @ 7% O2 Normal

1 3 7.4E-06 3.4E-05 lb/ton Lognormal

1 3 0.15 0.61 lb/ton Normal

1 3 6.4E-05 1.2E-04 lb/ton Normal

1 3 0.092 0.095 lb/ton Lognormal

1 3 0.23 0.37 ng/dscm @ 7% O2 Lognormal

1 3 3.3 3.8 lb/ton Lognormal

1 3 2.0 3.2 lb/ton Normal

1 3 1.3 2.2 lb/ton Lognormal
Floor Tile Floor Tile Floor Tile Floor Tile

Roller Kiln Roller Kiln Roller Kiln Press Dryer

Hg PM TEQ TEQ

lb/ton lb/ton ng/dscm @ 7% O2 ng/dscm @ 7% O2

Runs

36 77 84 PD-203

1 3.35E-05 1.85E-02 1.04E+00 7.55E-03

2 3.61E-05 2.11E-02 1.26E+00 7.57E-03

3 3.39E-05 1.99E-02 1.07E+00 5.89E-03

Average 3.45E-05 1.98E-02 1.12E+00 7.00E-03

Standard Deviation1.42E-06 1.32E-03 1.18E-01 9.66E-04

1 9.57E-25 3.34E-12 4.72E-05 8.75E-14

2 7.10E-24 2.83E-12 3.29E-04 1.07E-13

3 1.71E-25 9.27E-18 7.18E-06 1.55E-12

Testing NormalityRaw Data Raw Data Raw Data Raw Data

Sample Size 3.00 3.00 3.00 3.00

Kurtosis -2.00 -2.00 -2.00 -2.00

SE Kurtosis 2.83 2.83 2.83 2.83

Result Kurtosis normal normal normal normal

Skewness 1.62 -0.19 1.60 -1.73

SE Skewness 1.22 1.22 1.22 1.22

Result Skewness normal normal normal normal

1.321136685 0.152728268 1.303530926 1.412924941

Lognormal Normal Lognormal Normal

1 2 3 4
Floor Tile Wall Tile Wall Tile Wall Tile Wall Tile

Spray Dryer Roller Kiln Roller Kiln Roller Kiln Spray Dryer

TEQ Hg PM TEQ TEQ

ng/dscm @ 7% O2 lb/ton lb/ton ng/dscm @ 7% O2 ng/dscm @ 7% O2

SD-104 70 70 [n>3] 70 SD-105

9.11E-03 1.14E-06 3.06E-02 5.90E-02

1.03E-02 9.87E-07 2.96E-02 5.95E-02

1.09E-02 1.12E-06 1.07E-02 3.96E-02

1.01E-02 1.08E-06 #DIV/0! 2.36E-02 5.27E-02

8.98E-04 8.43E-08 #DIV/0! 1.12E-02 1.14E-02

9.25E-13 1.01E-29 #DIV/0! 2.35E-09 1.59E-09

1.61E-15 8.82E-29 #DIV/0! 1.26E-09 2.14E-09

3.71E-13 2.71E-30 #DIV/0! 2.79E-08 2.96E-08

Raw Data Raw Data Raw Data Raw Data Raw Data

3.00 3.00 0.00 3.00 3.00

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

2.83 2.83 #DIV/0! 2.83 2.83

normal normal #DIV/0! normal normal

-0.95 -1.66 #DIV/0! -1.72 -1.73

1.22 1.22 0.00 1.22 1.22

normal normal #DIV/0! normal normal

0.778657515 1.358786394 #DIV/0! 1.40154496 1.411216784

Normal Normal #DIV/0! Normal Normal

5 6 7 8 9
Tile Tile Sanitaryware Sanitaryware Sanitaryware

Glazing Glazing Tunnel Kiln Tunnel Kiln Tunnel Kiln

Hg PM Hg PM TEQ

lb/ton lb/ton lb/ton lb/ton ng/dscm @ 7% O2

GL-3, 4, 5 GL-1 TK #2 POC + EE + 

FE stack

TK #6 POC stack TK #10 POC + RCE 

stacks

2.87E-06 1.52E-01 7.38E-05 9.21E-02 2.87E-01

4.08E-06 2.33E-01 6.53E-05 9.33E-02 2.19E-01

1.52E-05 7.21E-02 5.41E-05 9.15E-02 1.99E-01

7.37E-06 1.52E-01 6.44E-05 9.23E-02 2.35E-01

6.79E-06 8.04E-02 9.92E-06 8.96E-04 4.62E-02

4.12E-22 1.36E-13 7.98E-21 2.39E-15 7.35E-06

1.19E-22 4.24E-05 5.88E-25 9.44E-13 6.63E-08

3.71E-21 4.11E-05 1.14E-20 3.42E-13 1.68E-06

Raw Data Raw Data Raw Data Raw Data Raw Data

3.00 3.00 3.00 3.00 3.00

-2.00 -2.00 -2.00 -2.00 -2.00

2.83 2.83 2.83 2.83 2.83

normal normal normal normal normal

1.67 0.03 -0.39 1.04 1.37

1.22 1.22 1.22 1.22 1.22

normal normal normal normal normal

1.364259977 0.02777046 0.321944145 0.851649774 1.122364916

Lognormal Normal Normal Lognormal Lognormal

10 11 12 13 14
Sanitaryware Sanitaryware Sanitaryware

Manual Glaze 

Application

Spray Machine 

Glaze Application

Robot Glaze 

Application

PM PM PM

lb/ton lb/ton lb/ton

SO3/SO4 EO14 (EPN014-

Bowl SM 1)

SR1 (EPN 104-1P) Example 18

3.52E+00 2.14E+00 1.68E+00

3.17E+00 1.72E+00 1.17E+00

3.29E+00 2.01E+00 1.10E+00

3.33E+00 1.95E+00 1.32E+00 #DIV/0!

1.75E-01 2.16E-01 3.19E-01 #DIV/0!

1.31E-03 1.18E-03 1.78E-02 #DIV/0!

5.69E-04 3.18E-03 4.38E-04 #DIV/0!

1.61E-06 7.32E-06 2.37E-03 #DIV/0!

Raw Data Raw Data Raw Data Raw Data

3.00 3.00 3.00 0.00

-2.00 -2.00 -2.00 #DIV/0!

2.83 2.83 2.83 #DIV/0!

normal normal normal #DIV/0!

0.88 -1.02 1.62 #DIV/0!

1.22 1.22 1.22 0.00

normal normal normal #DIV/0!

0.716530195 0.833009191 1.324485629 #DIV/0!

Lognormal

15 16 17 18
Example 1 Example 2 Example 3 Example 4 Example 5

-1.03E+01 -3.99E+00 3.68E-02 -4.89E+00 -4.70E+00

-1.02E+01 -3.86E+00 2.27E-01 -4.88E+00 -4.58E+00

-1.03E+01 -3.92E+00 6.63E-02 -5.14E+00 -4.52E+00

-1.03E+01 -3.92E+00 1.10E-01 -4.97E+00 -4.60E+00

4.08E-02 6.71E-02 1.02E-01 1.44E-01 9.04E-02

6.61E-07 2.29E-05 2.89E-05 4.42E-05 9.81E-05

4.78E-06 1.77E-05 1.88E-04 5.27E-05 2.49E-07

1.11E-07 3.36E-10 3.66E-06 7.73E-04 3.55E-05

Log Data Log Data Log Data Log Data Log Data

3.00 3.00 3.00 3.00 3.00

-2.00 -2.00 -2.00 -2.00 -2.00

2.83 2.83 2.83 2.83 2.83

normal normal normal normal normal

1.61 -0.29 1.57 -1.73 -1.04

1.22 1.22 1.22 1.22 1.22

normal normal normal normal normal

1.314519853 0.233288022 1.28273836 1.413204771 0.851958742
Example 6 Example 7 Example 8 Example 9 Example 10

-1.37E+01 #NUM! -3.49E+00 -2.83E+00 -1.28E+01

-1.38E+01 #NUM! -3.52E+00 -2.82E+00 -1.24E+01

-1.37E+01 #NUM! -4.54E+00 -3.23E+00 -1.11E+01

-1.37E+01 #NUM! -3.85E+00 -2.96E+00 -1.21E+01

7.97E-02 #NUM! 5.97E-01 2.33E-01 8.78E-01

7.76E-06 #NUM! 1.70E-02 2.88E-04 2.04E-01

7.05E-05 #NUM! 1.16E-02 3.69E-04 1.07E-02

2.28E-06 #NUM! 2.25E-01 5.23E-03 9.74E-01

Log Data Log Data Log Data Log Data Log Data

3.00 0.00 3.00 3.00 3.00

-2.00 #NUM! -2.00 -2.00 -2.00

2.83 #DIV/0! 2.83 2.83 2.83

normal #NUM! normal normal normal

-1.67 #NUM! -1.73 -1.73 1.43

1.22 0.00 1.22 1.22 1.22

normal #NUM! normal normal normal

1.365733998 #NUM! 1.409280588 1.412183555 1.165199367
Example 11 Example 12 Example 13 Example 14 Example 15

-1.89E+00 -9.51E+00 -2.39E+00 -1.25E+00 1.26E+00

-1.46E+00 -9.64E+00 -2.37E+00 -1.52E+00 1.16E+00

-2.63E+00 -9.83E+00 -2.39E+00 -1.62E+00 1.19E+00

-1.99E+00 -9.66E+00 -2.38E+00 -1.46E+00 1.20E+00

5.93E-01 1.57E-01 9.69E-03 1.90E-01 5.22E-02

1.19E-04 4.43E-04 3.14E-11 2.04E-03 1.01E-05

8.13E-02 2.18E-07 1.29E-08 1.17E-05 4.71E-06

1.66E-01 7.72E-04 4.72E-09 5.66E-04 9.37E-09

Log Data Log Data Log Data Log Data Log Data

3.00 3.00 3.00 3.00 3.00

-2.00 -2.00 -2.00 -2.00 -2.00

2.83 2.83 2.83 2.83 2.83

normal normal normal normal normal

-0.77 -0.61 1.03 1.25 0.82

1.22 1.22 1.22 1.22 1.22

normal normal normal normal normal

0.627460159 0.496426548 0.844039835 1.021912981 0.667740396
Example 16 Example 17 Example 18

7.60E-01 5.21E-01 #NUM!

5.40E-01 1.61E-01 #NUM!

6.96E-01 9.41E-02 #NUM!

6.65E-01 2.59E-01 #NUM!

1.13E-01 2.30E-01 #NUM!

8.11E-05 4.76E-03 #NUM!

2.47E-04 9.19E-05 #NUM!

8.65E-07 7.36E-04 #NUM!

Log Data Log Data Log Data

3.00 3.00 0.00

-2.00 -2.00 #NUM!

2.83 2.83 #DIV/0!

normal normal #NUM!

-1.12 1.57 #NUM!

1.22 1.22 0.00

normal normal #NUM!

0.917542655 1.281129305 #NUM!
Raw Data: Floor Tile - Roller Kiln - Hg lb/ton

Runs 36 1 3.35E-05 2 3.61E-05 3 3.39E-05 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 3.45E-05

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

Skewness 1.62 1.61

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 36 1 -1.03E+01 2 -1.02E+01 3 -1.03E+01 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 -1.03E+01

0.001666692

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





  

    

09000064819365f6 FT-RK-Hg 12/1/2014
3.026731076

8.34749E-06

362593.6779

7. Calculate

0.070780028

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

3.484 z value from "lognormal distribution 99% "

a- Calculate 3.4526E-05

b- Calculate 1.19204E-09

c- Calculate 0.001668082

d-calculate 0.000556259

e- Calculate 3.45E-06

f- Calculate UPL 3.85E-05

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 "lognormal distribution 99% "

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





  

    

  

09000064819365f6 FT-RK-Hg 12/1/2014
∆z z

0.101 362593.7 0.07078

-5 =f(a) 7221597

1 -4.899 6572083

2 -4.798 5965109

3 -4.697 5398863

4 -4.596 4871571

5 -4.495 4381499

6 -4.394 3926947

7 -4.293 3506256

8 -4.192 3117802

9 -4.091 2760002

10 -3.99 2431307

11 -3.889 2130208

12 -3.788 1855234

13 -3.687 1604950

14 -3.586 1377961

15 -3.485 1172908

16 -3.384 988469.8

17 -3.283 823363.6

18 -3.182 676344

19 -3.081 546203.6

20 -2.98 431772.3

21 -2.879 331918.2

22 -2.778 245546.8

23 -2.677 171601.4

24 -2.576 109063.1

25 -2.475 56950.79

26 -2.374 14320.88

27 -2.273 -19732.3

28 -2.172 -46076.8

29 -2.071 -65542.9

30 -1.97 -78923.2

31 -1.869 -86972.3

32 -1.768 -90407.5

33 -1.667 -89908

34 -1.566 -86115.6

35 -1.465 -79634

36 -1.364 -71029.4

37 -1.263 -60830.4

38 -1.162 -49527.4

39 -1.061 -37573.6

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -25384.2

41 -0.859 -13336.5

42 -0.758 -1770.54

43 -0.657 9011.804

44 -0.556 18746.19

45 -0.455 27206.03

46 -0.354 34202.5

47 -0.253 39584.48

48 -0.152 43238.58

49 -0.051 45089.16

50 0.05 45098.31

51 0.151 43265.83

52 0.252 39629.26

53 0.353 34263.89

54 0.454 27282.73

55 0.555 18836.5

56 0.656 9113.686

57 0.757 -1659.52

58 0.858 -13219.2

59 0.959 -25263.6

60 1.06 -37453.5

61 1.161 -49411.6

62 1.262 -60723.1

63 1.363 -70935.5

64 1.464 -79558.3

65 1.565 -86063.6

66 1.666 -89885.6

67 1.767 -90420.7

68 1.868 -87027.8

69 1.969 -79027.7

70 2.07 -65703.9

71 2.171 -46301.9

72 2.272 -20029.4

73 2.373 13943.29

74 2.474 56483.94

75 2.575 108497.9

76 2.676 170928.2

77 2.777 244755.8

78 2.878 330999.1

79 2.979 430714.5

80 3.08 544995.9

81 3.181 674975.3

82 3.282 821821.9

83 3.383 986743.1

84 3.484 1170984

85 3.585 1375827

86 3.686 1602592

87 3.787 1852638

88 3.888 2127361
89 3.989 2428194

90 4.09 2756608

91 4.191 3114113

92 4.292 3502256

93 4.393 3922620

94 4.494 4376829

95 4.595 4866541

96 4.696 5393456

97 4.797 5959308

98 4.898 6565871

99 4.999 7214955
distribution of Z

Normal 

distribution

1.48672E-06 10.73648974

2.45094E-06 16.1077804

3.9995E-06 23.85746918

6.46025E-06 34.87798606

1.03291E-05 50.31889059

1.63472E-05 71.62544432

2.56093E-05 100.5662095

3.97118E-05 139.2396071

6.09553E-05 190.0464178

9.26132E-05 255.6125344

0.000139285 338.644485

0.000207351 441.7000994

0.000305546 566.8591241

0.000445674 715.2845308

0.000643469 886.6755533

0.000919619 1078.628602

0.001300942 1285.942062

0.001821704 1499.924606

0.002525035 1707.792031

0.003464389 1892.261533

0.004704958 2031.470449

0.006324913 2099.353634

0.008416337 2066.604499

0.011085658 1902.314506

0.014453386 1576.331661

0.018652949 1062.300079

0.023828414 341.2437482

0.030130931 -594.5535287

0.03771375 -1737.730562

0.046725789 -3062.545433

0.057303789 -4522.595769

0.069563239 -6050.075597

0.083588399 -7557.017202

0.099421884 -8938.826343

0.117054396 -10080.20668

0.136415346 -10863.29884

0.157365126 -11177.55792

0.179689839 -10930.59789

0.203099244 -10058.98376

0.227228529 -8537.797812

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -6387.778425

0.275855243 -3678.953708

0.299326439 -529.9691644

0.321498297 2897.279649

0.341807853 6407.593656

0.359712192 9786.341653

0.37471238 12816.10021

0.386376486 15294.51087

0.394360216 17051.5761

0.398423793 17964.59582

0.398443914 17969.14645

0.394419966 17064.90576

0.386474058 15315.68213

0.374844865 12843.64488

0.359875719 9818.391247

0.34199778 6442.04169

0.32170943 2931.95862

0.299553264 -497.1141621

0.276092166 -3649.71155

0.25188591 -6363.552364

0.227469632 -8519.528826

0.203335281 -10047.12068

0.17991684 -10925.11289

0.157579839 -11178.00125

0.136615273 -10868.88333

0.117237788 -10089.91049

0.099587708 -8951.501856

0.083736272 -7571.495186

0.069693339 -6065.255472

0.05741676 -4537.515858

0.046822635 -3076.42995

0.037795734 -1750.013

0.030199481 -604.878573

0.023885038 333.0359363

0.018699163 1056.202315

0.014490659 1572.205774

0.011115369 1899.930118

0.008439746 2065.676485

0.006343145 2099.575251

0.004718997 2032.540229

0.003475077 1893.903092

0.002533081 1709.766853

0.001827693 1502.038506

0.001305351 1288.046452

0.000922829 1080.61787

0.00064578 888.4818841

0.00044732 716.8713341

0.000306705 568.2140508

0.000208159 442.8283549
0.000139842 339.5629115

9.29928E-05 256.3447138

6.12113E-05 190.6188728

3.98826E-05 139.6790522

2.5722E-05 100.8977243

1.64209E-05 71.87140348

1.03767E-05 50.49847029

6.49066E-06 35.00708314

4.01874E-06 23.94888848

2.46298E-06 16.17157618

1.49417E-06 10.78037585
Absolute values

10.73648974 0.542192732

16.1077804 1.62688582

23.85746918 2.409604387

34.87798606 3.522676592

50.31889059 5.08220795

71.62544432 7.234169876

100.5662095 10.15718715

139.2396071 14.06320032

190.0464178 19.19468819

255.6125344 25.81686597

338.644485 34.20309298

441.7000994 44.61171004

566.8591241 57.25277153

715.2845308 72.24373761

886.6755533 89.55423088

1078.628602 108.9414888

1285.942062 129.8801482

1499.924606 151.4923852

1707.792031 172.4869951

1892.261533 191.1184148

2031.470449 205.1785153

2099.353634 212.034717

2066.604499 208.7270544

1902.314506 192.1337651

1576.331661 159.2094977

1062.300079 107.2923079

341.2437482 34.46561857

594.5535287 60.0499064

1737.730562 175.5107867

3062.545433 309.3170888

4522.595769 456.7821726

6050.075597 611.0576353

7557.017202 763.2587374

8938.826343 902.8214606

10080.20668 1018.100875

10863.29884 1097.193183

11177.55792 1128.93335

10930.59789 1103.990387

10058.98376 1015.95736

8537.797812 862.317579

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
6387.778425 645.1656209

3678.953708 371.5743245

529.9691644 53.5268856

2897.279649 292.6252446

6407.593656 647.1669592

9786.341653 988.420507

12816.10021 1294.426122

15294.51087 1544.745598

17051.5761 1722.209186

17964.59582 1814.424178

17969.14645 1814.883791

17064.90576 1723.555482

15315.68213 1546.883895

12843.64488 1297.208133

9818.391247 991.6575159

6442.04169 650.6462107

2931.95862 296.1278206

497.1141621 50.20853037

3649.71155 368.6208666

6363.552364 642.7187888

8519.528826 860.4724115

10047.12068 1014.759189

10925.11289 1103.436402

11178.00125 1128.978126

10868.88333 1097.757216

10089.91049 1019.080959

8951.501856 904.1016875

7571.495186 764.7210138

6065.255472 612.5908026

4537.515858 458.2891016

3076.42995 310.719425

1750.013 176.751313

604.878573 61.09273587

333.0359363 33.63662956

1056.202315 106.6764338

1572.205774 158.7927832

1899.930118 191.8929419

2065.676485 208.633325

2099.575251 212.0571004

2032.540229 205.2865631

1893.903092 191.2842123

1709.766853 172.6864521

1502.038506 151.7058891

1288.046452 130.0926916

1080.61787 109.1424049

888.4818841 89.73667029

716.8713341 72.40400475

568.2140508 57.38961913

442.8283549 44.72566384
339.5629115 34.29585406

256.3447138 25.8908161

190.6188728 19.25250616

139.6790522 14.10758427

100.8977243 10.19067016

71.87140348 7.259011752

50.49847029 5.1003455

35.00708314 3.535715397

23.94888848 2.418837736

16.17157618 1.633329195

10.78037585 0.544408981
Cumulative normalization

0.542192732 1.28191E-05

2.169078552 5.12837E-05

4.57868294 0.000108254

8.101359531 0.000191541

13.18356748 0.0003117

20.41773736 0.000482738

30.57492451 0.000722885

44.63812483 0.001055382

63.83281303 0.001509204

89.649679 0.002119594

123.852772 0.00292826

168.464482 0.003983018

225.7172536 0.00533665

297.9609912 0.007044714

387.5152221 0.009162051

496.4567108 0.011737763

626.3368591 0.014808528

777.8292443 0.018390274

950.3162394 0.022468397

1141.434654 0.026987024

1346.61317 0.031838074

1558.647887 0.036851227

1767.374941 0.041786176

1959.508706 0.046328809

2118.718204 0.050093011

2226.010512 0.052629731

2260.47613 0.053444604

2320.526037 0.054864368

2496.036823 0.059013982

2805.353912 0.066327189

3262.136085 0.077126924

3873.19372 0.091574204

4636.452458 0.109619987

5539.273918 0.130965461

6557.374793 0.155036495

7654.567976 0.180977515

8783.501326 0.207668969

9887.491713 0.233770695

10903.44907 0.25779105

11765.76665 0.278178888

( f x ) ( f b)



 




12410.93227 0.293432586

12782.5066 0.302217745

12836.03348 0.303483285

13128.65873 0.310401845

13775.82569 0.325702862

14764.24619 0.34907216

16058.67232 0.379676373

17603.41791 0.416198907

19325.6271 0.456917225

21140.05128 0.49981579

22954.93507 0.542725221

24678.49055 0.58347537

26225.37445 0.62004846

27522.58258 0.650718447

28514.24009 0.674164279

29164.8863 0.689547555

29461.01413 0.696548927

29511.22266 0.697736011

29879.84352 0.706451342

30522.56231 0.721647189

31383.03472 0.741991402

32397.79391 0.765983428

33501.23031 0.792072056

34630.20844 0.818764569

35727.96565 0.844718924

36747.04661 0.86881313

37651.1483 0.890188873

38415.86932 0.908269228

39028.46012 0.922752758

39486.74922 0.933588121

39797.46864 0.940934483

39974.21996 0.945113427

40035.31269 0.946557847

40068.94932 0.947353121

40175.62576 0.949875279

40334.41854 0.953629629

40526.31148 0.958166568

40734.94481 0.963099301

40947.00191 0.968112983

41152.28847 0.972966588

41343.57268 0.977489135

41516.25913 0.981571973

41667.96502 0.985158767

41798.05772 0.988234558

41907.20012 0.99081502

41996.93679 0.99293667

42069.3408 0.994648524

42126.73041 0.996005391

42171.45608 0.997062843
42205.75193 0.997873702

42231.64275 0.998485841

42250.89525 0.998941029

42265.00284 0.999274576

42275.19351 0.999515514

42282.45252 0.99968714

42287.55287 0.999807728

42291.08858 0.999891323

42293.50742 0.999948512

42295.14075 0.999987128

42295.68516 1

2135744.65
Raw Data: Floor Tile - Roller Kiln - PM lb/ton

Runs 77

1 1.85E-02

2 2.11E-02

3 1.99E-02

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -0.19

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

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

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 1.98E-02 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1.98E-02

Mean =

Pooled Variance = 1.76E-06

m= number future runs = 3

Term1 6.67E-01

Term2 1.17E-06

Squared root Term2 1.08E-03

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

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

UPL POOLED VARIANCE = 2.73E-02

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

 

     

09000064819365f6 FT-RK-PM 12/1/2014
Raw Data: Floor Tile - Roller Kiln - TEQ ng/dscm @ 7% O2

Runs 84 1 1.04E+00 2 1.26E+00 3 1.07E+00 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 3.00 3.00

Kurtosis #DIV/0! #DIV/0!

SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 1.12E+00

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

Skewness 1.60 1.57

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 84 1 3.68E-02 2 2.27E-01 3 6.63E-02 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 1.10E-01

0.010495869

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





  

    

09000064819365f6 FT-RK-TEQ 12/1/2014
3.170495265

0.00033398

9495.071416

7. Calculate

0.178539847

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

3.484 z value from "lognormal distribution 99% "

a- Calculate 1.122251385

b- Calculate 1.259448172

c- Calculate 0.010551144

d-calculate 0.003526164

e- Calculate 0.282551

f- Calculate UPL 1.45E+00

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 "lognormal distribution 99% "

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





  

    

  

09000064819365f6 FT-RK-TEQ 12/1/2014
∆z z

0.101 9495.071 0.17854

-5 =f(a) 189048.1

1 -4.899 172045.1

2 -4.798 156155.6

3 -4.697 141332.3

4 -4.596 127528.8

5 -4.495 114699.7

6 -4.394 102800.4

7 -4.293 91787.45

8 -4.192 81618.48

9 -4.091 72251.95

10 -3.99 63647.36

11 -3.889 55765.18

12 -3.788 48566.9

13 -3.687 42014.98

14 -3.586 36072.86

15 -3.485 30704.98

16 -3.384 25876.77

17 -3.283 21554.63

18 -3.182 17705.96

19 -3.081 14299.16

20 -2.98 11303.6

21 -2.879 8689.638

22 -2.778 6428.628

23 -2.677 4492.909

24 -2.576 2855.805

25 -2.475 1491.631

26 -2.374 375.6882

27 -2.273 -515.734

28 -2.172 -1205.36

29 -2.071 -1714.92

30 -1.97 -2065.17

31 -1.869 -2275.86

32 -1.768 -2365.76

33 -1.667 -2352.67

34 -1.566 -2253.38

35 -1.465 -2083.69

36 -1.364 -1858.42

37 -1.263 -1591.42

38 -1.162 -1295.52

39 -1.061 -982.588

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -663.486

41 -0.859 -348.099

42 -0.758 -45.3221

43 -0.657 236.9388

44 -0.556 491.7645

45 -0.455 713.2237

46 -0.354 896.3731

47 -0.253 1037.257

48 -0.152 1132.907

49 -0.051 1181.344

50 0.05 1181.574

51 0.151 1133.594

52 0.252 1038.386

53 0.353 897.9205

54 0.454 715.1567

55 0.555 494.0407

56 0.656 239.5066

57 0.757 -42.5239

58 0.858 -345.141

59 0.959 -660.448

60 1.06 -979.56

61 1.161 -1292.6

62 1.262 -1588.72

63 1.363 -1856.05

64 1.464 -2081.78

65 1.565 -2252.07

66 1.666 -2352.11

67 1.767 -2366.1

68 1.868 -2277.25

69 1.969 -2067.8

70 2.07 -1718.98

71 2.171 -1211.03

72 2.272 -523.222

73 2.373 366.1717

74 2.474 1479.865

75 2.575 2841.558

76 2.676 4475.942

77 2.777 6408.692

78 2.878 8666.473

79 2.979 11276.94

80 3.08 14268.73

81 3.181 17671.46

82 3.282 21515.77

83 3.383 25833.25

84 3.484 30656.48

85 3.585 36019.06

86 3.686 41955.54

87 3.787 48501.48

88 3.888 55693.43
89 3.989 63568.91

90 4.09 72166.43

91 4.191 81525.51

92 4.292 91686.64

93 4.393 102691.3

94 4.494 114582

95 4.595 127402.1

96 4.696 141196.1

97 4.797 156009.4

98 4.898 171888.5

99 4.999 188880.7
distribution of Z

Normal 

distribution

1.48672E-06 0.281061573

2.45094E-06 0.421672107

3.9995E-06 0.624544769

6.46025E-06 0.913041785

1.03291E-05 1.317256601

1.63472E-05 1.875023676

2.56093E-05 2.632641018

3.97118E-05 3.645041944

6.09553E-05 4.97507532

9.26132E-05 6.691483567

0.000139285 8.865120769

0.000207351 11.56294953

0.000305546 14.8394188

0.000445674 18.72498041

0.000643469 23.2117715

0.000919619 28.23688558

0.001300942 33.66417483

0.001821704 39.26614469

0.002525035 44.70816606

0.003464389 49.53785638

0.004704958 53.18295458

0.006324913 54.96120246

0.008416337 54.10550473

0.011085658 49.80684942

0.014453386 41.27604955

0.018652949 27.82331266

0.023828414 8.952055043

0.030130931 -15.5395327

0.03771375 -45.45855476

0.046725789 -80.13094333

0.057303789 -118.3417902

0.069563239 -158.3159057

0.083588399 -197.750342

0.099421884 -233.906954

0.117054396 -263.7675041

0.136415346 -284.2467408

0.157365126 -292.4509854

0.179689839 -285.9620217

0.203099244 -263.1196187

0.227228529 -223.2719498

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -166.9624628

0.275855243 -96.0249832

0.299326439 -13.5660893

0.321498297 76.17542357

0.341807853 168.0889568

0.359712192 256.5552684

0.37471238 335.8821059

0.386376486 400.7717015

0.394360216 446.7735695

0.398423793 470.6755305

0.398443914 470.7911266

0.394419966 447.1121017

0.386474058 401.3091544

0.374844865 336.5808792

0.359875719 257.3675337

0.34199778 168.9608329

0.32170943 77.05152667

0.299553264 -12.73818613

0.276092166 -95.29081438

0.25188591 -166.35761

0.227469632 -222.8200906

0.203335281 -262.8317819

0.17991684 -285.8370103

0.157579839 -292.4768089

0.136615273 -284.4028098

0.117237788 -264.0272882

0.099587708 -234.2407938

0.083736272 -198.1281292

0.069693339 -158.7094005

0.05741676 -118.7264174

0.046822635 -80.48698388

0.037795734 -45.77172754

0.030199481 -15.80102209

0.023885038 8.746024642

0.018699163 27.67222808

0.014490659 41.17605202

0.011115369 49.75174434

0.008439746 54.08773332

0.006343145 54.97269775

0.004718997 53.21583324

0.003475077 49.58492546

0.002533081 44.76324671

0.001827693 39.32423185

0.001305351 33.72146266

0.000922829 28.29069302

0.00064578 23.26040237

0.00044732 18.76754876

0.000306705 14.87566457

0.000208159 11.59306301
0.000139842 8.889587867

9.29928E-05 6.710958295

6.12113E-05 4.990281253

3.98826E-05 3.65670131

2.5722E-05 2.641427952

1.64209E-05 1.881537182

1.03767E-05 1.322008539

6.49066E-06 0.916455513

4.01874E-06 0.62696068

2.46298E-06 0.423357077

1.49417E-06 0.282220104
Absolute values

0.281061573 0.014193609

0.421672107 0.042588883

0.624544769 0.063079022

0.913041785 0.09221722

1.317256601 0.133042917

1.875023676 0.189377391

2.632641018 0.265896743

3.645041944 0.368149236

4.97507532 0.502482607

6.691483567 0.67583984

8.865120769 0.895377198

11.56294953 1.167857903

14.8394188 1.498781299

18.72498041 1.891223022

23.2117715 2.344388922

28.23688558 2.851925444

33.66417483 3.400081658

39.26614469 3.965880614

44.70816606 4.515524773

49.53785638 5.003323494

53.18295458 5.371478413

54.96120246 5.551081449

54.10550473 5.464655978

49.80684942 5.030491792

41.27604955 4.168881004

27.82331266 2.810154579

8.952055043 0.904157559

15.5395327 1.569492803

45.45855476 4.591314031

80.13094333 8.093225276

118.3417902 11.95252081

158.3159057 15.98990648

197.750342 19.97278454

233.906954 23.62460235

263.7675041 26.64051791

284.2467408 28.70892082

292.4509854 29.53754952

285.9620217 28.88216419

263.1196187 26.57508149

223.2719498 22.55046693

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
166.9624628 16.86320875

96.0249832 9.698523303

13.5660893 1.370175019

76.17542357 7.69371778

168.0889568 16.97698463

256.5552684 25.91208211

335.8821059 33.9240927

400.7717015 40.47794185

446.7735695 45.12413052

470.6755305 47.53822858

470.7911266 47.54990378

447.1121017 45.15832227

401.3091544 40.53222459

336.5808792 33.99466879

257.3675337 25.9941209

168.9608329 17.06504412

77.05152667 7.782204194

12.73818613 1.286556799

95.29081438 9.624372253

166.35761 16.80211861

222.8200906 22.50482915

262.8317819 26.54600997

285.8370103 28.86953804

292.4768089 29.5401577

284.4028098 28.72468379

264.0272882 26.66675611

234.2407938 23.65832018

198.1281292 20.01094105

158.7094005 16.02964946

118.7264174 11.99136816

80.48698388 8.129185372

45.77172754 4.622944482

15.80102209 1.595903231

8.746024642 0.883348489

27.67222808 2.794895037

41.17605202 4.158781254

49.75174434 5.024926179

54.08773332 5.462861065

54.97269775 5.552242473

53.21583324 5.374799157

49.58492546 5.008077472

44.76324671 4.521087918

39.32423185 3.971747417

33.72146266 3.405867729

28.29069302 2.857359995

23.26040237 2.349300639

18.76754876 1.895522425

14.87566457 1.502442121

11.59306301 1.170899364
8.889587867 0.897848375

6.710958295 0.677806788

4.990281253 0.504018407

3.65670131 0.369326832

2.641427952 0.266784223

1.881537182 0.190035255

1.322008539 0.133522862

0.916455513 0.092562007

0.62696068 0.063323029

0.423357077 0.042759065

0.282220104 0.014252115
Cumulative normalization

0.014193609 1.2818E-05

0.056782492 5.12794E-05

0.119861514 0.000108245

0.212078734 0.000191525

0.345121651 0.000311674

0.534499042 0.000482698

0.800395785 0.000722825

1.168545021 0.001055295

1.671027629 0.001509079

2.346867469 0.00211942

3.242244667 0.002928021

4.410102569 0.003982696

5.908883868 0.005336222

7.80010689 0.007044156

10.14449581 0.009161338

12.99642126 0.011736868

16.39650291 0.014807429

20.36238353 0.018388955

24.8779083 0.022466856

29.8812318 0.02698528

35.25271021 0.031836179

40.80379166 0.036849275

46.26844763 0.041784322

51.29893943 0.046327282

55.46782043 0.050092134

58.27797501 0.052629941

59.18213257 0.053446472

60.75162537 0.054863857

65.3429394 0.059010201

73.43616468 0.066319068

85.38868549 0.077113206

101.378592 0.091553444

121.3513765 0.109590558

144.9759789 0.130925572

171.6164968 0.154984213

200.3254176 0.180910797

229.8629671 0.207585703

258.7451313 0.23366874

285.3202128 0.257668287

307.8706797 0.278033266

( f x ) ( f b)



 




324.7338885 0.29326217

334.4324118 0.302020757

335.8025868 0.303258141

343.4963046 0.310206218

360.4732892 0.325537872

386.3853713 0.348938674

420.309464 0.379574999

460.7874059 0.416130004

505.9115364 0.456880911

553.449765 0.499811953

600.9996688 0.542753538

646.157991 0.583535323

686.6902156 0.62013935

720.6848844 0.650839411

746.6790053 0.674314301

763.7440494 0.68972548

771.5262536 0.696753468

772.8128104 0.697915338

782.4371827 0.70660696

799.2393013 0.721780695

821.7441304 0.742104459

848.2901404 0.766077751

877.1596785 0.792149386

906.6998362 0.818826647

935.4245199 0.844767467

962.0912761 0.868849803

985.7495962 0.890215267

1005.760537 0.90828684

1021.790187 0.922762969

1033.781555 0.933592189

1041.91074 0.940933531

1046.533685 0.945108441

1048.129588 0.946549676

1049.012936 0.947347415

1051.807832 0.949871442

1055.966613 0.953627173

1060.991539 0.958165106

1066.4544 0.963098532

1072.006642 0.968112676

1077.381442 0.972966575

1082.389519 0.977489292

1086.910607 0.981572217

1090.882354 0.985159041

1094.288222 0.988234828

1097.145582 0.990815265

1099.494883 0.992936883

1101.390405 0.9946487

1102.892847 0.996005532

1104.063747 0.997062953
1104.961595 0.997873786

1105.639402 0.998485903

1106.14342 0.998941074

1106.512747 0.999274608

1106.779531 0.999515536

1106.969567 0.999687154

1107.103089 0.999807737

1107.195651 0.999891328

1107.258974 0.999948514

1107.301734 0.999987129

1107.315986 1

55914.28786
Raw Data: Floor Tile - Press Dryer - TEQ ng/dscm @ 7% O2

Runs PD-203

1 7.55E-03

2 7.57E-03

3 5.89E-03

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -1.73

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

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

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 7.00E-03 #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!

7.00E-03

Mean =

Pooled Variance = 9.33E-07

m= number future runs = 3

Term1 6.67E-01

Term2 6.22E-07

Squared root Term2 7.89E-04

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

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

UPL POOLED VARIANCE = 1.25E-02

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

 

     

09000064819365f6 FT-PD-TEQ 12/1/2014
Raw Data: Floor Tile - Spray Dryer - TEQ ng/dscm @ 7% O2

Runs SD-104

1 9.11E-03

2 1.03E-02

3 1.09E-02

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -0.95

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

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

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 1.01E-02 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1.01E-02

Mean =

Pooled Variance = 8.06E-07

m= number future runs = 3

Term1 6.67E-01

Term2 5.37E-07

Squared root Term2 7.33E-04

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

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

UPL POOLED VARIANCE = 1.52E-02

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

 

     

09000064819365f6 FT-SD-TEQ 12/1/2014
Raw Data: Wall Tile - Roller Kiln - Hg lb/ton

Runs

64, 65, 66, 67, 68, 69, 70, 

71, 72, 73, 74, 75

1 1.03E-04

2 4.77E-05

3 1.49E-04

4 1.14E-06

5 9.87E-07

6 1.12E-06

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 6.00

Kurtosis -0.92

SE Kurtosis 2.06

Result Kurtosis normal

Skewness 0.89

SE Skewness 0.85 Result Raw Data

Result Skewness normal normal

ni = number test runs = 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 1

n =Total # test runs = 6

5

This step is for QC only:

means 5.05E-05 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

5.05E-05

Mean =

Pooled Variance = 3.95E-09

m= number future runs = 3

Term1 5.00E-01

Term2 1.98E-09

Squared root Term2 4.44E-05

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 = 3.364929999

UPL POOLED VARIANCE = 2.00E-04

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

N ni

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

 

     

09000064819365f6 WT-RK-Hg 12/1/2014
Raw Data: Wall Tile - Roller Kiln - PM lb/ton

Runs

64, 65, 66, 67, 68, 69, 70, 

71, 72, 73, 74, 75

1 1.93E-01

2 1.21E-01

3 1.59E-01

4 2.29E-02

5 2.35E-02

6 5.01E-02

7 2.45E-02

8 1.64E-02

9 2.82E-02

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 9.00

Kurtosis -0.64

SE Kurtosis 1.56

Result Kurtosis normal

Skewness 1.05

SE Skewness 0.72 Result Raw Data

Result Skewness normal normal

ni = number test runs = 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 1

n =Total # test runs = 9

8

This step is for QC only:

means 7.10E-02 #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!

7.10E-02

Mean =

Pooled Variance = 4.65E-03

m= number future runs = 3

Term1 4.44E-01

Term2 2.07E-03

Squared root Term2 4.55E-02

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 = 2.896459448

UPL POOLED VARIANCE = 0.2027

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

N ni

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

 

     

09000064819365f6 WT-RK-PM 12/1/2014
Raw Data: Wall Tile - Roller Kiln - TEQ ng/dscm @ 7% O2

Runs

64, 65, 66, 67, 68, 69, 70, 

71, 72, 73, 74, 75

1 1.19E-01

2 9.98E-02

3 9.89E-02

4 3.06E-02

5 2.96E-02

6 1.07E-02

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 6.00

Kurtosis -2.73

SE Kurtosis 2.06

Result Kurtosis normal

Skewness 0.00

SE Skewness 0.85 Result Raw Data

Result Skewness normal normal

ni = number test runs = 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Total Number of sources = 1

n =Total # test runs = 6

5

This step is for QC only:

means 6.47E-02 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

6.47E-02

Mean =

Pooled Variance = 2.13E-03

m= number future runs = 3

Term1 5.00E-01

Term2 1.06E-03

Squared root Term2 3.26E-02

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 = 3.364929999

UPL POOLED VARIANCE = 1.74E-01

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

N ni

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

 

     

09000064819365f6 WT-RK-TEQ 12/1/2014
Raw Data: Wall Tile - Spray Dryer - TEQ ng/dscm @ 7% O2

Runs SD-105

1 5.90E-02

2 5.95E-02

3 3.96E-02

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -1.73

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

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

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 5.27E-02 #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!

5.27E-02

Mean =

Pooled Variance = 1.29E-04

m= number future runs = 3

Term1 6.67E-01

Term2 8.60E-05

Squared root Term2 9.27E-03

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

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

UPL POOLED VARIANCE = 1.17E-01

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

 

     

09000064819365f6 WT-SD-TEQ 12/1/2014
Raw Data: Tile - Glazing - Hg lb/ton

Runs GL-3, 4, 5 1 2.87E-06 2 4.08E-06 3 1.52E-05 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 7.37E-06

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

Skewness 1.67 1.43

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 GL-3, 4, 5 1 -1.28E+01 2 -1.24E+01 3 -1.11E+01 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 -1.21E+01

0.770857237

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





  

    

09000064819365f6 T-Glaze-Hg 12/1/2014
53.05165528

4.048072476

15.10541143

7. Calculate

2.589604208

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

3.484 z value from "lognormal distribution 99% "

a- Calculate 8.26398E-06

b- Calculate 6.82934E-11

c- Calculate 1.16161848

d-calculate 0.405507632

e- Calculate 2.21E-05

f- Calculate UPL 3.39E-05

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 "lognormal distribution 99% "

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





  

    

  

09000064819365f6 T-Glaze-Hg 12/1/2014
∆z z

0.101 15.10541 2.589604

-5 =f(a) 194.6234

1 -4.899 176.0118

2 -4.798 158.691

3 -4.697 142.6032

4 -4.596 127.6919

5 -4.495 113.9017

6 -4.394 101.1787

7 -4.293 89.47007

8 -4.192 78.72433

9 -4.091 68.89123

10 -3.99 59.92178

11 -3.889 51.76825

12 -3.788 44.38419

13 -3.687 37.72437

14 -3.586 31.74486

15 -3.485 26.40296

16 -3.384 21.65726

17 -3.283 17.46758

18 -3.182 13.79501

19 -3.081 10.60191

20 -2.98 7.851881

21 -2.879 5.509805

22 -2.778 3.541808

23 -2.677 1.91528

24 -2.576 0.598872

25 -2.475 -0.43751

26 -2.374 -1.22269

27 -2.273 -1.78424

28 -2.172 -2.14847

29 -2.071 -2.34044

30 -1.97 -2.38394

31 -1.869 -2.30149

32 -1.768 -2.11438

33 -1.667 -1.84261

34 -1.566 -1.50493

35 -1.465 -1.11885

36 -1.364 -0.70059

37 -1.263 -0.26512

38 -1.162 0.173834

39 -1.061 0.603823

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 1.013651

41 -0.859 1.393383

42 -0.758 1.734344

43 -0.657 2.029119

44 -0.556 2.271551

45 -0.455 2.456743

46 -0.354 2.58106

47 -0.253 2.642125

48 -0.152 2.638819

49 -0.051 2.571286

50 0.05 2.440928

51 0.151 2.250406

52 0.252 2.003642

53 0.353 1.705817

54 0.454 1.363373

55 0.555 0.984009

56 0.656 0.576687

57 0.757 0.151625

58 0.858 -0.2797

59 0.959 -0.70454

60 1.06 -1.1089

61 1.161 -1.47752

62 1.262 -1.79389

63 1.363 -2.04022

64 1.464 -2.19748

65 1.565 -2.24537

66 1.666 -2.16233

67 1.767 -1.92555

68 1.868 -1.51094

69 1.969 -0.89318

70 2.07 -0.04567

71 2.171 1.059452

72 2.272 2.451299

73 2.373 4.160245

74 2.474 6.217926

75 2.575 8.657235

76 2.676 11.51233

77 2.777 14.81862

78 2.878 18.61278

79 2.979 22.93274

80 3.08 27.8177

81 3.181 33.30811

82 3.282 39.44568

83 3.383 46.27338

84 3.484 53.83545

85 3.585 62.17738

86 3.686 71.34591

87 3.787 81.38907

88 3.888 92.35611
89 3.989 104.2976

90 4.09 117.2652

91 4.191 131.3122

92 4.292 146.4927

93 4.393 162.8623

94 4.494 180.4779

95 4.595 199.3976

96 4.696 219.6806

97 4.797 241.3877

98 4.898 264.5806

99 4.999 289.3226
distribution of Z

Normal 

distribution

1.48672E-06 0.00028935

2.45094E-06 0.000431394

3.9995E-06 0.000634685

6.46025E-06 0.000921252

1.03291E-05 0.00131894

1.63472E-05 0.001861979

2.56093E-05 0.002591111

3.97118E-05 0.003553015

6.09553E-05 0.004798662

9.26132E-05 0.006380237

0.000139285 0.008346204

0.000207351 0.010734183

0.000305546 0.013561406

0.000445674 0.016812767

0.000643469 0.020426835

0.000919619 0.024280667

0.001300942 0.028174839

0.001821704 0.031820749

0.002525035 0.03483287

0.003464389 0.036729123

0.004704958 0.036942767

0.006324913 0.034849035

0.008416337 0.029809048

0.011085658 0.021232139

0.014453386 0.008655724

0.018652949 -0.008160799

0.023828414 -0.029134698

0.030130931 -0.053760791

0.03771375 -0.081027031

0.046725789 -0.109359108

0.057303789 -0.136608819

0.069563239 -0.160099436

0.083588399 -0.17673764

0.099421884 -0.183195608

0.117054396 -0.17615906

0.136415346 -0.152627999

0.157365126 -0.110247691

0.179689839 -0.047639348

0.203099244 0.035305615

0.227228529 0.137205758

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 0.255079465

0.275855243 0.384372016

0.299326439 0.519135114

0.321498297 0.652358269

0.341807853 0.776433863

0.359712192 0.883720548

0.37471238 0.967155268

0.386376486 1.020854837

0.394360216 1.040645245

0.398423793 1.024461502

0.398443914 0.972572724

0.394419966 0.887604919

0.386474058 0.774355573

0.374844865 0.639416814

0.359875719 0.490644759

0.34199778 0.336528948

0.32170943 0.185525529

0.299553264 0.045419827

0.276092166 -0.077221715

0.25188591 -0.17746266

0.227469632 -0.252240409

0.203335281 -0.300431991

0.17991684 -0.32275046

0.157579839 -0.321497362

0.136615273 -0.300209101

0.117237788 -0.263241944

0.099587708 -0.215341467

0.083736272 -0.161238177

0.069693339 -0.105302766

0.05741676 -0.051283741

0.046822635 -0.002138419

0.037795734 0.040042762

0.030199481 0.074027951

0.023885038 0.099367614

0.018699163 0.116270005

0.014490659 0.125449043

0.011115369 0.127963767

0.008439746 0.125065364

0.006343145 0.118063543

0.004718997 0.108219524

0.003475077 0.096668657

0.002533081 0.084372127

0.001827693 0.072094608

0.001305351 0.060403024

0.000922829 0.049680921

0.00064578 0.040152934

0.00044732 0.031914448

0.000306705 0.024962463

0.000208159 0.019224714
0.000139842 0.014585156

9.29928E-05 0.010904823

6.12113E-05 0.008037787

3.98826E-05 0.005842508

2.5722E-05 0.004189148

1.64209E-05 0.002963607

1.03767E-05 0.002069082

6.49066E-06 0.001425872

4.01874E-06 0.000970074

2.46298E-06 0.000651656

1.49417E-06 0.000432297
Absolute values

0.00028935 1.46122E-05

0.000431394 4.35708E-05

0.000634685 6.41032E-05

0.000921252 9.30464E-05

0.00131894 0.000133213

0.001861979 0.00018806

0.002591111 0.000261702

0.003553015 0.000358854

0.004798662 0.000484665

0.006380237 0.000644404

0.008346204 0.000842967

0.010734183 0.001084153

0.013561406 0.001369702

0.016812767 0.001698089

0.020426835 0.00206311

0.024280667 0.002452347

0.028174839 0.002845659

0.031820749 0.003213896

0.03483287 0.00351812

0.036729123 0.003709641

0.036942767 0.003731219

0.034849035 0.003519753

0.029809048 0.003010714

0.021232139 0.002144446

0.008655724 0.000874228

0.008160799 0.000824241

0.029134698 0.002942604

0.053760791 0.00542984

0.081027031 0.00818373

0.109359108 0.01104527

0.136608819 0.013797491

0.160099436 0.016170043

0.17673764 0.017850502

0.183195608 0.018502756

0.17615906 0.017792065

0.152627999 0.015415428

0.110247691 0.011135017

0.047639348 0.004811574

0.035305615 0.003565867

0.137205758 0.013857782

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.255079465 0.025763026

0.384372016 0.038821574

0.519135114 0.052432647

0.652358269 0.065888185

0.776433863 0.07841982

0.883720548 0.089255775

0.967155268 0.097682682

1.020854837 0.103106339

1.040645245 0.10510517

1.024461502 0.103470612

0.972572724 0.098229845

0.887604919 0.089648097

0.774355573 0.078209913

0.639416814 0.064581098

0.490644759 0.049555121

0.336528948 0.033989424

0.185525529 0.018738078

0.045419827 0.004587403

0.077221715 0.007799393

0.17746266 0.017923729

0.252240409 0.025476281

0.300431991 0.030343631

0.32275046 0.032597796

0.321497362 0.032471234

0.300209101 0.030321119

0.263241944 0.026587436

0.215341467 0.021749488

0.161238177 0.016285056

0.105302766 0.010635579

0.051283741 0.005179658

0.002138419 0.00021598

0.040042762 0.004044319

0.074027951 0.007476823

0.099367614 0.010036129

0.116270005 0.01174327

0.125449043 0.012670353

0.127963767 0.012924341

0.125065364 0.012631602

0.118063543 0.011924418

0.108219524 0.010930172

0.096668657 0.009763534

0.084372127 0.008521585

0.072094608 0.007281555

0.060403024 0.006100705

0.049680921 0.005017773

0.040152934 0.004055446

0.031914448 0.003223359

0.024962463 0.002521209

0.019224714 0.001941696
0.014585156 0.001473101

0.010904823 0.001101387

0.008037787 0.000811816

0.005842508 0.000590093

0.004189148 0.000423104

0.002963607 0.000299324

0.002069082 0.000208977

0.001425872 0.000144013

0.000970074 9.79774E-05

0.000651656 6.58172E-05

0.000432297 2.1831E-05
Cumulative normalization

1.46122E-05 8.10525E-06

5.8183E-05 3.22736E-05

0.000122286 6.78311E-05

0.000215333 0.000119443

0.000348546 0.000193335

0.000536605 0.00029765

0.000798308 0.000442814

0.001157162 0.000641868

0.001641827 0.000910707

0.002286231 0.001268152

0.003129198 0.001735738

0.00421335 0.002337108

0.005583052 0.003096869

0.007281142 0.004038784

0.009344252 0.005183173

0.011796599 0.006543468

0.014642258 0.008121929

0.017856154 0.009904648

0.021374273 0.011856118

0.025083915 0.013913823

0.028815134 0.015983497

0.032334887 0.017935872

0.035345601 0.019605888

0.037490047 0.020795393

0.038364275 0.021280319

0.039188516 0.021737518

0.04213112 0.023369755

0.04756096 0.026381638

0.05574469 0.03092108

0.06678996 0.037047792

0.080587451 0.044701137

0.096757494 0.053670515

0.114607996 0.063572028

0.133110752 0.073835341

0.150902817 0.08370444

0.166318245 0.09225524

0.177453262 0.098431734

0.182264836 0.101100671

0.185830703 0.103078625

0.199688484 0.110765413

( f x ) ( f b)



 




0.22545151 0.125055932

0.264273084 0.146589911

0.316705731 0.17567383

0.382593916 0.212221416

0.461013736 0.255720187

0.550269511 0.305229566

0.647952193 0.359413274

0.751058532 0.416605436

0.856163702 0.474906332

0.959634314 0.532300553

1.057864159 0.586787767

1.147512255 0.636514763

1.225722168 0.679897101

1.290303267 0.715719657

1.339858387 0.743207438

1.373847811 0.762061067

1.392585889 0.772454911

1.397173292 0.774999502

1.404972685 0.779325756

1.422896414 0.789267887

1.448372695 0.803399352

1.478716326 0.820230692

1.511314123 0.838312398

1.543785356 0.8563239

1.574106475 0.873142753

1.600693912 0.887890566

1.6224434 0.899954812

1.638728456 0.908987987

1.649364035 0.914887447

1.654543693 0.917760557

1.654759673 0.917880359

1.658803992 0.920123707

1.666280815 0.924271034

1.676316944 0.929837984

1.688060215 0.936351871

1.700730568 0.943380003

1.713654909 0.950549018

1.726286511 0.957555655

1.738210928 0.964170022

1.7491411 0.970232891

1.758904635 0.975648636

1.76742622 0.980375483

1.774707775 0.984414496

1.78080848 0.987798503

1.785826253 0.990581816

1.7898817 0.992831336

1.793105059 0.994619305

1.795626268 0.996017797

1.797567964 0.997094838
1.799041065 0.997911954

1.800142452 0.998522883

1.800954268 0.998973191

1.801544361 0.99930051

1.801967465 0.999535202

1.80226679 0.999701235

1.802475767 0.999817152

1.80261978 0.999897035

1.802717757 0.999951382

1.802783575 0.999987891

1.802805406 1

89.06411024
Raw Data: Tile - Glazing - PM lb/ton

Runs GL-1

1 1.52E-01

2 2.33E-01

3 7.21E-02

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness 0.03

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

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

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 1.52E-01 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1.52E-01

Mean =

Pooled Variance = 6.46E-03

m= number future runs = 3

Term1 6.67E-01

Term2 4.31E-03

Squared root Term2 6.56E-02

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

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

UPL POOLED VARIANCE = 6.09E-01

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

 

     

09000064819365f6 T-Glaze-PM 12/1/2014
Raw Data: Sanitaryware - Tunnel Kiln - Hg lb/ton

Runs TK #2 POC + EE + FE stack

1 7.38E-05

2 6.53E-05

3 5.41E-05

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -0.39

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

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

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 6.44E-05 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

6.44E-05

Mean =

Pooled Variance = 9.84E-11

m= number future runs = 3

Term1 6.67E-01

Term2 6.56E-11

Squared root Term2 8.10E-06

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

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

UPL POOLED VARIANCE = 1.21E-04

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

09000064819365f6 S-TK-Hg 12/1/2014
Raw Data: Sanitaryware - Tunnel Kiln - PM lb/ton

Runs TK #6 POC stack 1 9.21E-02 2 9.33E-02 3 9.15E-02 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 9.23E-02

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

Skewness 1.04 1.03

SE Skewness 1.22 1.22 Result Raw Data Result Log Data

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

1. Take LN( ) of Raw Data

Runs TK #6 POC stack 1 -2.39E+00 2 -2.37E+00 3 -2.39E+00 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

3

3. Calculate -2.38E+00

9.39312E-05

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





  

    

09000064819365f6 S-TK-PM 12/1/2014
3.001503102

2.64717E-08

113385346.5

7. Calculate

0.01678763

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

3.484 z value from "lognormal distribution 99% "

a- Calculate 0.092280273

b- Calculate 0.008515649

c- Calculate 9.39356E-05

d-calculate 3.13126E-05

e- Calculate 0.002191

f- Calculate UPL 9.48E-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 "lognormal distribution 99% "

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





  

    

  

09000064819365f6 S-TK-PM 12/1/2014
∆z z

0.101 1.13E+08 0.016788

-5 =f(a) 2.26E+09

1 -4.899 2.06E+09

2 -4.798 1.87E+09

3 -4.697 1.69E+09

4 -4.596 1.52E+09

5 -4.495 1.37E+09

6 -4.394 1.23E+09

7 -4.293 1.1E+09

8 -4.192 9.75E+08

9 -4.091 8.63E+08

10 -3.99 7.6E+08

11 -3.889 6.66E+08

12 -3.788 5.8E+08

13 -3.687 5.02E+08

14 -3.586 4.31E+08

15 -3.485 3.67E+08

16 -3.384 3.09E+08

17 -3.283 2.57E+08

18 -3.182 2.11E+08

19 -3.081 1.71E+08

20 -2.98 1.35E+08

21 -2.879 1.04E+08

22 -2.778 76784406

23 -2.677 53661019

24 -2.576 34104740

25 -2.475 17808736

26 -2.374 4477977

27 -2.273 -6170770

28 -2.172 -1.4E+07

29 -2.071 -2E+07

30 -1.97 -2.5E+07

31 -1.869 -2.7E+07

32 -1.768 -2.8E+07

33 -1.667 -2.8E+07

34 -1.566 -2.7E+07

35 -1.465 -2.5E+07

36 -1.364 -2.2E+07

37 -1.263 -1.9E+07

38 -1.162 -1.5E+07

39 -1.061 -1.2E+07

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -7938170

41 -0.859 -4170773

42 -0.758 -553982

43 -0.657 2817754

44 -0.556 5861783

45 -0.455 8507253

46 -0.354 10695112

47 -0.253 12378105

48 -0.152 13520777

49 -0.051 14099472

50 0.05 14102333

51 0.151 13529300

52 0.252 12392116

53 0.353 10714318

54 0.454 8531246

55 0.555 5890036

56 0.656 2849625

57 0.757 -519251

58 0.858 -4134060

59 0.959 -7900467

60 1.06 -1.2E+07

61 1.161 -1.5E+07

62 1.262 -1.9E+07

63 1.363 -2.2E+07

64 1.464 -2.5E+07

65 1.565 -2.7E+07

66 1.666 -2.8E+07

67 1.767 -2.8E+07

68 1.868 -2.7E+07

69 1.969 -2.5E+07

70 2.07 -2.1E+07

71 2.171 -1.4E+07

72 2.272 -6263713

73 2.373 4359856

74 2.474 17662691

75 2.575 33927909

76 2.676 53450425

77 2.777 76536951

78 2.878 1.04E+08

79 2.979 1.35E+08

80 3.08 1.7E+08

81 3.181 2.11E+08

82 3.282 2.57E+08

83 3.383 3.09E+08

84 3.484 3.66E+08

85 3.585 4.3E+08

86 3.686 5.01E+08

87 3.787 5.79E+08

88 3.888 6.65E+08
89 3.989 7.59E+08

90 4.09 8.62E+08

91 4.191 9.74E+08

92 4.292 1.1E+09

93 4.393 1.23E+09

94 4.494 1.37E+09

95 4.595 1.52E+09

96 4.696 1.69E+09

97 4.797 1.86E+09

98 4.898 2.05E+09

99 4.999 2.26E+09
distribution of Z

Normal 

distribution

1.48672E-06 3357.396374

2.45094E-06 5037.046963

3.9995E-06 7460.443865

6.46025E-06 10906.65787

1.03291E-05 15735.16646

1.63472E-05 22397.91572

2.56093E-05 31447.95068

3.97118E-05 43541.46609

6.09553E-05 59429.20696

9.26132E-05 79932.31282

0.000139285 105897.1381

0.000207351 138123.5375

0.000305546 177261.8642

0.000445674 223675.7866

0.000643469 277271.2481

0.000919619 337296.6189

0.001300942 402125.2997

0.001821704 469039.4245

0.002525035 534041.2451

0.003464389 591726.2577

0.004704958 635257.7907

0.006324913 656485.0151

0.008416337 646243.4679

0.011085658 594867.7368

0.014453386 492928.983

0.018652949 332185.4401

0.023828414 106703.09

0.030130931 -185931.0438

0.03771375 -543415.1043

0.046725789 -957699.4152

0.057303789 -1414273.586

0.069563239 -1891934.123

0.083588399 -2363172.761

0.099421884 -2795281.96

0.117054396 -3152207.519

0.136415346 -3397093.723

0.157365126 -3495372.149

0.179689839 -3418152.773

0.203099244 -3145598.806

0.227228529 -2669918.126

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -1997595.656

0.275855243 -1150529.645

0.299326439 -165821.4223

0.321498297 905903.0098

0.341807853 2003603.336

0.359712192 3060162.631

0.37471238 4007590.797

0.386376486 4782608.658

0.394360216 5332056.558

0.398423793 5617565.146

0.398443914 5618988.607

0.394419966 5336226.156

0.386474058 4789231.18

0.374844865 4016207.027

0.359875719 3070188.142

0.34199778 2014379.253

0.32170943 916751.3666

0.299553264 -155543.3833

0.276092166 -1141381.483

0.25188591 -1990016.323

0.227469632 -2664201.993

0.203335281 -3141886.317

0.17991684 -3416435.281

0.157579839 -3495509.041

0.136615273 -3398838.844

0.117237788 -3155241.299

0.099587708 -2799245.486

0.083736272 -2367700.338

0.069693339 -1896681.517

0.05741676 -1418939.998

0.046822635 -962042.1766

0.037795734 -547256.9948

0.030199481 -189160.8925

0.023885038 104135.3169

0.018699163 330277.5365

0.014490659 491637.7715

0.011115369 594121.1961

0.008439746 645952.4486

0.006343145 656553.6007

0.004718997 635591.7078

0.003475077 592239.073

0.002533081 534658.3646

0.001827693 469700.1149

0.001305351 402783.0852

0.000922829 337918.4636

0.00064578 277835.9353

0.00044732 224171.8651

0.000306705 177685.4645

0.000208159 138476.2807
0.000139842 106184.2851

9.29928E-05 80161.23333

6.12113E-05 59608.1912

3.98826E-05 43678.865

2.5722E-05 31551.60476

1.64209E-05 22474.81999

1.03767E-05 15791.31627

6.49066E-06 10947.0234

4.01874E-06 7489.028652

2.46298E-06 5056.994608

1.49417E-06 3371.118743
Absolute values

3357.396374 169.5485169

5037.046963 508.7417432

7460.443865 753.5048304

10906.65787 1101.572445

15735.16646 1589.251813

22397.91572 2262.189488

31447.95068 3176.243019

43541.46609 4397.688075

59429.20696 6002.349903

79932.31282 8073.163595

105897.1381 10695.61095

138123.5375 13950.47729

177261.8642 17903.44828

223675.7866 22591.25445

277271.2481 28004.39606

337296.6189 34066.95851

402125.2997 40614.65527

469039.4245 47372.98187

534041.2451 53938.16576

591726.2577 59764.35203

635257.7907 64161.03686

656485.0151 66304.98652

646243.4679 65270.59026

594867.7368 60081.64142

492928.983 49785.82728

332185.4401 33550.72945

106703.09 10777.01209

185931.0438 18779.03542

543415.1043 54884.92553

957699.4152 96727.64093

1414273.586 142841.6322

1891934.123 191085.3464

2363172.761 238680.4489

2795281.96 282323.478

3152207.519 318372.9594

3397093.723 343106.466

3495372.149 353032.587

3418152.773 345233.4301

3145598.806 317705.4795

2669918.126 269661.7307

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
1997595.656 201757.1613

1150529.645 116203.4942

165821.4223 16747.96366

905903.0098 91496.20399

2003603.336 202363.9369

3060162.631 309076.4257

4007590.797 404766.6705

4782608.658 483043.4744

5332056.558 538537.7124

5617565.146 567374.0798

5618988.607 567517.8493

5336226.156 538958.8417

4789231.18 483712.3492

4016207.027 405636.9097

3070188.142 310089.0023

2014379.253 203452.3046

916751.3666 92591.88802

155543.3833 15709.88171

1141381.483 115279.5298

1990016.323 200991.6486

2664201.993 269084.4013

3141886.317 317330.5181

3416435.281 345059.9634

3495509.041 353046.4131

3398838.844 343282.7232

3155241.299 318679.3712

2799245.486 282723.7941

2367700.338 239137.7341

1896681.517 191564.8332

1418939.998 143312.9398

962042.1766 97166.25984

547256.9948 55272.95648

189160.8925 19105.25014

104135.3169 10517.66701

330277.5365 33358.03119

491637.7715 49655.41493

594121.1961 60006.2408

645952.4486 65241.19731

656553.6007 66311.91367

635591.7078 64194.76249

592239.073 59816.14638

534658.3646 54000.49483

469700.1149 47439.7116

402783.0852 40681.09161

337918.4636 34129.76482

277835.9353 28061.42946

224171.8651 22641.35838

177685.4645 17946.23192

138476.2807 13986.10435
106184.2851 10724.61279

80161.23333 8096.284566

59608.1912 6020.427311

43678.865 4411.565365

31551.60476 3186.71208

22474.81999 2269.956819

15791.31627 1594.922943

10947.0234 1105.649363

7489.028652 756.3918938

5056.994608 510.7564554

3371.118743 170.2414965
Cumulative normalization

169.5485169 1.28191E-05

678.2902601 5.12838E-05

1431.795091 0.000108254

2533.367535 0.000191541

4122.619348 0.000311701

6384.808836 0.000482739

9561.051855 0.000722887

13958.73993 0.001055384

19961.08983 0.001509207

28034.25343 0.002119598

38729.86438 0.002928266

52680.34167 0.003983025

70583.78995 0.005336659

93175.04439 0.007044726

121179.4405 0.009162066

155246.399 0.011737781

195861.0542 0.014808551

243234.0361 0.018390301

297172.2019 0.022468427

356936.5539 0.026987056

421097.5908 0.031838107

487402.5773 0.036851257

552673.1675 0.041786199

612754.809 0.046328817

662540.6362 0.050092996

696091.3657 0.05262968

706868.3778 0.053444502

725647.4132 0.054864337

780532.3387 0.059014045

877259.9797 0.066327374

1020101.612 0.077127263

1211186.958 0.091574735

1449867.407 0.109620751

1732190.885 0.130966504

2050563.844 0.155037865

2393670.31 0.180979264

2746702.898 0.207671151

3091936.328 0.233773364

3409641.807 0.257794259

3679303.538 0.27818269

( f x ) ( f b)



 




3881060.699 0.293437031

3997264.193 0.302222878

4014012.157 0.303489149

4105508.361 0.310406942

4307872.298 0.325707159

4616948.724 0.349075634

5021715.394 0.379679002

5504758.868 0.416200679

6043296.581 0.456918132

6610670.661 0.499815828

7178188.51 0.542724395

7717147.352 0.583473689

8200859.701 0.620045937

8606496.611 0.650715102

8916585.613 0.674160135

9120037.918 0.689542641

9212629.806 0.696543276

9228339.687 0.69773106

9343619.217 0.706447049

9544610.866 0.721643511

9813695.267 0.741988292

10131025.78 0.765980837

10476085.75 0.792069935

10829132.16 0.818762867

11172414.88 0.844717592

11491094.26 0.868812121

11773818.05 0.89018814

12012955.78 0.90826873

12204520.62 0.922752455

12347833.56 0.933587979

12444999.82 0.94093447

12500272.77 0.945113516

12519378.02 0.946558015

12529895.69 0.947353229

12563253.72 0.949875344

12612909.14 0.953629662

12672915.38 0.95816658

12738156.57 0.963099299

12804468.49 0.968112973

12868663.25 0.972966574

12928479.4 0.977489119

12982479.89 0.981571958

13029919.6 0.985158753

13070600.7 0.988234545

13104730.46 0.990815009

13132791.89 0.992936662

13155433.25 0.994648517

13173379.48 0.996005385

13187365.58 0.997062839
13198090.2 0.997873699

13206186.48 0.998485838

13212206.91 0.998941027

13216618.47 0.999274575

13219805.19 0.999515514

13222075.14 0.999687139

13223670.07 0.999807727

13224775.72 0.999891323

13225532.11 0.999948511

13226042.86 0.999987128

13226213.11 1

667865176.8
Raw Data: Sanitaryware - Tunnel Kiln - TEQ ng/dscm @ 7% O2

Runs TK #10 POC + RCE stacks 1 2.87E-01 2 2.19E-01 3 1.99E-01 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 2.35E-01

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

Skewness 1.37 1.25

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 TK #10 POC + RCE stacks 1 -1.25E+00 2 -1.52E+00 3 -1.62E+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 -1.46E+00

0.036209387

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





  

    

09000064819365f6 S-TK-TEQ 12/1/2014
3.610663737

0.004078839

887.2184741

7. Calculate

0.336681942

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

3.484 z value from "lognormal distribution 99% "

a- Calculate 0.23616946

b- Calculate 0.055776014

c- Calculate 0.036872931

d-calculate 0.012397575

e- Calculate 0.111325

f- Calculate UPL 3.65E-01

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

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

8. Go to tab distribution for 99 percentile "lognormal distribution 99% "

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





  

    

  

09000064819365f6 S-TK-TEQ 12/1/2014
∆z z

0.101 887.2185 0.336682

-5 =f(a) 17605.51

1 -4.899 16022

2 -4.798 14542.2

3 -4.697 13161.72

4 -4.596 11876.2

5 -4.495 10681.44

6 -4.394 9573.284

7 -4.293 8547.685

8 -4.192 7600.685

9 -4.091 6728.422

10 -3.99 5927.123

11 -3.889 5193.107

12 -3.788 4522.785

13 -3.687 3912.661

14 -3.586 3359.33

15 -3.485 2859.479

16 -3.384 2409.889

17 -3.283 2007.431

18 -3.182 1649.068

19 -3.081 1331.855

20 -2.98 1052.941

21 -2.879 809.5638

22 -2.778 599.0562

23 -2.677 418.8415

24 -2.576 266.4351

25 -2.475 139.4446

26 -2.374 35.56968

27 -2.273 -47.3982

28 -2.172 -111.575

29 -2.071 -158.986

30 -1.97 -191.563

31 -1.869 -211.146

32 -1.768 -219.484

33 -1.667 -218.233

34 -1.566 -208.957

35 -1.465 -193.129

36 -1.364 -172.129

37 -1.263 -147.245

38 -1.162 -119.673

39 -1.061 -90.5184

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -60.7927

41 -0.859 -31.4161

42 -0.758 -3.21699

43 -0.657 23.06851

44 -0.556 46.79617

45 -0.455 67.41383

46 -0.354 84.46131

47 -0.253 97.57045

48 -0.152 106.4651

49 -0.051 110.9612

50 0.05 110.9665

51 0.151 106.481

52 0.252 97.59651

53 0.353 84.49703

54 0.454 67.45845

55 0.555 46.84871

56 0.656 23.12777

57 0.757 -3.1524

58 0.858 -31.3478

59 0.959 -60.7226

60 1.06 -90.4485

61 1.161 -119.606

62 1.262 -147.182

63 1.363 -172.074

64 1.464 -193.085

65 1.565 -208.927

66 1.666 -218.22

67 1.767 -219.492

68 1.868 -211.178

69 1.969 -191.624

70 2.07 -159.08

71 2.171 -111.706

72 2.272 -47.571

73 2.373 35.35002

74 2.474 139.1731

75 2.575 266.1063

76 2.676 418.4499

77 2.777 598.5961

78 2.878 809.0291

79 2.979 1052.325

80 3.08 1331.152

81 3.181 1648.271

82 3.282 2006.534

83 3.383 2408.885

84 3.484 2858.36

85 3.585 3358.088

86 3.686 3911.289

87 3.787 4521.275

88 3.888 5191.45
89 3.989 5925.312

90 4.09 6726.448

91 4.191 7598.539

92 4.292 8545.358

93 4.393 9570.767

94 4.494 10678.72

95 4.595 11873.28

96 4.696 13158.57

97 4.797 14538.83

98 4.898 16018.38

99 4.999 17601.65
distribution of Z

Normal 

distribution

1.48672E-06 0.026174458

2.45094E-06 0.039268952

3.9995E-06 0.058161586

6.46025E-06 0.085027928

1.03291E-05 0.12267037

1.63472E-05 0.174612169

2.56093E-05 0.245164733

3.97118E-05 0.33944365

6.09553E-05 0.463301697

9.26132E-05 0.623140642

0.000139285 0.825559206

0.000207351 1.076794276

0.000305546 1.381918343

0.000445674 1.743770679

0.000643469 2.161624805

0.000919619 2.629631696

0.001300942 3.135126544

0.001821704 3.656944495

0.002525035 4.163952607

0.003464389 4.614063516

0.004704958 4.954040402

0.006324913 5.120420174

0.008416337 5.041859407

0.011085658 4.643133802

0.014453386 3.850889559

0.018652949 2.601053709

0.023828414 0.847569168

0.030130931 -1.428150627

0.03771375 -4.207922039

0.046725789 -7.428748381

0.057303789 -10.97727572

0.069563239 -14.68799766

0.083588399 -18.34630149

0.099421884 -21.69710668

0.117054396 -24.45932762

0.136415346 -26.34573356

0.157365126 -27.08704299

0.179689839 -26.45836928

0.203099244 -24.30553195

0.227228529 -20.56837045

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -15.29813113

0.275855243 -8.666299915

0.299326439 -0.96292955

0.321498297 7.416485128

0.341807853 15.99529887

0.359712192 24.24957629

0.37471238 31.64869828

0.386376486 37.69892943

0.394360216 41.9856079

0.398423793 44.2095716

0.398443914 44.21392378

0.394419966 41.99822077

0.386474058 37.71851907

0.374844865 31.6732761

0.359875719 24.27665692

0.34199778 16.0221556

0.32170943 7.440423246

0.299553264 -0.94431222

0.276092166 -8.654893328

0.25188591 -15.29515592

0.227469632 -20.57429664

0.203335281 -24.32007705

0.17991684 -26.48057001

0.157579839 -27.115387

0.136615273 -26.3783319

0.117237788 -24.49410634

0.099587708 -21.73199573

0.083736272 -18.37940249

0.069693339 -14.71771614

0.05741676 -11.00240962

0.046822635 -7.448530149

0.037795734 -4.222017161

0.030199481 -1.436619466

0.023885038 0.844336633

0.018699163 2.602419502

0.014490659 3.856055223

0.011115369 4.651224709

0.008439746 5.051998886

0.006343145 5.131788893

0.004718997 4.965918877

0.003475077 4.625857719

0.002533081 4.175204294

0.001827693 3.667328948

0.001305351 3.144440968

0.000922829 2.637777577

0.00064578 2.168587251

0.00044732 1.749597231

0.000306705 1.386699139

0.000208159 1.080644774
0.000139842 0.828606121

9.29928E-05 0.625511251

6.12113E-05 0.465116345

3.98826E-05 0.340811039

2.5722E-05 0.246179489

1.64209E-05 0.175354117

1.03767E-05 0.123205036

6.49066E-06 0.085407785

4.01874E-06 0.058427725

2.46298E-06 0.039452881

1.49417E-06 0.026299871
Absolute values

0.026174458 0.00132181

0.039268952 0.003966164

0.058161586 0.00587432

0.085027928 0.008587821

0.12267037 0.012389707

0.174612169 0.017635829

0.245164733 0.024761638

0.33944365 0.034283809

0.463301697 0.046793471

0.623140642 0.062937205

0.825559206 0.08338148

1.076794276 0.108756222

1.381918343 0.139573753

1.743770679 0.176120839

2.161624805 0.218324105

2.629631696 0.265592801

3.135126544 0.316647781

3.656944495 0.369351394

4.163952607 0.420559213

4.614063516 0.466020415

4.954040402 0.500358081

5.120420174 0.517162438

5.041859407 0.5092278

4.643133802 0.468956514

3.850889559 0.388939845

2.601053709 0.262706425

0.847569168 0.085604486

1.428150627 0.144243213

4.207922039 0.425000126

7.428748381 0.750303587

10.97727572 1.108704848

14.68799766 1.483487763

18.34630149 1.852976451

21.69710668 2.191407775

24.45932762 2.47039209

26.34573356 2.660919089

27.08704299 2.735791342

26.45836928 2.672295297

24.30553195 2.454858726

20.56837045 2.077405415

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
15.29813113 1.545111244

8.666299915 0.875296291

0.96292955 0.097255885

7.416485128 0.749064998

15.99529887 1.615525186

24.24957629 2.449207205

31.64869828 3.196518526

37.69892943 3.807591873

41.9856079 4.240546398

44.2095716 4.465166731

44.21392378 4.465606302

41.99822077 4.241820298

37.71851907 3.809570426

31.6732761 3.199000886

24.27665692 2.451942349

16.0221556 1.618237715

7.440423246 0.751482748

0.94431222 0.095375534

8.654893328 0.874144226

15.29515592 1.544810748

20.57429664 2.07800396

24.32007705 2.456327782

26.48057001 2.674537571

27.115387 2.738654087

26.3783319 2.664211522

24.49410634 2.47390474

21.73199573 2.194931569

18.37940249 1.856319652

14.71771614 1.48648933

11.00240962 1.111243372

7.448530149 0.752301545

4.222017161 0.426423733

1.436619466 0.145098566

0.844336633 0.085278

2.602419502 0.26284437

3.856055223 0.389461578

4.651224709 0.469773696

5.051998886 0.510251888

5.131788893 0.518310678

4.965918877 0.501557807

4.625857719 0.46721163

4.175204294 0.421695634

3.667328948 0.370400224

3.144440968 0.317588538

2.637777577 0.266415535

2.168587251 0.219027312

1.749597231 0.17670932

1.386699139 0.140056613

1.080644774 0.109145122
0.828606121 0.083689218

0.625511251 0.063176636

0.465116345 0.046976751

0.340811039 0.034421915

0.246179489 0.024864128

0.175354117 0.017710766

0.123205036 0.012443709

0.085407785 0.008626186

0.058427725 0.0059012

0.039452881 0.003984741

0.026299871 0.001328143
Cumulative normalization

0.00132181 1.28052E-05

0.005287974 5.12281E-05

0.011162294 0.000108136

0.019750115 0.000191332

0.032139822 0.000311359

0.049775652 0.000482209

0.07453729 0.000722091

0.108821098 0.001054221

0.15561457 0.00150754

0.218551774 0.002117253

0.301933254 0.002925024

0.410689476 0.003978616

0.550263229 0.005330758

0.726384067 0.007036955

0.944708173 0.009152003

1.210300974 0.011724974

1.526948755 0.014792547

1.896300149 0.018370694

2.316859362 0.022444925

2.782879777 0.026959569

3.283237858 0.031806863

3.800400295 0.036816952

4.309628096 0.041750174

4.77858461 0.046293261

5.167524455 0.050061174

5.43023088 0.052606183

5.515835366 0.053435489

5.660078579 0.054832867

6.085078705 0.05895012

6.835382291 0.066218799

7.94408714 0.076959545

9.427574903 0.091331057

11.28055135 0.109282047

13.47195913 0.130511641

15.94235122 0.154443938

18.60327031 0.180221994

21.33906165 0.206725386

24.01135695 0.232613651

26.46621567 0.256395466

28.54362109 0.276520645

( f x ) ( f b)



 




30.08873233 0.291489144

30.96402862 0.299968709

31.06128451 0.30091089

31.81034951 0.30816757

33.42587469 0.323818215

35.8750819 0.34754528

39.07160042 0.378512036

42.8791923 0.415398658

47.11973869 0.456479593

51.58490542 0.499736571

56.05051173 0.542997807

60.29233202 0.584091082

64.10190245 0.620996872

67.30090334 0.651987677

69.75284569 0.675741239

71.3710834 0.691418161

72.12256615 0.698698264

72.21794168 0.699622229

73.09208591 0.708090633

74.63689666 0.723056221

76.71490062 0.743187199

79.1712284 0.766983245

81.84576597 0.792893232

84.58442006 0.819424358

87.24863158 0.84523431

89.72253632 0.869200636

91.91746789 0.890464367

93.77378754 0.908447745

95.26027687 0.922848335

96.37152024 0.933613673

97.12382179 0.940901708

97.55024552 0.945032752

97.69534409 0.946438417

97.78062209 0.94726456

98.04346646 0.949810905

98.43292803 0.953583873

98.90270173 0.958134876

99.41295362 0.963078019

99.93126429 0.968099232

100.4328221 0.972958149

100.9000337 0.977484332

101.3217294 0.981569572

101.6921296 0.985157881

102.0097181 0.988234568

102.2761337 0.990815508

102.495161 0.992937369

102.6718703 0.994649267

102.8119269 0.996006086

102.921072 0.997063446
103.0047612 0.997874198

103.0679379 0.998486231

103.1149146 0.998941326

103.1493365 0.999274793

103.1742007 0.999515668

103.1919114 0.999687244

103.2043552 0.999807794

103.2129813 0.999891362

103.2188825 0.999948531

103.2228673 0.999987133

103.2241954 1

5211.862942
Raw Data: Sanitaryware - Manual Glaze Application - PM lb/ton

Runs SO3/SO4 1 3.52E+00 2 3.17E+00 3 3.29E+00 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 3.00 3.00

Kurtosis #DIV/0! #DIV/0!

SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 3.33E+00

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

Skewness 0.88 0.82

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 SO3/SO4 1 1.26E+00 2 1.16E+00 3 1.19E+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 1.20E+00

0.00272611

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





  

    

09000064819365f6 S-Manual-PM 12/1/2014
3.043789176

2.23559E-05

136153.433

7. Calculate

0.090578141

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

3.484 z value from "lognormal distribution 99% "

a- Calculate 3.330813203

b- Calculate 11.09431659

c- Calculate 0.00272983

d-calculate 0.000910561

e- Calculate 0.426351

f- Calculate UPL 3.83E+00

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 "lognormal distribution 99% "

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





  

    

  

09000064819365f6 S-Manual-PM 12/1/2014
∆z z

0.101 136153.4 0.090578

-5 =f(a) 2711662

1 -4.899 2467774

2 -4.798 2239859

3 -4.697 2027237

4 -4.596 1829243

5 -4.495 1645224

6 -4.394 1474543

7 -4.293 1316576

8 -4.192 1170714

9 -4.091 1036363

10 -3.99 912939.8

11 -3.889 799879.3

12 -3.788 696628.4

13 -3.687 602648.6

14 -3.586 517415.7

15 -3.485 440419.6

16 -3.384 371164.3

17 -3.283 309168

18 -3.182 253963.1

19 -3.081 205096.3

20 -2.98 162128.1

21 -2.879 124633.6

22 -2.778 92201.7

23 -2.677 64435.71

24 -2.576 40953.02

25 -2.475 21385.19

26 -2.374 5377.953

27 -2.273 -7408.79

28 -2.172 -17301

29 -2.071 -24610.3

30 -1.97 -29634.5

31 -1.869 -32656.9

32 -1.768 -33946.8

33 -1.667 -33759.3

34 -1.566 -32335.2

35 -1.465 -29901.4

36 -1.364 -26670.5

37 -1.263 -22840.8

38 -1.162 -18596.6

39 -1.061 -14108

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -9530.94

41 -0.859 -5007.14

42 -0.758 -664.182

43 -0.657 3384.516

44 -0.556 7039.711

45 -0.455 10216.33

46 -0.354 12843.45

47 -0.253 14864.34

48 -0.152 16236.43

49 -0.051 16931.31

50 0.05 16934.74

51 0.151 16246.66

52 0.252 14881.15

53 0.353 12866.48

54 0.454 10245.1

55 0.555 7073.592

56 0.656 3422.736

57 0.757 -622.533

58 0.858 -4963.11

59 0.959 -9485.72

60 1.06 -14062.9

61 1.161 -18553.1

62 1.262 -22800.5

63 1.363 -26635.2

64 1.464 -29873

65 1.565 -32315.7

66 1.666 -33750.8

67 1.767 -33951.8

68 1.868 -32677.7

69 1.969 -29673.8

70 2.07 -24670.7

71 2.171 -17385.4

72 2.272 -7520.25

73 2.373 5236.304

74 2.474 21210.06

75 2.575 40740.97

76 2.676 64183.17

77 2.777 91904.95

78 2.878 124288.8

79 2.979 161731.3

80 3.08 204643.3

81 3.181 253449.7

82 3.282 308589.7

83 3.383 370516.5

84 3.484 439697.7

85 3.585 516614.9

86 3.686 601763.9

87 3.787 695654.6

88 3.888 798811.3
89 3.989 911772.1

90 4.09 1035090

91 4.191 1169331

92 4.292 1315076

93 4.393 1472920

94 4.494 1643472

95 4.595 1827356

96 4.696 2025209

97 4.797 2237683

98 4.898 2465443

99 4.999 2709170
distribution of Z

Normal 

distribution

1.48672E-06 4.031481005

2.45094E-06 6.048365247

3.9995E-06 8.958322361

6.46025E-06 13.09645398

1.03291E-05 18.89441226

1.63472E-05 26.89488357

2.56093E-05 37.76195168

3.97118E-05 52.28356041

6.09553E-05 71.3611878

9.26132E-05 95.98084062

0.000139285 127.1588042

0.000207351 165.8555296

0.000305546 212.8519519

0.000445674 268.5847654

0.000643469 332.9410422

0.000919619 405.0182615

0.001300942 482.8632665

0.001821704 563.2125411

0.002525035 641.2657272

0.003464389 710.5332793

0.004704958 762.805953

0.006324913 788.2965098

0.008416337 776.0005805

0.011085658 714.312285

0.014453386 591.9098322

0.018652949 398.8968556

0.023828414 128.1480907

0.030130931 -223.2337352

0.03771375 -652.4844322

0.046725789 -1149.937971

0.057303789 -1698.170467

0.069563239 -2271.72114

0.083588399 -2837.558632

0.099421884 -3356.408468

0.117054396 -3784.977645

0.136415346 -4079.010797

0.157365126 -4196.999161

0.179689839 -4104.252546

0.203099244 -3776.951758

0.227228529 -3205.740627

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -2398.406597

0.275855243 -1381.245114

0.299326439 -198.8071609

0.321498297 1088.116162

0.341807853 2406.228512

0.359712192 3674.936935

0.37471238 4812.600058

0.386376486 5743.233362

0.394360216 6403.003714

0.398423793 6745.837944

0.398443914 6747.545402

0.394419966 6408.005158

0.386474058 5751.17698

0.374844865 4822.934858

0.359875719 3686.961705

0.34199778 2419.152737

0.32170943 1101.12645

0.299553264 -186.4819206

0.276092166 -1370.276165

0.25188591 -2389.320434

0.227469632 -3198.890238

0.203335281 -3772.505416

0.17991684 -4102.199631

0.157579839 -4197.170717

0.136615273 -4081.111259

0.117237788 -3788.623394

0.099587708 -3361.168725

0.083736272 -2842.994537

0.069693339 -2277.419639

0.05741676 -1703.770674

0.046822635 -1155.148797

0.037795734 -657.0933561

0.030199481 -227.1075141

0.023885038 125.0693275

0.018699163 396.6102932

0.014490659 590.3635011

0.011115369 713.4196203

0.008439746 775.6544861

0.006343145 788.3817932

0.004718997 763.2094176

0.003475077 711.1511603

0.002533081 642.0084875

0.001827693 564.0072941

0.001305351 483.6542486

0.000922829 405.7658469

0.00064578 333.619795

0.00044732 269.1809725

0.000306705 213.3609994

0.000208159 166.2793915
0.000139842 127.5038211

9.29928E-05 96.25588068

6.12113E-05 71.57622068

3.98826E-05 52.44862548

2.5722E-05 37.88647264

1.64209E-05 26.9872667

1.03767E-05 18.96186169

6.49066E-06 13.14494148

4.01874E-06 8.992657936

2.46298E-06 6.072325546

1.49417E-06 4.047963456
Absolute values

4.031481005 0.203589791

6.048365247 0.61088489

8.958322361 0.904790558

13.09645398 1.322741852

18.89441226 1.908335639

26.89488357 2.716383241

37.76195168 3.81395712

52.28356041 5.280639602

71.3611878 7.207479968

95.98084062 9.694064903

127.1588042 12.84303922

165.8555296 16.75140849

212.8519519 21.49804715

268.5847654 27.1270613

332.9410422 33.62704526

405.0182615 40.90684442

482.8632665 48.76918992

563.2125411 56.88446665

641.2657272 64.76783845

710.5332793 71.76386121

762.805953 77.04340126

788.2965098 79.61794749

776.0005805 78.37605863

714.312285 72.14554078

591.9098322 59.78289306

398.8968556 40.28858242

128.1480907 12.94295716

223.2337352 22.54660726

652.4844322 65.90092765

1149.937971 116.1437351

1698.170467 171.5152171

2271.72114 229.4438351

2837.558632 286.5934218

3356.408468 338.9972553

3784.977645 382.2827422

4079.010797 411.9800905

4196.999161 423.8969152

4104.252546 414.5295072

3776.951758 381.4721275

3205.740627 323.7798033

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
2398.406597 242.2390663

1381.245114 139.5057565

198.8071609 20.07952325

1088.116162 109.8997324

2406.228512 243.0290797

3674.936935 371.1686304

4812.600058 486.0726059

5743.233362 580.0665696

6403.003714 646.7033751

6745.837944 681.3296324

6747.545402 681.5020856

6408.005158 647.208521

5751.17698 580.868875

4822.934858 487.1164206

3686.961705 372.3831322

2419.152737 244.3344265

1101.12645 111.2137715

186.4819206 18.83467398

1370.276165 138.3978926

2389.320434 241.3213639

3198.890238 323.087914

3772.505416 381.023047

4102.199631 414.3221628

4197.170717 423.9142425

4081.111259 412.1922371

3788.623394 382.6509628

3361.168725 339.4780413

2842.994537 287.1424482

2277.419639 230.0193835

1703.770674 172.0808381

1155.148797 116.6700285

657.0933561 66.36642897

227.1075141 22.93785892

125.0693275 12.63200207

396.6102932 40.05763962

590.3635011 59.62671361

713.4196203 72.05538166

775.6544861 78.3411031

788.3817932 79.62656111

763.2094176 77.08415118

711.1511603 71.82626719

642.0084875 64.84285724

564.0072941 56.96473671

483.6542486 48.8490791

405.7658469 40.98235053

333.619795 33.6955993

269.1809725 27.18727823

213.3609994 21.54946094

166.2793915 16.79421855
127.5038211 12.87788593

96.25588068 9.721843948

71.57622068 7.229198289

52.44862548 5.297311174

37.88647264 3.826533737

26.9872667 2.725713937

18.96186169 1.915148031

13.14494148 1.327639089

8.992657936 0.908258451

6.072325546 0.61330488

4.047963456 0.204422155
Cumulative normalization

0.203589791 1.28191E-05

0.814474681 5.12835E-05

1.719265239 0.000108254

3.042007091 0.00019154

4.95034273 0.000311699

7.66672597 0.000482736

11.48068309 0.000722883

16.76132269 0.001055379

23.96880266 0.001509199

33.66286756 0.002119587

46.50590679 0.002928251

63.25731527 0.003983006

84.75536242 0.005336633

111.8824237 0.007044692

145.509469 0.009162024

186.4163134 0.011737729

235.1855033 0.014808487

292.06997 0.018390226

356.8378084 0.022468341

428.6016696 0.026986963

505.6450709 0.031838011

585.2630184 0.036851166

663.639077 0.041786125

735.7846178 0.046328779

795.5675109 0.050093017

835.8560933 0.052629793

848.7990504 0.053444748

871.3456577 0.054864398

937.2465853 0.059013859

1053.39032 0.066326866

1224.905538 0.077126345

1454.349373 0.091573307

1740.942795 0.109618701

2079.94005 0.13096371

2462.222792 0.155034195

2874.202882 0.180974578

3298.099798 0.207665305

3712.629305 0.233766212

4094.101432 0.257785657

4417.881236 0.278172496

( f x ) ( f b)



 




4660.120302 0.293425112

4799.626058 0.302209111

4819.705582 0.303473421

4929.605314 0.310393273

5172.634394 0.325695632

5543.803024 0.349066316

6029.87563 0.379671944

6609.9422 0.416195915

7256.645575 0.456915681

7937.975207 0.499815694

8619.477293 0.542726566

9266.685814 0.58347814

9847.554689 0.620052628

10334.67111 0.65072398

10707.05424 0.674171135

10951.38867 0.689555685

11062.60244 0.696558275

11081.43711 0.697744203

11219.83501 0.706458446

11461.15637 0.721653278

11784.24428 0.741996553

12165.26733 0.765987721

12579.58949 0.792075573

13003.50374 0.818767391

13415.69597 0.844721132

13798.34694 0.868814802

14137.82498 0.890190083

14424.96743 0.908270047

14654.98681 0.922753249

14827.06765 0.933588342

14943.73768 0.940934486

15010.1041 0.945113258

15033.04196 0.946557543

15045.67397 0.947352919

15085.73161 0.949875154

15145.35832 0.953629558

15217.4137 0.958166535

15295.7548 0.963099293

15375.38136 0.96811299

15452.46552 0.972966604

15524.29178 0.977489155

15589.13464 0.981571994

15646.09938 0.985158787

15694.94846 0.988234576

15735.93081 0.990815035

15769.62641 0.992936683

15796.81368 0.994648534

15818.36315 0.996005399

15835.15736 0.997062849
15848.03525 0.997873707

15857.75709 0.998485844

15864.98629 0.998941031

15870.2836 0.999274577

15874.11014 0.999515516

15876.83585 0.999687141

15878.751 0.999807728

15880.07864 0.999891323

15880.9869 0.999948512

15881.6002 0.999987129

15881.80462 1

801960.6529
Raw Data: Sanitaryware - Spray Machine Glaze Application - PM lb/ton

Runs EO14 (EPN014-Bowl SM 1)

1 2.14E+00

2 1.72E+00

3 2.01E+00

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -1.02

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

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

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 1.95E+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!

1.95E+00

Mean =

Pooled Variance = 4.67E-02

m= number future runs = 3

Term1 6.67E-01

Term2 3.11E-02

Squared root Term2 1.76E-01

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

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

UPL POOLED VARIANCE = 3.18E+00

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

 

     

09000064819365f6 S-SprayMachine-PM 12/1/2014
Raw Data: Sanitaryware - Robot Glaze Application - PM lb/ton

Runs SR1 (EPN 104-1P) 1 1.68E+00 2 1.17E+00 3 1.10E+00 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 3.00 3.00

Kurtosis #DIV/0! #DIV/0!

SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 1.32E+00

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

Skewness 1.62 1.57

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 SR1 (EPN 104-1P) 1 5.21E-01 2 1.61E-01 3 9.41E-02 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

3

3. Calculate 2.59E-01

0.052842818

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





  

    

09000064819365f6 S-Robot-PM 12/1/2014
3.913358704

0.008833719

445.0023834

7. Calculate

0.41077265

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

3.484 z value from "lognormal distribution 99% "

a- Calculate 1.330015688

b- Calculate 1.768941731

c- Calculate 0.054263921

d-calculate 0.018312364

e- Calculate 0.761256

f- Calculate UPL 2.21E+00

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 "lognormal distribution 99% "

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





  

    

  

09000064819365f6 S-Robot-PM 12/1/2014
∆z z

0.101 445.0024 0.410773

-5 =f(a) 8796.683

1 -4.899 8005.405

2 -4.798 7265.963

3 -4.697 6576.149

4 -4.596 5933.801

5 -4.495 5336.805

6 -4.394 4783.09

7 -4.293 4270.633

8 -4.192 3797.456

9 -4.091 3361.627

10 -3.99 2961.262

11 -3.889 2594.52

12 -3.788 2259.609

13 -3.687 1954.779

14 -3.586 1678.33

15 -3.485 1428.606

16 -3.384 1203.998

17 -3.283 1002.941

18 -3.182 823.9184

19 -3.081 665.4583

20 -2.98 526.135

21 -2.879 404.5689

22 -2.778 299.4263

23 -2.677 209.4195

24 -2.576 133.307

25 -2.475 69.89296

26 -2.374 18.02778

27 -2.273 -23.3922

28 -2.172 -55.4248

29 -2.071 -79.0816

30 -1.97 -95.3283

31 -1.869 -105.085

32 -1.768 -109.224

33 -1.667 -108.575

34 -1.566 -103.918

35 -1.465 -95.9904

36 -1.364 -85.4806

37 -1.263 -73.0327

38 -1.162 -59.2445

39 -1.061 -44.6678

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -29.8082

41 -0.859 -15.1256

42 -0.758 -1.03369

43 -0.657 12.09978

44 -0.556 23.95301

45 -0.455 34.25024

46 -0.354 42.76169

47 -0.253 49.30358

48 -0.152 53.7381

49 -0.051 55.97348

50 0.05 55.9639

51 0.151 53.70957

52 0.252 49.25668

53 0.353 42.6974

54 0.454 34.16993

55 0.555 23.85844

56 0.656 11.99309

57 0.757 -1.14994

58 0.858 -15.2485

59 0.959 -29.9344

60 1.06 -44.7936

61 1.161 -59.3658

62 1.262 -73.145

63 1.363 -85.579

64 1.464 -96.0696

65 1.565 -103.973

66 1.666 -108.598

67 1.767 -109.21

68 1.868 -105.027

69 1.969 -95.2188

70 2.07 -78.913

71 2.171 -55.1892

72 2.272 -23.0811

73 2.373 18.42318

74 2.474 70.38183

75 2.575 133.8989

76 2.676 210.1245

77 2.777 300.2546

78 2.878 405.5313

79 2.979 527.2427

80 3.08 666.7228

81 3.181 825.3517

82 3.282 1004.555

83 3.383 1205.806

84 3.484 1430.621

85 3.585 1680.566

86 3.686 1957.249

87 3.787 2262.327

88 3.888 2597.502
89 3.989 2964.522

90 4.09 3365.181

91 4.191 3801.318

92 4.292 4274.821

93 4.393 4787.621

94 4.494 5341.695

95 4.595 5939.068

96 4.696 6581.81

97 4.797 7272.037

98 4.898 8011.911

99 4.999 8803.639
distribution of Z

Normal 

distribution

1.48672E-06 0.013078201

2.45094E-06 0.019620766

3.9995E-06 0.029060239

6.46025E-06 0.042483545

1.03291E-05 0.061290758

1.63472E-05 0.087242068

2.56093E-05 0.122491391

3.97118E-05 0.169594361

6.09553E-05 0.231474862

9.26132E-05 0.311331014

0.000139285 0.412459327

0.000207351 0.537975608

0.000305546 0.690414156

0.000445674 0.871194011

0.000643469 1.079953604

0.000919619 1.313773542

0.001300942 1.56633144

0.001821704 1.827061505

0.002525035 2.080422425

0.003464389 2.305406221

0.004704958 2.475442692

0.006324913 2.558862722

0.008416337 2.520072443

0.011085658 2.321553371

0.014453386 1.926737401

0.018652949 1.303709889

0.023828414 0.429573486

0.030130931 -0.704829979

0.03771375 -2.09027684

0.046725789 -3.695148895

0.057303789 -5.462671206

0.069563239 -7.310024956

0.083588399 -9.129880838

0.099421884 -10.79472745

0.117054396 -12.16410983

0.136415346 -13.09456353

0.157365126 -13.45166219

0.179689839 -13.1232363

0.203099244 -12.03252038

0.227228529 -10.14979695

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -7.501073506

0.275855243 -4.172479605

0.299326439 -0.309410026

0.321498297 3.890057793

0.341807853 8.18732786

0.359712192 12.32023057

0.37471238 16.02333601

0.386376486 19.04974229

0.394360216 21.19216922

0.398423793 22.30116466

0.398443914 22.29847568

0.394419966 21.18412711

0.386474058 19.03642773

0.374844865 16.00490254

0.359875719 12.29692901

0.34199778 8.159532851

0.32170943 3.858290395

0.299553264 -0.344469529

0.276092166 -4.209993231

0.25188591 -7.540064044

0.227469632 -10.18918348

0.203335281 -12.07116845

0.17991684 -13.16001836

0.157579839 -13.48552206

0.136615273 -13.12457851

0.117237788 -12.18954394

0.099587708 -10.81507064

0.083736272 -9.144870707

0.069693339 -7.319649489

0.05741676 -5.467153308

0.046822635 -3.694915216

0.037795734 -2.08591484

0.030199481 -0.697037982

0.023885038 0.44003835

0.018699163 1.316081305

0.014490659 1.940283374

0.011115369 2.335610987

0.008439746 2.534072515

0.006343145 2.572344056

0.004718997 2.488056719

0.003475077 2.316913432

0.002533081 2.090682518

0.001827693 1.836019268

0.001305351 1.574000364

0.000922829 1.320218888

0.00064578 1.085276294

0.00044732 0.875516308

0.000306705 0.693867728

0.000208159 0.540692188
0.000139842 0.414563943

9.29928E-05 0.312937549

6.12113E-05 0.232683571

3.98826E-05 0.170490958

2.5722E-05 0.123147285

1.64209E-05 0.087715363

1.03767E-05 0.061627722

6.49066E-06 0.042720286

4.01874E-06 0.029224401

2.46298E-06 0.019733138

1.49417E-06 0.013154142
Absolute values

0.013078201 0.000660449

0.019620766 0.001981697

0.029060239 0.002935084

0.042483545 0.004290838

0.061290758 0.006190367

0.087242068 0.008811449

0.122491391 0.01237163

0.169594361 0.01712903

0.231474862 0.023378961

0.311331014 0.031444432

0.412459327 0.041658392

0.537975608 0.054335536

0.690414156 0.06973183

0.871194011 0.087990595

1.079953604 0.109075314

1.313773542 0.132691128

1.56633144 0.158199475

1.827061505 0.184533212

2.080422425 0.210122665

2.305406221 0.232846028

2.475442692 0.250019712

2.558862722 0.258445135

2.520072443 0.254527317

2.321553371 0.23447689

1.926737401 0.194600478

1.303709889 0.131674699

0.429573486 0.043386922

0.704829979 0.071187828

2.09027684 0.211117961

3.695148895 0.373210038

5.462671206 0.551729792

7.310024956 0.738312521

9.129880838 0.922117965

10.79472745 1.090267472

12.16410983 1.228575092

13.09456353 1.322550917

13.45166219 1.358617881

13.1232363 1.325446867

12.03252038 1.215284559

10.14979695 1.025129492

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
7.501073506 0.757608424

4.172479605 0.42142044

0.309410026 0.031250413

3.890057793 0.392895837

8.18732786 0.826920114

12.32023057 1.244343288

16.02333601 1.618356937

19.04974229 1.924023971

21.19216922 2.140409091

22.30116466 2.252417631

22.29847568 2.252146043

21.18412711 2.139596839

19.03642773 1.922679201

16.00490254 1.616495157

12.29692901 1.24198983

8.159532851 0.824112818

3.858290395 0.38968733

0.344469529 0.034791422

4.209993231 0.425209316

7.540064044 0.761546468

10.18918348 1.029107531

12.07116845 1.219188014

13.16001836 1.329161854

13.48552206 1.362037728

13.12457851 1.325582429

12.18954394 1.231143938

10.81507064 1.092322135

9.144870707 0.923631941

7.319649489 0.739284598

5.467153308 0.552182484

3.694915216 0.373186437

2.08591484 0.210677399

0.697037982 0.070400836

0.44003835 0.044443873

1.316081305 0.132924212

1.940283374 0.195968621

2.335610987 0.23589671

2.534072515 0.255941324

2.572344056 0.25980675

2.488056719 0.251293729

2.316913432 0.234008257

2.090682518 0.211158934

1.836019268 0.185437946

1.574000364 0.158974037

1.320218888 0.133342108

1.085276294 0.109612906

0.875516308 0.088427147

0.693867728 0.070080641

0.540692188 0.054609911
0.414563943 0.041870958

0.312937549 0.031606692

0.232683571 0.023501041

0.170490958 0.017219587

0.123147285 0.012437876

0.087715363 0.008859252

0.061627722 0.0062244

0.042720286 0.004314749

0.029224401 0.002951665

0.019733138 0.001993047

0.013154142 0.000664284
Cumulative normalization

0.000660449 1.27894E-05

0.002642147 5.11643E-05

0.005577231 0.000108001

0.009868069 0.000191092

0.016058435 0.000310966

0.024869884 0.000481597

0.037241515 0.00072117

0.054370545 0.001052868

0.077749506 0.001505594

0.109193939 0.002114505

0.150852331 0.002921206

0.205187867 0.003973395

0.274919697 0.005323729

0.362910292 0.007027638

0.471985606 0.009139845

0.604676734 0.011709365

0.762876209 0.014772846

0.947409421 0.018346271

1.157532086 0.022415227

1.390378114 0.026924213

1.640397826 0.031765762

1.898842961 0.036770467

2.153370278 0.041699305

2.387847168 0.046239872

2.582447646 0.050008246

2.714122344 0.052558083

2.757509267 0.053398257

2.828697094 0.054776785

3.039815055 0.058865015

3.413025094 0.066092104

3.964754885 0.07677617

4.703067406 0.091073348

5.625185371 0.108929858

6.715452843 0.130042527

7.944027935 0.153833479

9.266578852 0.17944424

10.62519673 0.205753427

11.9506436 0.231420268

13.16592816 0.254953853

14.19105765 0.274805147

( f x ) ( f b)



 




14.94866607 0.289475984

15.37008651 0.297636652

15.40133693 0.298241805

15.79423276 0.305850104

16.62115288 0.321863139

17.86549617 0.345959436

19.4838531 0.377298384

21.40787707 0.414556473

23.54828616 0.456004788

25.8007038 0.499622112

28.05284984 0.543234177

30.19244668 0.584666763

32.11512588 0.621898811

33.73162104 0.653201706

34.97361087 0.677252429

35.79772368 0.693211102

36.18741101 0.700757268

36.22220244 0.701430993

36.64741175 0.709665031

37.40895822 0.724412127

38.43806575 0.744340455

39.65725377 0.767949628

40.98641562 0.793688408

42.34845335 0.820063819

43.67403578 0.845733286

44.90517971 0.869573982

45.99750185 0.890726439

46.92113379 0.908612267

47.66041839 0.922928269

48.21260087 0.9336211

48.58578731 0.940847733

48.79646471 0.944927431

48.86686554 0.94629072

48.91130942 0.947151361

49.04423363 0.949725394

49.24020225 0.953520262

49.47609896 0.958088324

49.73204028 0.963044543

49.99184703 0.968075615

50.24314076 0.972941835

50.47714902 0.977473328

50.68830795 0.981562351

50.8737459 0.985153295

51.03271994 0.988231775

51.16606204 0.990813901

51.27567495 0.992936519

51.3641021 0.994648881

51.43418274 0.99600597

51.48879265 0.997063473
51.53066361 0.99787429

51.5622703 0.998486343

51.58577134 0.998941433

51.60299093 0.999274885

51.6154288 0.99951574

51.62428805 0.999687297

51.63051245 0.99980783

51.6348272 0.999891384

51.63777887 0.999948542

51.63977191 0.999987136

51.6404362 1

2607.000507
Raw Data

Runs

1

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Testing Normality Raw Data

Sample Size 0.00

Kurtosis #DIV/0!

SE Kurtosis 0.00

Result Kurtosis #DIV/0!

Skewness #DIV/0!

SE Skewness 0.00 Result Raw Data

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

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

Total Number of sources = 0

n =Total # test runs = 0

-1

This step is for QC only:

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

#DIV/0!

Mean =

Pooled Variance = #DIV/0!

m= number future runs = 3

Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0!

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

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

UPL POOLED VARIANCE = #DIV/0!

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

09000064819365f6 UPL Pooled Template 12/1/2014
Raw Data

Runs 123456789

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Testing Normality Raw Data Logtransformed Data

Sample Size 0.00 0.00

Kurtosis #DIV/0! #DIV/0!

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

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

Skewness #DIV/0! #DIV/0!

SE Skewness 0.00 0.00 Result Raw Data Result Log Data

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

1. Take LN( ) of Raw Data

Runs 123456789

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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





  

    

09000064819365f6 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 "lognormal distribution 99% "

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 "lognormal distribution 99% "

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





  

    

  

09000064819365f6 Lognormal Template 12/1/2014
∆z z

0.101 23.80249 1.377354

-5 =f(a) 390.0648

1 -4.899 354.4355

2 -4.798 321.1779

3 -4.697 290.1895

4 -4.596 261.37

5 -4.495 234.6213

6 -4.394 209.8473

7 -4.293 186.9542

8 -4.192 165.8504

9 -4.091 146.4462

10 -3.99 128.6545

11 -3.889 112.3899

12 -3.788 97.56945

13 -3.687 84.11235

14 -3.586 71.93989

15 -3.485 60.97555

16 -3.384 51.14496

17 -3.283 42.37594

18 -3.182 34.59846

19 -3.081 27.74465

20 -2.98 21.74883

21 -2.879 16.54744

22 -2.778 12.07914

23 -2.677 8.284705

24 -2.576 5.107107

25 -2.475 2.491468

26 -2.374 0.38508

27 -2.273 -1.2626

28 -2.172 -2.49996

29 -2.071 -3.37321

30 -1.97 -3.92639

31 -1.869 -4.20141

32 -1.768 -4.23797

33 -1.667 -4.07363

34 -1.566 -3.74379

35 -1.465 -3.28168

36 -1.364 -2.71835

37 -1.263 -2.0827

38 -1.162 -1.40147

39 -1.061 -0.69922

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 0.001637

41 -0.859 0.68087

42 -0.758 1.320402

43 -0.657 1.904325

44 -0.556 2.418893

45 -0.455 2.852526

46 -0.354 3.19581

47 -0.253 3.441494

48 -0.152 3.584492

49 -0.051 3.621883

50 0.05 3.55291

51 0.151 3.378983

52 0.252 3.103674

53 0.353 2.73272

54 0.454 2.274026

55 0.555 1.737657

56 0.656 1.135845

57 0.757 0.482988

58 0.858 -0.20435

59 0.959 -0.90745

60 1.06 -1.60542

61 1.161 -2.2752

62 1.262 -2.89157

63 1.363 -3.42715

64 1.464 -3.85238

65 1.565 -4.13556

66 1.666 -4.2428

67 1.767 -4.13806

68 1.868 -3.78313

69 1.969 -3.13764

70 2.07 -2.15904

71 2.171 -0.80265

72 2.272 0.978414

73 2.373 3.233181

74 2.474 6.012847

75 2.575 9.370774

76 2.676 13.36249

77 2.777 18.04568

78 2.878 23.4802

79 2.979 29.72808

80 3.08 36.8535

81 3.181 44.92281

82 3.282 54.00452

83 3.383 64.16932

84 3.484 75.49004

85 3.585 88.04169

86 3.686 101.9015

87 3.787 117.1487

88 3.888 133.8648
89 3.989 152.1336

90 4.09 172.0408

91 4.191 193.6745

92 4.292 217.1248

93 4.393 242.484

94 4.494 269.8467

95 4.595 299.3094

96 4.696 330.971

97 4.797 364.9326

98 4.898 401.2971

99 4.999 440.17
distribution of Z

Normal 

distribution

1.48672E-06 0.000579917

2.45094E-06 0.0008687

3.9995E-06 0.001284552

6.46025E-06 0.001874696

1.03291E-05 0.002699714

1.63472E-05 0.003835413

2.56093E-05 0.005374035

3.97118E-05 0.007424283

6.09553E-05 0.010109452

9.26132E-05 0.013562852

0.000139285 0.017919633

0.000207351 0.023304116

0.000305546 0.029811945

0.000445674 0.037486681

0.000643469 0.0462911

0.000919619 0.056074278

0.001300942 0.066536639

0.001821704 0.077196417

0.002525035 0.087362307

0.003464389 0.09611827

0.004704958 0.102327303

0.006324913 0.104661137

0.008416337 0.101662098

0.011085658 0.091841411

0.014453386 0.073814986

0.018652949 0.046473228

0.023828414 0.00917585

0.030130931 -0.03804338

0.03771375 -0.094282831

0.046725789 -0.1576157

0.057303789 -0.224997187

0.069563239 -0.292263436

0.083588399 -0.354244929

0.099421884 -0.405008262

0.117054396 -0.438227575

0.136415346 -0.447671321

0.157365126 -0.427773224

0.179689839 -0.374240219

0.203099244 -0.284637455

0.227228529 -0.158883344

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 0.000411873

0.275855243 0.18782147

0.299326439 0.395231257

0.321498297 0.612237125

0.341807853 0.8267965

0.359712192 1.026088455

0.37471238 1.197509618

0.386376486 1.329712319

0.394360216 1.413580947

0.398423793 1.443044227

0.398443914 1.415635461

0.394419966 1.33273831

0.386474058 1.199489353

0.374844865 1.024346204

0.359875719 0.818366595

0.34199778 0.594274668

0.32170943 0.365412073

0.299553264 0.144680625

0.276092166 -0.056420378

0.25188591 -0.228574626

0.227469632 -0.365184292

0.203335281 -0.462628185

0.17991684 -0.520241996

0.157579839 -0.540049159

0.136615273 -0.526293985

0.117237788 -0.484843542

0.099587708 -0.422530427

0.083736272 -0.346505441

0.069693339 -0.263658786

0.05741676 -0.180152901

0.046822635 -0.101092161

0.037795734 -0.030336761

0.030199481 0.029547598

0.023885038 0.07722465

0.018699163 0.112435203

0.014490659 0.135788687

0.011115369 0.148528978

0.008439746 0.152300952

0.006343145 0.148938347

0.004718997 0.140286733

0.003475077 0.128068772

0.002533081 0.113793105

0.001827693 0.098703708

0.001305351 0.083763503

0.000922829 0.0696644

0.00064578 0.0568556

0.00044732 0.045582551

0.000306705 0.035930124

0.000208159 0.027865109
0.000139842 0.021274633

9.29928E-05 0.015998559

6.12113E-05 0.011855066

3.98826E-05 0.008659501

2.5722E-05 0.00623718

1.64209E-05 0.004431121

1.03767E-05 0.003105833

6.49066E-06 0.00214822

4.01874E-06 0.001466568

2.46298E-06 0.000988385

1.49417E-06 0.000657689
Absolute values

0.000579917 2.92858E-05

0.0008687 8.77387E-05

0.001284552 0.00012974

0.001874696 0.000189344

0.002699714 0.000272671

0.003835413 0.000387377

0.005374035 0.000542778

0.007424283 0.000749853

0.010109452 0.001021055

0.013562852 0.001369848

0.017919633 0.001809883

0.023304116 0.002353716

0.029811945 0.003011006

0.037486681 0.003786155

0.0462911 0.004675401

0.056074278 0.005663502

0.066536639 0.006720201

0.077196417 0.007796838

0.087362307 0.008823593

0.09611827 0.009707945

0.102327303 0.010335058

0.104661137 0.010570775

0.101662098 0.010267872

0.091841411 0.009275983

0.073814986 0.007455314

0.046473228 0.004693796

0.00917585 0.000926761

0.03804338 0.003842381

0.094282831 0.009522566

0.1576157 0.015919186

0.224997187 0.022724716

0.292263436 0.029518607

0.354244929 0.035778738

0.405008262 0.040905835

0.438227575 0.044260985

0.447671321 0.045214803

0.427773224 0.043205096

0.374240219 0.037798262

0.284637455 0.028748383

0.158883344 0.016047218

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.000411873 4.15991E-05

0.18782147 0.018969969

0.395231257 0.039918357

0.612237125 0.06183595

0.8267965 0.083506447

1.026088455 0.103634934

1.197509618 0.120948471

1.329712319 0.134300944

1.413580947 0.142771676

1.443044227 0.145747467

1.415635461 0.142979182

1.33273831 0.134606569

1.199489353 0.121148425

1.024346204 0.103458967

0.818366595 0.082655026

0.594274668 0.060021741

0.365412073 0.036906619

0.144680625 0.014612743

0.056420378 0.005698458

0.228574626 0.023086037

0.365184292 0.036883613

0.462628185 0.046725447

0.520241996 0.052544442

0.540049159 0.054544965

0.526293985 0.053155692

0.484843542 0.048969198

0.422530427 0.042675573

0.346505441 0.03499705

0.263658786 0.026629537

0.180152901 0.018195443

0.101092161 0.010210308

0.030336761 0.003064013

0.029547598 0.002984307

0.07722465 0.00779969

0.112435203 0.011355955

0.135788687 0.013714657

0.148528978 0.015001427

0.152300952 0.015382396

0.148938347 0.015042773

0.140286733 0.01416896

0.128068772 0.012934946

0.113793105 0.011493104

0.098703708 0.009969074

0.083763503 0.008460114

0.0696644 0.007036104

0.0568556 0.005742416

0.045582551 0.004603838

0.035930124 0.003628943

0.027865109 0.002814376
0.021274633 0.002148738

0.015998559 0.001615854

0.011855066 0.001197362

0.008659501 0.00087461

0.00623718 0.000629955

0.004431121 0.000447543

0.003105833 0.000313689

0.00214822 0.00021697

0.001466568 0.000148123

0.000988385 9.98269E-05

0.000657689 3.32133E-05
Cumulative normalization

2.92858E-05 1.10037E-05

0.000117025 4.39703E-05

0.000246764 9.27182E-05

0.000436108 0.000163862

0.00070878 0.000266314

0.001096156 0.000411865

0.001638934 0.000615806

0.002388786 0.000897553

0.003409841 0.0012812

0.004779689 0.001795901

0.006589572 0.002475939

0.008943288 0.003360314

0.011954294 0.004491657

0.015740449 0.005914251

0.02041585 0.007670967

0.026079352 0.009798948

0.032799553 0.012323968

0.040596391 0.01525352

0.049419984 0.01856886

0.059127929 0.022216484

0.069462986 0.026099735

0.080033761 0.030071555

0.090301633 0.033929562

0.099577616 0.037414882

0.107032929 0.04021611

0.111726725 0.041979738

0.112653486 0.042327955

0.116495868 0.043771676

0.126018434 0.047349645

0.141937619 0.053331054

0.164662335 0.061869545

0.194180942 0.072960745

0.22995968 0.0864041

0.270865514 0.101773889

0.3151265 0.118404328

0.360341303 0.135393152

0.403546399 0.151626856

0.441344661 0.165829018

0.470093044 0.176630817

0.486140262 0.182660332

( f x ) ( f b)



 




0.486181861 0.182675963

0.505151829 0.18980366

0.545070186 0.204802419

0.606906136 0.228036403

0.690412582 0.25941277

0.794047516 0.298352131

0.914995988 0.34379681

1.049296932 0.394258492

1.192068608 0.447902931

1.337816075 0.502665482

1.480795256 0.556387888

1.615401825 0.606964404

1.73655025 0.652484212

1.840009217 0.691357457

1.922664243 0.722413915

1.982685984 0.744966236

2.019592604 0.758833376

2.034205347 0.764323908

2.039903805 0.766465023

2.062989842 0.775139275

2.099873456 0.788997772

2.146598902 0.806554198

2.199143344 0.826297029

2.253688309 0.846791529

2.306844002 0.866764029

2.355813199 0.885163513

2.398488772 0.901198257

2.433485822 0.914347903

2.460115359 0.924353576

2.478310802 0.931190256

2.488521111 0.935026635

2.491585124 0.936177895

2.494569431 0.937299206

2.502369121 0.940229829

2.513725076 0.944496669

2.527439734 0.949649758

2.54244116 0.955286332

2.557823556 0.96106605

2.572866329 0.966718159

2.58703529 0.972041945

2.599970236 0.976902069

2.611463339 0.98122044

2.621432414 0.984966179

2.629892527 0.988144947

2.636928632 0.990788664

2.642671047 0.992946295

2.647274885 0.994676122

2.650903828 0.996039646

2.653718204 0.997097108
2.655866942 0.997904466

2.657482796 0.9985116

2.658680158 0.998961492

2.659554767 0.999290114

2.660184722 0.999526811

2.660632266 0.999694969

2.660945955 0.999812833

2.661162925 0.999894357

2.661311048 0.999950012

2.661410875 0.999987521

2.661444089 1

132.8984354
Raw Data

Runs

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

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

Sample size 0

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY

Testing Normality Raw Data Logtransformed

Sample Size 0.00 0.00

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

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

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

Skewness #DIV/0! #DIV/0!

SE Skewness 0.00 0.00 Result Raw Data Result Log Data

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

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n = 3 ONLY

Average #DIV/0!

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

1

2

3
4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 0.00

Kurtosis -3.00

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness #DIV/0!

SE Skewness 0.00 Result Raw Data

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

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

kurtosis= #DIV/0! depending on sample size

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

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

Total Number of sources = 0

n =Total # test runs = 0

1

N

i df n

 

    

1

N

i

i

n





N

1

N

i

i

n




-1

This step is for QC only:

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

Mean = #DIV/0!

Pooled Variance = #DIV/0!

m= number future runs = 3

Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0!

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

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

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

0

#NUM!

Current Confidence Level is #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 i

i

df n



      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 
Conclusion

interval of differences for t-values used 0.103

3.982929999

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

#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

Calculations Calculations

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 ,

 

x

I a b 

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )



( )

w a

e w





a w  
Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

Result #DIV/0!

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



1 1 1 1  1 2   1 3  4 1   1

( ) , , ,

n n n n n n n n

P t I I I

             

        
4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

Corrected probability is::

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

3 4 3/2

2

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

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

#NUM! e.1.7

Calculations Calculations

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



 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
#NUM! Result3:

3 4 1 1

,

n

I

 

   

 
#NUM!

0

3 4 1 1

,

24 2 2

u

n

I

 

   

  

 

1 1

2 2

5 1

2 2



 








a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

#NUM! f.1.7 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
#NUM! Result4: #NUM!

a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.7 #NUM! h.1.7

h.1.) Calculate 

Calculations Calculations

g.1.) Calculate 0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
Result5: #NUM! Result6:

#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

3 1

2 6

w a

    

 
#NUM!
#NUM!

0.103

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

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

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

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

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

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

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

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

1) Start with initial t-statistic from Template

2) Determine interval of differences for t-values

2) Candidate values for t-statistic

3) Corrected p-values

Acutal significance level with t-value

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

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 