Subcategory Pollutant

No. of 

Sources 

Industryw

ide

No. of 

Sources 

with Data

Determination 

of No. of 

Sources in 

MACT Floor

No. of 

Sources 

in MACT 

Floor (N)

No. Data 

Points in 

MACT 

Floor (n)

Average 

of Top 

Sources

UPL Unit of Measure

All Kilns PM 225 31 12% Sources 

Industrywide 27 148 0.041 0.16 lb/ton

Equivalent PM 225 31 12% Sources 

Industrywide 27 148 0.011 0.040 gr/dscf @ 7% O2

Large Kilns Mercury 122 19 12% Sources with 

Data 3 9 1.6E-05 2.2E-05 lb/ton

Equivalent Mercury 122 19 12% Sources with 

Data 3 9 14 29 μg/dscm @ 7% O2

Equivalent Mercury 122 19 12% Sources with 

Data 3 9 1.7E-04 2.7E-04 lb/hr

Small Kilns Mercury 103 2

12% Sources with 

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

Equivalent Mercury 103 2

12% Sources with 

Data 1 3 62 70 μg/dscm @ 7% O2

Equivalent Mercury 103 2

12% Sources with 

Data 1 3 0.0010 0.0011 lb/hr
Data 

Distribution

lognormal

lognormal

normal

normal

skewed/unknown

lognormal

normal

lognormal
Runs 531 lb/ton 531 conc 531 lb/hr Example 4

1 1.84E-04 6.03E+01 1.04E-03

2 1.69E-04 6.21E+01 9.55E-04

3 1.76E-04 6.32E+01 9.96E-04

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

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

1 3.16E-21 6.01E+00 3.23E-18 #DIV/0!

2 2.91E-21 1.60E-03 2.98E-18 #DIV/0!

3 5.41E-28 3.48E+00 5.53E-25 #DIV/0!

Testing Normality Raw Data Raw Data Raw Data Raw Data

Sample Size 3.00 3.00 3.00 0.00

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

SE Kurtosis 2.83 2.83 2.83 #DIV/0!

Result Kurtosis normal normal normal #DIV/0!

Skewness 0.09 -0.60 0.09 #DIV/0!

SE Skewness 1.22 1.22 1.22 0.00

Result Skewness normal normal normal #DIV/0!

lb/ton 0.075456359 0.488415014 0.075456359 #DIV/0!

Lognormal Normal Lognormal #DIV/0!

1 2 3 4
Example 5 Example 6 Example 7 Example 8 Example 9

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

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

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

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

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

Raw Data Raw Data Raw Data Raw Data Raw Data

0.00 0.00 0.00 0.00 0.00

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

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

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

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

0.00 0.00 0.00 0.00 0.00

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

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

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

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

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

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

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

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

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

Raw Data Raw Data Raw Data Raw Data Raw Data

0.00 0.00 0.00 0.00 0.00

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

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

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

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

0.00 0.00 0.00 0.00 0.00

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

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

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

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

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

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

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

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

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

Raw Data Raw Data Raw Data Raw Data Raw Data

0.00 0.00 0.00 0.00 0.00

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

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

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

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

0.00 0.00 0.00 0.00 0.00

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

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

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

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

-8.60E+00 4.10E+00

-8.69E+00 4.13E+00

-8.64E+00 4.15E+00

#DIV/0! #DIV/0! -8.64E+00 4.12E+00

#DIV/0! #DIV/0! 4.21E-02 2.39E-02

#DIV/0! #DIV/0! 3.19E-06 4.19E-07

#DIV/0! #DIV/0! 3.11E-06 1.36E-10

#DIV/0! #DIV/0! 5.67E-15 2.35E-07

Raw Data Raw Data Log Data Log Data

0.00 0.00 3.00 3.00

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

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

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

#DIV/0! #DIV/0! 0.03 -0.63

0.00 0.00 1.22 1.22

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

#DIV/0! #DIV/0! 0.023935519 0.513997038

#DIV/0! #DIV/0!

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

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

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

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

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

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

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

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

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

Log Data Log Data Log Data Log Data Log Data

3.00 0.00 0.00 0.00 0.00

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

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

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

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

1.22 0.00 0.00 0.00 0.00

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

0.023935519 #NUM! #NUM! #NUM! #NUM!
Example 8 Example 9 Example 10 Example 11 Example 12

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

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

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

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

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

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

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

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

Log Data Log Data Log Data Log Data Log Data

0.00 0.00 0.00 0.00 0.00

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

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

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

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

0.00 0.00 0.00 0.00 0.00

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

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

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

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

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

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

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

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

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

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

Log Data Log Data Log Data Log Data Log Data

0.00 0.00 0.00 0.00 0.00

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

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

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

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

0.00 0.00 0.00 0.00 0.00

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

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

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

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

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

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

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

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

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

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

Log Data Log Data Log Data Log Data

0.00 0.00 0.00 0.00

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

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

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

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

0.00 0.00 0.00 0.00

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

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

Runs 514 686 687 538 599 508 503 600 604 568 688 603 558 610 543 606 502 544 654 653 547 548 506 566 526 527 567

1 2.99E-03 8.07E-03 0.008071368 1.10E-02 8.11E-03 0.00985 0.014691 0.029395 0.011364 0.026644 0.044467 0.025719 0.063886 0.043787 0.027694 0.08631 0.003761 0.01295 0.03875 0.045215 0.08865 0.08865 0.087039 0.113399 0.05275 0.040802 0.065536

2 9.53E-03 1.02E-02 0.010195412 1.26E-02 1.53E-02 0.013537 0.013642 0.020209 0.023485 0.026077 0.031368 0.049924 0.064942 0.043907 0.045821 0.004523 0.005575 0.021133 0.05 0.044636 0.050124 0.050124 0.076493 0.041379 0.069585 0.035075 0.10717

3 5.61E-03 6.37E-03 0.006372133 6.74E-03 4.53E-03 0.009217 0.013642 0.0459296 0.020455 0.014172 0.032403 0.022693 0.063886 0.047333 0.091641 0.02619 0.01585 0.033007 0.04625 0.052993 0.090307 0.090307 0.060864 0.023596 0.075196 0.037223 0.106399

4 0.008756567 0.014378 0.0041162 0.04589 0.006585 0.028407 0.089596 0.052179 0.058712 0.042327 0.042327 0.023585 0.157302 0.110354 0.024479

5 0.004108 0.0041298 0.011616 0.016438 0.005768 0.028911 0.01413 0.076806 0.055327 0.055871 0.041089 0.041089 0.053774 0.107421 0.115804 0.013599

6 0.004300271 0.008901 0.0041838 0.022559 0.018836 0.003212 0.031548 0.010195 0.063728 0.036207 0.052083 0.004505 0.004505 0.034906 0.019087 0.091281 0.004932

7 0.027478 0.072889 0.012142 0.024 0.17612

8 0.021659 0.048889 0.05862 0.212343

9 0.043956 0.043556 0.007607 0.029607 0.148328

10 0.02997

11 0.012615

12 0.116444

13 0.050074

14 0.115111

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 148.00 148.00

Kurtosis 3.67 -0.77

SE Kurtosis 0.40 0.40 Average of Raw Data 4.11E-02

Result Kurtosis non normal normal

Skewness 1.68 -0.32

SE Skewness 0.20 0.20 Result Raw Data Result Log Data

Result Skewness non normal normal non normal normal

1. Take LN( ) of Raw Data

Runs 514 686 687 538 599 508 503 600 604 568 688 603 558 610 543 606 502 544 654 653 547 548 506 566 526 527 567

1 -5.81E+00 -4.82E+00 -4.82E+00 -4.51E+00 -4.81E+00 -4.62E+00 -4.22E+00 -3.53E+00 -4.48E+00 -3.63E+00 -3.11E+00 -3.66E+00 -2.75E+00 -3.13E+00 -3.59E+00 -2.45E+00 -5.58E+00 -4.35E+00 -3.25E+00 -3.10E+00 -2.42E+00 -2.42E+00 -2.44E+00 -2.18E+00 -2.94E+00 -3.20E+00 -2.73E+00

2 -4.65E+00 -4.59E+00 -4.59E+00 -4.37E+00 -4.18E+00 -4.30E+00 -4.29E+00 -3.90E+00 -3.75E+00 -3.65E+00 -3.46E+00 -3.00E+00 -2.73E+00 -3.13E+00 -3.08E+00 -5.40E+00 -5.19E+00 -3.86E+00 -3.00E+00 -3.11E+00 -2.99E+00 -2.99E+00 -2.57E+00 -3.18E+00 -2.67E+00 -3.35E+00 -2.23E+00

3 -5.18E+00 -5.06E+00 -5.06E+00 -5.00E+00 -5.40E+00 -4.69E+00 -4.29E+00 -3.08E+00 -3.89E+00 -4.26E+00 -3.43E+00 -3.79E+00 -2.75E+00 -3.05E+00 -2.39E+00 -3.64E+00 -4.14E+00 -3.41E+00 -3.07E+00 -2.94E+00 -2.40E+00 -2.40E+00 -2.80E+00 -3.75E+00 -2.59E+00 -3.29E+00 -2.24E+00

4 -4.74E+00 -4.24E+00 -5.49E+00 -3.08E+00 -5.02E+00 -3.56E+00 -2.41E+00 -2.95E+00 -2.84E+00 -3.16E+00 -3.16E+00 -3.75E+00 -1.85E+00 -2.20E+00 -3.71E+00

5 -5.49E+00 -5.49E+00 -4.46E+00 -4.11E+00 -5.16E+00 -3.54E+00 -4.26E+00 -2.57E+00 -2.89E+00 -2.88E+00 -3.19E+00 -3.19E+00 -2.92E+00 -2.23E+00 -2.16E+00 -4.30E+00

6 -5.45E+00 -4.72E+00 -5.48E+00 -3.79E+00 -3.97E+00 -5.74E+00 -3.46E+00 -4.59E+00 -2.75E+00 -3.32E+00 -2.95E+00 -5.40E+00 -5.40E+00 -3.36E+00 -3.96E+00 -2.39E+00 -5.31E+00

7 -3.59E+00 -2.62E+00 -4.41E+00 -3.73E+00 -1.74E+00

8 -3.83E+00 -3.02E+00 -2.84E+00 -1.55E+00

9 -3.12E+00 -3.13E+00 -4.88E+00 -3.52E+00 -1.91E+00

10 -3.51E+00

11 -4.37E+00

12 -2.15E+00

13 -2.99E+00

14 -2.16E+00

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

148

3. Calculate -3.63E+00

1.019643371

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

2

4 3 ˆˆˆ 2

2 2

ˆ

233 1

3 1

1

z

e e e

m m e







  

       

0900006481934216 PM-lbton 12/1/2014
121.7260378

9.422141661

14.91914749

7. Calculate

3.66788443

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

3.484 z value from "z-stat "

a- Calculate 0.044251848

b- Calculate 0.001958226

c- Calculate 1.772205938

d-calculate 0.010425784

e- Calculate 0.102931

f- Calculate UPL 0.16

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

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

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

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

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1

 

2z 

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z 1z

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

m e   1 

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

0.101 14.91915 3.667884

-5 =f(a) 171.1451

1 -4.899 154.1468

2 -4.798 138.3634

3 -4.697 123.7394

4 -4.596 110.2202

5 -4.495 97.75249

6 -4.394 86.28422

7 -4.293 75.76462

8 -4.192 66.14412

9 -4.091 57.3744

10 -3.99 49.40837

11 -3.889 42.20021

12 -3.788 35.7053

13 -3.687 29.8803

14 -3.586 24.68308

15 -3.485 20.07277

16 -3.384 16.00972

17 -3.283 12.45555

18 -3.182 9.37308

19 -3.081 6.726409

20 -2.98 4.480855

21 -2.879 2.602983

22 -2.778 1.060595

23 -2.677 -0.17726

24 -2.576 -1.14031

25 -2.475 -1.85701

26 -2.374 -2.35461

27 -2.273 -2.6591

28 -2.172 -2.79524

29 -2.071 -2.78654

30 -1.97 -2.65527

31 -1.869 -2.42246

32 -1.768 -2.10792

33 -1.667 -1.73018

34 -1.566 -1.30657

35 -1.465 -0.85316

36 -1.364 -0.38477

37 -1.263 0.084995

38 -1.162 0.543791

39 -1.061 0.980505

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 1.385267

41 -0.859 1.749446

42 -0.758 2.065654

43 -0.657 2.32774

44 -0.556 2.530794

45 -0.455 2.671148

46 -0.354 2.746374

47 -0.253 2.755283

48 -0.152 2.697926

49 -0.051 2.575596

50 0.05 2.390826

51 0.151 2.147389

52 0.252 1.850297

53 0.353 1.505805

54 0.454 1.121405

55 0.555 0.705833

56 0.656 0.269063

57 0.757 -0.17769

58 0.858 -0.62197

59 0.959 -1.05009

60 1.06 -1.4471

61 1.161 -1.79684

62 1.262 -2.08188

63 1.363 -2.28357

64 1.464 -2.38201

65 1.565 -2.35606

66 1.666 -2.18335

67 1.767 -1.84026

68 1.868 -1.30194

69 1.969 -0.54227

70 2.07 0.466074

71 2.171 1.751672

72 2.272 3.344346

73 2.373 5.275154

74 2.474 7.576397

75 2.575 10.28162

76 2.676 13.42559

77 2.777 17.04434

78 2.878 21.17513

79 2.979 25.85646

80 3.08 31.12807

81 3.181 37.03094

82 3.282 43.6073

83 3.383 50.90061

84 3.484 58.95557

85 3.585 67.81813

86 3.686 77.53547

87 3.787 88.15601

88 3.888 99.72942
89 3.989 112.3066

90 4.09 125.9397

91 4.191 140.6821

92 4.292 156.5884

93 4.393 173.7146

94 4.494 192.1176

95 4.595 211.8559

96 4.696 232.989

97 4.797 255.5777

98 4.898 279.6842

99 4.999 305.3718
distribution of Z

Normal 

distribution

1.48672E-06 0.000254445

2.45094E-06 0.000377804

3.9995E-06 0.000553385

6.46025E-06 0.000799387

1.03291E-05 0.001138474

1.63472E-05 0.001597984

2.56093E-05 0.002209675

3.97118E-05 0.003008747

6.09553E-05 0.004031831

9.26132E-05 0.005313626

0.000139285 0.006881844

0.000207351 0.008750242

0.000305546 0.01090961

0.000445674 0.013316871

0.000643469 0.015882803

0.000919619 0.018459303

0.001300942 0.020827725

0.001821704 0.022690317

0.002525035 0.023667351

0.003464389 0.023302895

0.004704958 0.021082232

0.006324913 0.016463637

0.008416337 0.008926329

0.011085658 -0.001965069

0.014453386 -0.016481283

0.018652949 -0.034638733

0.023828414 -0.056106679

0.030130931 -0.080121292

0.03771375 -0.105419042

0.046725789 -0.130203175

0.057303789 -0.152156836

0.069563239 -0.168514263

0.083588399 -0.176197271

0.099421884 -0.172017858

0.117054396 -0.152939866

0.136415346 -0.11638371

0.157365126 -0.060549568

0.179689839 0.015272756

0.203099244 0.110443481

0.227228529 0.222798658

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 0.348594543

0.275855243 0.482593977

0.299326439 0.618304831

0.321498297 0.748364303

0.341807853 0.865045289

0.359712192 0.960844678

0.37471238 1.029100385

0.386376486 1.064576417

0.394360216 1.063954656

0.398423793 1.026178861

0.398443914 0.95261022

0.394419966 0.84697308

0.386474058 0.715091895

0.374844865 0.564443205

0.359875719 0.403566586

0.34199778 0.241393398

0.32170943 0.086559994

0.299553264 -0.053228079

0.276092166 -0.171722226

0.25188591 -0.264502926

0.227469632 -0.329172129

0.203335281 -0.365360668

0.17991684 -0.374565027

0.157579839 -0.359844169

0.136615273 -0.325418619

0.117237788 -0.276219533

0.099587708 -0.217435206

0.083736272 -0.154096878

0.069693339 -0.090736244

0.05741676 -0.031135345

0.046822635 0.021822793

0.037795734 0.066205731

0.030199481 0.1009975

0.023885038 0.12599725

0.018699163 0.141672277

0.014490659 0.148987379

0.011115369 0.149230376

0.008439746 0.143849904

0.006343145 0.134316918

0.004718997 0.122016541

0.003475077 0.108172443

0.002533081 0.093802365

0.001827693 0.079700779

0.001305351 0.066443182

0.000922829 0.054405917

0.00064578 0.043795621

0.00044732 0.034683159

0.000306705 0.02703792

0.000208159 0.020759531
0.000139842 0.015705153

9.29928E-05 0.011711485

6.12113E-05 0.008611332

3.98826E-05 0.006245153

2.5722E-05 0.00446829

1.64209E-05 0.003154741

1.03767E-05 0.002198357

6.49066E-06 0.001512252

4.01874E-06 0.0010271

2.46298E-06 0.000688855

1.49417E-06 0.000456278
Absolute values

0.000254445 1.28495E-05

0.000377804 3.81582E-05

0.000553385 5.58919E-05

0.000799387 8.07381E-05

0.001138474 0.000114986

0.001597984 0.000161396

0.002209675 0.000223177

0.003008747 0.000303883

0.004031831 0.000407215

0.005313626 0.000536676

0.006881844 0.000695066

0.008750242 0.000883774

0.01090961 0.001101871

0.013316871 0.001345004

0.015882803 0.001604163

0.018459303 0.00186439

0.020827725 0.0021036

0.022690317 0.002291722

0.023667351 0.002390402

0.023302895 0.002353592

0.021082232 0.002129305

0.016463637 0.001662827

0.008926329 0.000901559

0.001965069 0.000198472

0.016481283 0.00166461

0.034638733 0.003498512

0.056106679 0.005666775

0.080121292 0.00809225

0.105419042 0.010647323

0.130203175 0.013150521

0.152156836 0.01536784

0.168514263 0.017019941

0.176197271 0.017795924

0.172017858 0.017373804

0.152939866 0.015446927

0.11638371 0.011754755

0.060549568 0.006115506

0.015272756 0.001542548

0.110443481 0.011154792

0.222798658 0.022502664

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.348594543 0.035208049

0.482593977 0.048741992

0.618304831 0.062448788

0.748364303 0.075584795

0.865045289 0.087369574

0.960844678 0.097045312

1.029100385 0.103939139

1.064576417 0.107522218

1.063954656 0.10745942

1.026178861 0.103644065

0.95261022 0.096213632

0.84697308 0.085544281

0.715091895 0.072224281

0.564443205 0.057008764

0.403566586 0.040760225

0.241393398 0.024380733

0.086559994 0.008742559

0.053228079 0.005376036

0.171722226 0.017343945

0.264502926 0.026714796

0.329172129 0.033246385

0.365360668 0.036901428

0.374565027 0.037831068

0.359844169 0.036344261

0.325418619 0.032867281

0.276219533 0.027898173

0.217435206 0.021960956

0.154096878 0.015563785

0.090736244 0.009164361

0.031135345 0.00314467

0.021822793 0.002204102

0.066205731 0.006686779

0.1009975 0.010200748

0.12599725 0.012725722

0.141672277 0.0143089

0.148987379 0.015047725

0.149230376 0.015072268

0.143849904 0.01452884

0.134316918 0.013566009

0.122016541 0.012323671

0.108172443 0.010925417

0.093802365 0.009474039

0.079700779 0.008049779

0.066443182 0.006710761

0.054405917 0.005494998

0.043795621 0.004423358

0.034683159 0.003502999

0.02703792 0.00273083

0.020759531 0.002096713
0.015705153 0.00158622

0.011711485 0.00118286

0.008611332 0.000869745

0.006245153 0.00063076

0.00446829 0.000451297

0.003154741 0.000318629

0.002198357 0.000222034

0.001512252 0.000152737

0.0010271 0.000103737

0.000688855 6.95744E-05

0.000456278 2.3042E-05
Cumulative normalization

1.28495E-05 6.77666E-06

5.10077E-05 2.69009E-05

0.0001069 5.63776E-05

0.000187638 9.8958E-05

0.000302624 0.0001596

0.00046402 0.000244719

0.000687197 0.00036242

0.000991081 0.000522685

0.001398296 0.000737445

0.001934972 0.001020482

0.002630038 0.001387052

0.003513813 0.001853145

0.004615683 0.002434259

0.005960687 0.003143599

0.00756485 0.003989617

0.00942924 0.004972875

0.01153284 0.00608229

0.013824562 0.007290918

0.016214965 0.008551589

0.018568557 0.009792847

0.020697862 0.010915819

0.02236069 0.011792775

0.023262249 0.012268247

0.023460721 0.012372919

0.025125331 0.013250815

0.028623843 0.015095891

0.034290617 0.018084484

0.042382868 0.022352245

0.053030191 0.027967523

0.066180711 0.034902959

0.081548552 0.043007785

0.098568492 0.051983909

0.116364417 0.061369278

0.13373822 0.070532026

0.149185147 0.078678561

0.160939902 0.084877885

0.167055408 0.088103134

0.168597956 0.088916657

0.179752748 0.094799568

0.202255412 0.106667219

( f x ) ( f b)



 




0.237463461 0.125235546

0.286205453 0.150941522

0.348654241 0.183876307

0.424239036 0.223738873

0.51160861 0.269816599

0.608653922 0.3209972

0.712593061 0.375813528

0.820115279 0.43251953

0.927574699 0.489192414

1.031218764 0.543853123

1.127432397 0.594595105

1.212976678 0.639710192

1.285200959 0.677800462

1.342209723 0.70786624

1.382969948 0.729362722

1.407350681 0.742220845

1.41609324 0.746831572

1.421469276 0.749666833

1.438813221 0.758813833

1.465528017 0.772902915

1.498774402 0.790436682

1.535675829 0.809898078

1.573506897 0.829849756

1.609851158 0.849017308

1.642718439 0.866351141

1.670616612 0.881064322

1.692577567 0.892646282

1.708141352 0.900854446

1.717305713 0.905687626

1.720450382 0.907346089

1.722654485 0.908508508

1.729341263 0.91203504

1.739542011 0.917414799

1.752267733 0.924126201

1.766576633 0.931672553

1.781624358 0.939608553

1.796696626 0.947557497

1.811225467 0.955219843

1.824791475 0.962374403

1.837115146 0.968873767

1.848040563 0.974635709

1.857514602 0.97963221

1.86556438 0.983877572

1.872275142 0.987416752

1.877770139 0.990314751

1.882193497 0.99264758

1.885696496 0.994495022

1.888427326 0.995935231

1.890524038 0.997041014
1.892110259 0.997877568

1.893293119 0.998501395

1.894162863 0.998960089

1.894793624 0.999292745

1.895244921 0.999530754

1.89556355 0.999698795

1.895785584 0.999815893

1.895938322 0.999896445

1.896042059 0.999951155

1.896111633 0.999987848

1.896134675 1

93.15042416
Raw Data rank by lb/ton use gr/dscf @ 7% O2 values

Runs 514 686 687 538 599 508 503 600 604 568 688 603 558 610 543 606 502 544 654 653 547 548 506 566 526 527 567

1 0.001024602 0.002138462 0.002138462 0.0278 0.002025429 0.002349 0.003141 0.0085662 0.00417 0.007784 0.015078 0.008084 0.035522 0.011024 0.004633 0.022757 0.000982 0.003353 0.009119 0.009785 0.015886 0.015886 0.021789 0.02445 0.009678 0.008486 0.013187

2 0.003182773 0.002527273 0.002527273 0.0278 0.003772857 0.003132 0.002514 0.0054668 0.007359 0.007784 0.009468 0.013974 0.035522 0.011024 0.007841 0.001008 0.001324 0.005523 0.010615 0.009812 0.011405 0.011405 0.019535 0.008715 0.016809 0.008096 0.021741

3 0.001782597 0.001782051 0.001782051 0.0139 0.001123347 0.00182 0.002547 0.0147517 0.00695 0.004544 0.011308 0.006758 0.035522 0.008554 0.014969 0.006904 0.003734 0.008176 0.009856 0.012184 0.020672 0.020672 0.016154 0.004808 0.015617 0.008034 0.022097

4 0.002494872 0.003282 0.0016079 0.021482 0.003742 0.023167 0.025608 0.012209 0.018215 0.010692 0.010692 0.007174 0.024241 0.023879 0.005673

5 0.000927 0.0014766 0.003375 0.00891 0.003451 0.011705 0.003569 0.021898 0.014 0.016466 0.010239 0.010239 0.016776 0.016555 0.022114 0.003159

6 0.00119472 0.001782 0.001456 0.006155 0.009979 0.002099 0.012437 0.002529 0.018266 0.009165 0.016113 0.00106 0.00106 0.009688 0.00339 0.023712 0.001137

7 0.005505 0.016395 0.003061 0.004026 0.02889

8 0.004526 0.012311 0.009449 0.036601

9 0.005272 0.013003 0.002033 0.00532 0.025076

10 0.003834

11 0.001554

12 0.027525

13 0.011851

14 0.02306

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 148.00 148.00

Kurtosis 0.52 -0.87

SE Kurtosis 0.40 0.40 Average of Raw Data 1.05E-02

Result Kurtosis normal non normal Note: Not ideal to use the Skewed/Unknown template, because the "current confidence level" is 100%, which is a higher confidence level than expected. Performed more statistical tests in R to determine the distribution, and results indicated a lognormal distribution was the best fit for the data.

Skewness 1.04 -0.38

SE Skewness 0.20 0.20 Result Raw Data Result Log Data

Result Skewness non normal normal non normal non normal

1. Take LN( ) of Raw Data

Runs 514 686 687 538 599 508 503 600 604 568 688 603 558 610 543 606 502 544 654 653 547 548 506 566 526 527 567

1 -6.88E+00 -6.15E+00 -6.15E+00 -3.58E+00 -6.20E+00 -6.05E+00 -5.76E+00 -4.76E+00 -5.48E+00 -4.86E+00 -4.19E+00 -4.82E+00 -3.34E+00 -4.51E+00 -5.37E+00 -3.78E+00 -6.93E+00 -5.70E+00 -4.70E+00 -4.63E+00 -4.14E+00 -4.14E+00 -3.83E+00 -3.71E+00 -4.64E+00 -4.77E+00 -4.33E+00

2 -5.75E+00 -5.98E+00 -5.98E+00 -3.58E+00 -5.58E+00 -5.77E+00 -5.99E+00 -5.21E+00 -4.91E+00 -4.86E+00 -4.66E+00 -4.27E+00 -3.34E+00 -4.51E+00 -4.85E+00 -6.90E+00 -6.63E+00 -5.20E+00 -4.55E+00 -4.62E+00 -4.47E+00 -4.47E+00 -3.94E+00 -4.74E+00 -4.09E+00 -4.82E+00 -3.83E+00

3 -6.33E+00 -6.33E+00 -6.33E+00 -4.28E+00 -6.79E+00 -6.31E+00 -5.97E+00 -4.22E+00 -4.97E+00 -5.39E+00 -4.48E+00 -5.00E+00 -3.34E+00 -4.76E+00 -4.20E+00 -4.98E+00 -5.59E+00 -4.81E+00 -4.62E+00 -4.41E+00 -3.88E+00 -3.88E+00 -4.13E+00 -5.34E+00 -4.16E+00 -4.82E+00 -3.81E+00

4 -5.99E+00 -5.72E+00 -6.43E+00 -3.84E+00 -5.59E+00 -3.77E+00 -3.66E+00 -4.41E+00 -4.01E+00 -4.54E+00 -4.54E+00 -4.94E+00 -3.72E+00 -3.73E+00 -5.17E+00

5 -6.98E+00 -6.52E+00 -5.69E+00 -4.72E+00 -5.67E+00 -4.45E+00 -5.64E+00 -3.82E+00 -4.27E+00 -4.11E+00 -4.58E+00 -4.58E+00 -4.09E+00 -4.10E+00 -3.81E+00 -5.76E+00

6 -6.73E+00 -6.33E+00 -6.53E+00 -5.09E+00 -4.61E+00 -6.17E+00 -4.39E+00 -5.98E+00 -4.00E+00 -4.69E+00 -4.13E+00 -6.85E+00 -6.85E+00 -4.64E+00 -5.69E+00 -3.74E+00 -6.78E+00

7 -5.20E+00 -4.11E+00 -5.79E+00 -5.51E+00 -3.54E+00

8 -5.40E+00 -4.40E+00 -4.66E+00 -3.31E+00

9 -5.25E+00 -4.34E+00 -6.20E+00 -5.24E+00 -3.69E+00

10 -5.56E+00

11 -6.47E+00

12 -3.59E+00

13 -4.44E+00

14 -3.77E+00

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

148

3. Calculate -4.95E+00

0.946447098

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

2

4 3 ˆˆˆ 2

2 2

ˆ

233 1

3 1

1

z

e e e

m m e







  

       

0900006481934216 PM-conc 12/1/2014
95.19505418

7.456427425

14.76684513

7. Calculate

3.31763759

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

3.484 z value from "z-stat "

a- Calculate 0.011379046

b- Calculate 0.000129483

c- Calculate 1.576539187

d-calculate 0.009441723

e- Calculate 0.024968

f- Calculate UPL 0.040

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

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

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

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

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z 1z

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )





2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

1

z m m e 

    

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

0.101 14.76685 3.317638

-5 =f(a) 174.533

1 -4.899 157.3918

2 -4.798 141.4653

3 -4.697 126.6982

4 -4.596 113.0362

5 -4.495 100.4266

6 -4.394 88.81752

7 -4.293 78.1586

8 -4.192 68.40058

9 -4.091 59.49545

10 -3.99 51.3964

11 -3.889 44.05788

12 -3.788 37.43554

13 -3.687 31.48627

14 -3.586 26.16816

15 -3.485 21.44056

16 -3.384 17.26402

17 -3.283 13.60032

18 -3.182 10.41246

19 -3.081 7.66468

20 -2.98 5.322424

21 -2.879 3.352374

22 -2.778 1.722431

23 -2.677 0.401725

24 -2.576 -0.6394

25 -2.475 -1.42935

26 -2.374 -1.99534

27 -2.273 -2.36335

28 -2.172 -2.55812

29 -2.071 -2.60317

30 -1.97 -2.52081

31 -1.869 -2.33212

32 -1.768 -2.05695

33 -1.667 -1.71393

34 -1.566 -1.32046

35 -1.465 -0.89272

36 -1.364 -0.44567

37 -1.263 0.006973

38 -1.162 0.45269

39 -1.061 0.880204

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 1.279457

41 -0.859 1.641618

42 -0.758 1.959078

43 -0.657 2.225453

44 -0.556 2.435585

45 -0.455 2.585539

46 -0.354 2.672604

47 -0.253 2.695295

48 -0.152 2.653351

49 -0.051 2.547734

50 0.05 2.380633

51 0.151 2.155459

52 0.252 1.876849

53 0.353 1.550664

54 0.454 1.18399

55 0.555 0.785136

56 0.656 0.363637

57 0.757 -0.06975

58 0.858 -0.50304

59 0.959 -0.92302

60 1.06 -1.31527

61 1.161 -1.66412

62 1.262 -1.95269

63 1.363 -2.16289

64 1.464 -2.27538

65 1.565 -2.2696

66 1.666 -2.12377

67 1.767 -1.8149

68 1.868 -1.31875

69 1.969 -0.60987

70 2.07 0.338411

71 2.171 1.554

72 2.272 3.066022

73 2.373 4.904827

74 2.474 7.101989

75 2.575 9.690307

76 2.676 12.70381

77 2.777 16.17773

78 2.878 20.14856

79 2.979 24.65399

80 3.08 29.73294

81 3.181 35.42555

82 3.282 41.7732

83 3.383 48.81848

84 3.484 56.60521

85 3.585 65.17843

86 3.686 74.58442

87 3.787 84.87066

88 3.888 96.08587
89 3.989 108.28

90 4.09 121.5042

91 4.191 135.8109

92 4.292 151.2537

93 4.393 167.8873

94 4.494 185.768

95 4.595 204.953

96 4.696 225.5007

97 4.797 247.471

98 4.898 270.9248

99 4.999 295.9242
distribution of Z

Normal 

distribution

1.48672E-06 0.000259482

2.45094E-06 0.000385758

3.9995E-06 0.000565791

6.46025E-06 0.000818501

1.03291E-05 0.001167561

1.63472E-05 0.001641698

2.56093E-05 0.002274551

3.97118E-05 0.003103816

6.09553E-05 0.004169375

9.26132E-05 0.005510063

0.000139285 0.007158747

0.000207351 0.009135432

0.000305546 0.011438276

0.000445674 0.014032607

0.000643469 0.016838405

0.000919619 0.019717151

0.001300942 0.022459495

0.001821704 0.024775757

0.002525035 0.026291831

0.003464389 0.026553433

0.004704958 0.02504178

0.006324913 0.021203472

0.008416337 0.014496564

0.011085658 0.004453382

0.014453386 -0.009241426

0.018652949 -0.026661628

0.023828414 -0.047545907

0.030130931 -0.071209917

0.03771375 -0.096476136

0.046725789 -0.121635173

0.057303789 -0.144452185

0.069563239 -0.162230129

0.083588399 -0.171937529

0.099421884 -0.170402355

0.117054396 -0.154565776

0.136415346 -0.12178082

0.157365126 -0.070132299

0.179689839 0.001252934

0.203099244 0.091940952

0.227228529 0.200007356

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 0.321968128

0.275855243 0.452848877

0.299326439 0.586403756

0.321498297 0.715479414

0.341807853 0.832502123

0.359712192 0.930049835

0.37471238 1.001457829

0.386376486 1.041398655

0.394360216 1.046375954

0.398423793 1.015077836

0.398443914 0.948548573

0.394419966 0.85015594

0.386474058 0.725353361

0.374844865 0.581258401

0.359875719 0.426089097

0.34199778 0.268514675

0.32170943 0.116985384

0.299553264 -0.020893351

0.276092166 -0.138884407

0.25188591 -0.232495603

0.227469632 -0.299182933

0.203335281 -0.338373769

0.17991684 -0.35132257

0.157579839 -0.340827901

0.136615273 -0.310851037

0.117237788 -0.266082314

0.099587708 -0.21150142

0.083736272 -0.151972751

0.069693339 -0.091907998

0.05741676 -0.035016841

0.046822635 0.015845302

0.037795734 0.058734586

0.030199481 0.092592277

0.023885038 0.117151971

0.018699163 0.132801241

0.014490659 0.140418941

0.011115369 0.141207497

0.008439746 0.136535971

0.006343145 0.127805262

0.004718997 0.1163421

0.003475077 0.103324255

0.002533081 0.089735775

0.001827693 0.076348598

0.001305351 0.063725265

0.000922829 0.05223693

0.00064578 0.042090954

0.00044732 0.033363095

0.000306705 0.026030285

0.000208159 0.020001096
0.000139842 0.015142066

9.29928E-05 0.011299016

6.12113E-05 0.008313159

3.98826E-05 0.006032388

2.5722E-05 0.004318402

1.64209E-05 0.003050476

1.03767E-05 0.002126728

6.49066E-06 0.001463648

4.01874E-06 0.000994521

2.46298E-06 0.000667281

1.49417E-06 0.000442161
Absolute values

0.000259482 1.31038E-05

0.000385758 3.89615E-05

0.000565791 5.71449E-05

0.000818501 8.26686E-05

0.001167561 0.000117924

0.001641698 0.000165811

0.002274551 0.00022973

0.003103816 0.000313485

0.004169375 0.000421107

0.005510063 0.000556516

0.007158747 0.000723033

0.009135432 0.000922679

0.011438276 0.001155266

0.014032607 0.001417293

0.016838405 0.001700679

0.019717151 0.001991432

0.022459495 0.002268409

0.024775757 0.002502351

0.026291831 0.002655475

0.026553433 0.002681897

0.02504178 0.00252922

0.021203472 0.002141551

0.014496564 0.001464153

0.004453382 0.000449792

0.009241426 0.000933384

0.026661628 0.002692824

0.047545907 0.004802137

0.071209917 0.007192202

0.096476136 0.00974409

0.121635173 0.012285152

0.144452185 0.014589671

0.162230129 0.016385243

0.171937529 0.01736569

0.170402355 0.017210638

0.154565776 0.015611143

0.12178082 0.012299863

0.070132299 0.007083362

0.001252934 0.000126546

0.091940952 0.009286036

0.200007356 0.020200743

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.321968128 0.032518781

0.452848877 0.045737737

0.586403756 0.059226779

0.715479414 0.072263421

0.832502123 0.084082714

0.930049835 0.093935033

1.001457829 0.101147241

1.041398655 0.105181264

1.046375954 0.105683971

1.015077836 0.102522861

0.948548573 0.095803406

0.85015594 0.08586575

0.725353361 0.073260689

0.581258401 0.058707098

0.426089097 0.043034999

0.268514675 0.027119982

0.116985384 0.011815524

0.020893351 0.002110228

0.138884407 0.014027325

0.232495603 0.023482056

0.299182933 0.030217476

0.338373769 0.034175751

0.35132257 0.03548358

0.340827901 0.034423618

0.310851037 0.031395955

0.266082314 0.026874314

0.21150142 0.021361643

0.151972751 0.015349248

0.091907998 0.009282708

0.035016841 0.003536701

0.015845302 0.001600375

0.058734586 0.005932193

0.092592277 0.00935182

0.117151971 0.011832349

0.132801241 0.013412925

0.140418941 0.014182313

0.141207497 0.014261957

0.136535971 0.013790133

0.127805262 0.012908331

0.1163421 0.011750552

0.103324255 0.01043575

0.089735775 0.009063313

0.076348598 0.007711208

0.063725265 0.006436252

0.05223693 0.00527593

0.042090954 0.004251186

0.033363095 0.003369673

0.026030285 0.002629059

0.020001096 0.002020111
0.015142066 0.001529349

0.011299016 0.001141201

0.008313159 0.000839629

0.006032388 0.000609271

0.004318402 0.000436159

0.003050476 0.000308098

0.002126728 0.0002148

0.001463648 0.000147828

0.000994521 0.000100447

0.000667281 6.73954E-05

0.000442161 2.23292E-05
Cumulative normalization

1.31038E-05 7.12293E-06

5.20654E-05 2.83015E-05

0.00010921 5.93641E-05

0.000191879 0.000104301

0.000309803 0.000168401

0.000475614 0.000258533

0.000705344 0.000383408

0.001018829 0.000553811

0.001439936 0.000782715

0.001996452 0.001085224

0.002719486 0.001478248

0.003642164 0.001979794

0.00479743 0.002607769

0.006214724 0.003378176

0.007915402 0.004302625

0.009906835 0.00538512

0.012175244 0.006618174

0.014677595 0.007978393

0.01733307 0.009421846

0.020014967 0.010879661

0.022544187 0.012254485

0.024685737 0.013418581

0.02614989 0.01421446

0.026599682 0.014458956

0.027533066 0.014966322

0.03022589 0.016430077

0.035028027 0.019040405

0.042220228 0.022949915

0.051964318 0.028246571

0.064249471 0.034924488

0.078839141 0.042855087

0.095224384 0.051761717

0.112590075 0.061201294

0.129800713 0.070556589

0.145411856 0.079042436

0.157711719 0.08572835

0.164795081 0.089578697

0.164921627 0.089647485

0.174207663 0.094695154

0.194408406 0.105675799

( f x ) ( f b)



 




0.226927187 0.123352237

0.272664924 0.148214186

0.331891703 0.180408458

0.404155124 0.21968914

0.488237838 0.265394509

0.582172872 0.316455366

0.683320113 0.371436607

0.788501377 0.428610647

0.894185348 0.486057947

0.996708209 0.541786943

1.092511615 0.593863403

1.178377365 0.640537988

1.251638055 0.680360761

1.310345153 0.712272547

1.353380152 0.735665352

1.380500134 0.750407131

1.392315658 0.75682977

1.394425886 0.75797684

1.408453211 0.765601761

1.431935267 0.77836605

1.462152744 0.794791553

1.496328494 0.813368681

1.531812074 0.832656713

1.566235692 0.851368575

1.597631647 0.868434672

1.62450596 0.883042911

1.645867604 0.894654594

1.661216852 0.902998081

1.67049956 0.908043941

1.67403626 0.909966407

1.675636636 0.910836334

1.681568829 0.914060933

1.690920649 0.919144361

1.702752998 0.925576145

1.716165923 0.932867093

1.730348236 0.940576262

1.744610194 0.948328724

1.758400327 0.955824713

1.771308658 0.962841376

1.78305921 0.969228697

1.79349496 0.974901323

1.802558273 0.979827926

1.810269482 0.984019555

1.816705734 0.987518149

1.821981663 0.99038602

1.82623285 0.992696864

1.829602522 0.994528538

1.832231581 0.995957632

1.834251692 0.997055716
1.835781041 0.997887034

1.836922241 0.998507363

1.83776187 0.998963766

1.838371141 0.999294951

1.8388073 0.999532037

1.839115398 0.999699511

1.839330198 0.999816271

1.839478026 0.999896627

1.839578473 0.999951228

1.839645868 0.999987862

1.839668197 1

90.51431457
Raw Data rank by lb/ton use lb/ton

Runs 568 558 655

1 1.37E-05 1.75E-05 2.04E-05

2 1.50E-05 1.57E-05 2.04E-05

3 1.56E-05 1.80E-05 1.08E-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 9.00

Kurtosis -0.10

SE Kurtosis 1.56

Result Kurtosis normal

Skewness -0.29

SE Skewness 0.72 Result Raw Data

Result Skewness normal normal

ni = number test runs = 3 3 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

Total Number of sources = 3

n =Total # test runs = 9

8

This step is for QC only:

means 1.48E-05 1.70E-05 1.72E-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!

1.63E-05

Mean =

Pooled Variance = 9.70E-12

m= number future runs = 3

Term1 4.44E-01

Term2 4.31E-12

Squared root Term2 2.08E-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 = 2.896459448

UPL POOLED VARIANCE = 2.2E-05

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs 568 558 655

1 1.17E+01 2.31E+01 9.79E+00

2 9.98E+00 2.19E+01 1.14E+01

3 9.71E+00 2.69E+01 5.67E+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 9.00

Kurtosis -1.09

SE Kurtosis 1.56

Result Kurtosis normal

Skewness 0.76

SE Skewness 0.72 Result Raw Data

Result Skewness normal normal

ni = number test runs = 3 3 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

Total Number of sources = 3

n =Total # test runs = 9

8

This step is for QC only:

means 1.05E+01 2.40E+01 8.94E+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!

1.45E+01

Mean =

Pooled Variance = 5.55E+01

m= number future runs = 3

Term1 4.44E-01

Term2 2.47E+01

Squared root Term2 4.97E+00

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

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.896459448

UPL POOLED VARIANCE = 29

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs 568 558 655

1 1.63E-04 1.99E-04 1.63E-04

2 1.78E-04 1.79E-04 1.62E-04

3 1.85E-04 2.05E-04 8.57E-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

n for each source 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0

Sample size 9

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY

Testing Normality Raw Data Logtransformed

Sample Size 9.00 9.00

Kurtosis 4.67 6.46 http://www.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics

SE Kurtosis 1.56 1.56 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 non normal non normal

Skewness -1.90 -2.39

SE Skewness 0.72 0.72 Result Raw Data Result Log Data

Result Skewness non normal non normal non normal non normal

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n = 3 ONLY

Average 1.69E-04

Standard Deviation 3.48E-05 http://www.spiderfinancial.com/support/documentation/numxl/reference-manual/statistical-tests/test_xkurt

568 558 655

1 6.78E-04 5.49E-01 8.97E-04

2 4.99E-03 6.74E-03 1.33E-03

3 4.77E-02 1.15E+00 3.27E+01
4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 9.00

Kurtosis 1.31

SE Kurtosis 1.63

Result Kurtosis normal

Skewness -1.90

SE Skewness 0.72 Result Raw Data

Result Skewness non normal non normal

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

kurtosis= 4.67 depending on sample size

skewness= -1.90 same for n>3 and n=3

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

Total Number of sources = 3

n =Total # test runs = 9

1

N

i df n

 

    

1

N

i

i

n





N

1

N

i

i

n




8

This step is for QC only:

means 0.00018 0.00019 0.00014 #DIV/0! #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 = 1.69E-04

Pooled Variance = 1.2091E-09

m= number future runs = 3

Term1 0.4444

Term2 5.3737E-10

Squared root Term2 2.31813E-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 = 2.896459

UPL based on t-statistic with df at 99% confidence level 0.000236

9

0.488118

98.07%

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

a) n = number of test-runs =

b) Calculate u0

Current Confidence Level is

2

1

0

1

1

u

t

n







 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1 i

i

df n



      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 
Conclusion

interval of differences for t-values used 0.236

4.312459448

Adjusted UPL based on recalculated t-statistic 2.7E-04

new t-statistic from tab Recalculate t-statistic

Confidence Level is not 99%, Go to Recalculate t-stat n=3 tab to fix the Confidence Level
Logtransformed

568 558 655

-8.72E+00 -8.52E+00 -8.72E+00

-8.63E+00 -8.63E+00 -8.73E+00

-8.59E+00 -8.49E+00 -9.36E+00

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!






Raw Data rank by lb/ton use lb/ton

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

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 3.00 3.00

Kurtosis #DIV/0! #DIV/0!

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

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

Skewness 0.09 0.03

SE Skewness 1.22 1.22 Result Raw Data Result Log Data

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

1. Take LN( ) of Raw Data

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

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

3

3. Calculate -8.64E+00

0.001773919

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

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

9.45713E-06

320231.8612

7. Calculate

0.07302593

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

3.484 z value from "z-stat "

a- Calculate 0.000176319

b- Calculate 3.10885E-08

c- Calculate 0.001775493

d-calculate 0.000592093

e- Calculate 1.82E-05

f- Calculate UPL 2.0E-04

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

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

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

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

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

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

0.101 320231.9 0.073026

-5 =f(a) 6377891

1 -4.899 5804260

2 -4.798 5268199

3 -4.697 4768108

4 -4.596 4302421

5 -4.495 3869604

6 -4.394 3468158

7 -4.293 3096616

8 -4.192 2753546

9 -4.091 2437548

10 -3.99 2147255

11 -3.889 1881334

12 -3.788 1638485

13 -3.687 1417443

14 -3.586 1216973

15 -3.485 1035876

16 -3.384 872986

17 -3.283 727169.3

18 -3.182 597326.2

19 -3.081 482390.2

20 -2.98 381328

21 -2.879 293140

22 -2.778 216859.4

23 -2.677 151553.1

24 -2.576 96321.29

25 -2.475 50297.27

26 -2.374 12647.86

27 -2.273 -17426.9

28 -2.172 -40693.5

29 -2.071 -57885.4

30 -1.97 -69702.4

31 -1.869 -76811.1

32 -1.768 -79845

33 -1.667 -79403.9

34 -1.566 -76054.5

35 -1.465 -70330.2

36 -1.364 -62730.9

37 -1.263 -53723.4

38 -1.162 -43741

39 -1.061 -33183.7

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -22418.4

41 -0.859 -11778.3

42 -0.758 -1563.57

43 -0.657 7959.066

44 -0.556 16556.17

45 -0.455 24027.64

46 -0.354 30206.71

47 -0.253 34959.9

48 -0.152 38187.09

49 -0.051 39821.47

50 0.05 39829.55

51 0.151 38211.15

52 0.252 34999.45

53 0.353 30260.93

54 0.454 24095.38

55 0.555 16635.93

56 0.656 8049.039

57 0.757 -1465.52

58 0.858 -11674.7

59 0.959 -22311.9

60 1.06 -33077.6

61 1.161 -43638.7

62 1.262 -53628.7

63 1.363 -62647.9

64 1.464 -70263.3

65 1.565 -76008.6

66 1.666 -79384.1

67 1.767 -79856.7

68 1.868 -76860.1

69 1.969 -69794.7

70 2.07 -58027.5

71 2.171 -40892.3

72 2.272 -17689.3

73 2.373 12314.41

74 2.474 49884.99

75 2.575 95822.1

76 2.676 150958.6

77 2.777 216160.8

78 2.878 292328.3

79 2.979 380393.8

80 3.08 481323.7

81 3.181 596117.4

82 3.282 725807.8

83 3.383 871461.1

84 3.484 1034177

85 3.585 1215088

86 3.686 1415360

87 3.787 1636193

88 3.888 1878820
89 3.989 2144506

90 4.09 2434551

91 4.191 2750289

92 4.292 3093084

93 4.393 3464337

94 4.494 3865480

95 4.595 4297979

96 4.696 4763334

97 4.797 5263077

98 4.898 5798774

99 4.999 6372025
distribution of Z

Normal 

distribution

1.48672E-06 9.482135231

2.45094E-06 14.22589281

3.9995E-06 21.07017796

6.46025E-06 30.80315723

1.03291E-05 44.44008603

1.63472E-05 63.25737464

2.56093E-05 88.8169625

3.97118E-05 122.9721103

6.09553E-05 167.8431131

9.26132E-05 225.7490785

0.000139285 299.0803286

0.000207351 390.0958602

0.000305546 500.6324415

0.000445674 631.7171745

0.000643469 783.0844331

0.000919619 952.6114436

0.001300942 1135.704304

0.001821704 1324.687107

0.002525035 1508.269244

0.003464389 1671.187086

0.004704958 1794.132213

0.006324913 1854.084705

0.008416337 1825.161903

0.011085658 1680.066356

0.014453386 1392.168772

0.018652949 938.1924676

0.023828414 301.3784363

0.030130931 -525.0880094

0.03771375 -1534.705674

0.046725789 -2704.740072

0.057303789 -3994.210046

0.069563239 -5343.231447

0.083588399 -6674.1139

0.099421884 -7894.483006

0.117054396 -8902.512616

0.136415346 -9594.112976

0.157365126 -9871.654295

0.179689839 -9653.544101

0.203099244 -8883.758627

0.227228529 -7540.291364

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -5641.457864

0.275855243 -3249.104849

0.299326439 -468.0164235

0.321498297 2558.8262

0.341807853 5659.029172

0.359712192 8643.036797

0.37471238 11318.82738

0.386376486 13507.68414

0.394360216 15059.47049

0.398423793 15865.82151

0.398443914 15869.84028

0.394419966 15071.24222

0.386474058 13526.38094

0.374844865 11343.15265

0.359875719 8671.340423

0.34199778 5689.45084

0.32170943 2589.451717

0.299553264 -439.0018051

0.276092166 -3223.280932

0.25188591 -5620.06389

0.227469632 -7524.158318

0.203335281 -8873.282863

0.17991684 -9648.701009

0.157579839 -9872.046658

0.136615273 -9599.045599

0.117237788 -8911.083046

0.099587708 -7905.677738

0.083736272 -6686.900326

0.069693339 -5356.637602

0.05741676 -4007.386645

0.046822635 -2717.001999

0.037795734 -1545.55264

0.030199481 -534.2062408

0.023885038 294.1300802

0.018699163 932.807623

0.014490659 1388.525396

0.011115369 1677.960975

0.008439746 1824.342699

0.006343145 1854.280771

0.004718997 1795.0773

0.003475077 1672.637104

0.002533081 1510.013548

0.001827693 1326.554202

0.001305351 1137.562967

0.000922829 954.3684071

0.00064578 784.6798087

0.00044732 633.1186513

0.000306705 501.829117

0.000208159 391.0923349
0.000139842 299.8914797

9.29928E-05 226.395735

6.12113E-05 168.3487008

3.98826E-05 123.3602238

2.5722E-05 89.10975263

1.64209E-05 63.47460261

1.03767E-05 44.59868829

6.49066E-06 30.91717381

4.01874E-06 21.15091801

2.46298E-06 14.28223616

1.49417E-06 9.520894664
Absolute values

9.482135231 0.478847829

14.22589281 1.436815174

21.07017796 2.128087974

30.80315723 3.11111888

44.44008603 4.488448689

63.25737464 6.388994839

88.8169625 8.970513213

122.9721103 12.42018314

167.8431131 16.95215442

225.7490785 22.80065693

299.0803286 30.20711319

390.0958602 39.39968188

500.6324415 50.56387659

631.7171745 63.80343462

783.0844331 79.09152774

952.6114436 96.21375581

1135.704304 114.7061347

1324.687107 133.7933978

1508.269244 152.3351937

1671.187086 168.7898957

1794.132213 181.2073535

1854.084705 187.2625552

1825.161903 184.3413522

1680.066356 169.6867019

1392.168772 140.609046

938.1924676 94.75743923

301.3784363 30.43922207

525.0880094 53.03388895

1534.705674 155.005273

2704.740072 273.1787473

3994.210046 403.4152146

5343.231447 539.6663761

6674.1139 674.0855039

7894.483006 797.3427836

8902.512616 899.1537742

9594.112976 969.0054106

9871.654295 997.0370838

9653.544101 975.0079542

8883.758627 897.2596213

7540.291364 761.5694277

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
5641.457864 569.7872442

3249.104849 328.1595898

468.0164235 47.26965877

2558.8262 258.4414462

5659.029172 571.5619464

8643.036797 872.9467164

11318.82738 1143.201565

13507.68414 1364.276098

15059.47049 1521.006519

15865.82151 1602.447973

15869.84028 1602.853868

15071.24222 1522.195464

13526.38094 1366.164475

11343.15265 1145.658418

8671.340423 875.8053828

5689.45084 574.6345348

2589.451717 261.5346234

439.0018051 44.33918231

3223.280932 325.5513742

5620.06389 567.6264529

7524.158318 759.9399901

8873.282863 896.2015692

9648.701009 974.5188019

9872.046658 997.0767124

9599.045599 969.5036055

8911.083046 900.0193876

7905.677738 798.4734515

6686.900326 675.3769329

5356.637602 541.0203978

4007.386645 404.7460512

2717.001999 274.4172019

1545.55264 156.1008167

534.2062408 53.95483033

294.1300802 29.7071381

932.807623 94.21356992

1388.525396 140.241065

1677.960975 169.4740585

1824.342699 184.2586126

1854.280771 187.2823578

1795.0773 181.3028073

1672.637104 168.9363475

1510.013548 152.5113683

1326.554202 133.9819744

1137.562967 114.8938597

954.3684071 96.39120912

784.6798087 79.25266068

633.1186513 63.94498378

501.829117 50.68474082

391.0923349 39.50032582
299.8914797 30.28903945

226.395735 22.86596924

168.3487008 17.00321878

123.3602238 12.4593826

89.10975263 9.000085016

63.47460261 6.410934864

44.59868829 4.504467517

30.91717381 3.122634555

21.15091801 2.136242719

14.28223616 1.442505852

9.520894664 0.480805181
Cumulative normalization

0.478847829 1.28191E-05

1.915663003 5.12837E-05

4.043750977 0.000108254

7.154869857 0.000191541

11.64331855 0.0003117

18.03231338 0.000482738

27.0028266 0.000722885

39.42300974 0.001055382

56.37516416 0.001509204

79.17582108 0.002119593

109.3829343 0.002928259

148.7826162 0.003983017

199.3464927 0.005336648

263.1499274 0.007044712

342.2414551 0.009162049

438.4552109 0.01173776

553.1613456 0.014808525

686.9547434 0.01839027

839.2899371 0.022468393

1008.079833 0.026987019

1189.287186 0.03183807

1376.549741 0.036851222

1560.891094 0.041786172

1730.577796 0.046328807

1871.186842 0.050093012

1965.944281 0.052629736

1996.383503 0.053444616

2049.417392 0.054864371

2204.422665 0.059013973

2477.601412 0.066327164

2881.016627 0.077126878

3420.683003 0.091574133

4094.768507 0.109619885

4892.11129 0.130965322

5791.265065 0.155036313

6760.270475 0.180977282

7757.307559 0.207668678

8732.315513 0.233770339

9629.575134 0.257790622

10391.14456 0.278178381

( f x ) ( f b)



 




10960.93181 0.293431993

11289.0914 0.302217061

11336.36105 0.303482503

11594.8025 0.310401166

12166.36445 0.325702288

13039.31116 0.349071697

14182.51273 0.379676022

15546.78883 0.41619867

17067.79535 0.456917103

18670.24332 0.499815783

20273.09719 0.542725329

21795.29265 0.583475591

23161.45713 0.620048793

24307.11554 0.650718889

25182.92093 0.674164826

25757.55546 0.689548204

26019.09009 0.696549673

26063.42927 0.697736665

26388.98064 0.706451908

26956.6071 0.721647675

27716.54709 0.741991813

28612.74865 0.765983771

29587.26746 0.792072337

30584.34417 0.818764794

31553.84777 0.8447191

32453.86716 0.868813264

33252.34061 0.89018897

33927.71755 0.908269294

34468.73794 0.922752797

34873.484 0.933588139

35147.9012 0.940934484

35304.00201 0.945113414

35357.95684 0.946557824

35387.66398 0.947353105

35481.87755 0.94987527

35622.11862 0.953629624

35791.59268 0.958166566

35975.85129 0.963099301

36163.13365 0.968112983

36344.43645 0.972966589

36513.3728 0.977489136

36665.88417 0.981571975

36799.86614 0.985158768

36914.76 0.988234559

37011.15121 0.990815021

37090.40387 0.992936671

37154.34886 0.994648524

37205.0336 0.996005391

37244.53392 0.997062843
37274.82296 0.997873703

37297.68893 0.998485841

37314.69215 0.998941029

37327.15153 0.999274576

37336.15162 0.999515515

37342.56255 0.99968714

37347.06702 0.999807728

37350.18966 0.999891323

37352.3259 0.999948512

37353.7684 0.999987129

37354.24921 1

1886224.019
Raw Data rank by lb/ton using concentration data

Runs 531

1 6.03E+01

2 6.21E+01

3 6.32E+01

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -0.60

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

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

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

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

6.19E+01

Mean =

Pooled Variance = 2.18E+00

m= number future runs = 3

Term1 6.67E-01

Term2 1.45E+00

Squared root Term2 1.21E+00

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

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

UPL POOLED VARIANCE = 70

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

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

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 3.00 3.00

Kurtosis #DIV/0! #DIV/0!

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

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

Skewness 0.09 0.03

SE Skewness 1.22 1.22 Result Raw Data Result Log Data

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

1. Take LN( ) of Raw Data

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

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

3

3. Calculate -6.91E+00

0.001773919

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

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

9.45713E-06

320231.8612

7. Calculate

0.07302593

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

3.484 z value from "z-stat "

a- Calculate 0.000997086

b- Calculate 9.9418E-07

c- Calculate 0.001775493

d-calculate 0.000592093

e- Calculate 0.000103

f- Calculate UPL 0.0011

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

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

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

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

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

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

0.101 320231.9 0.073026

-5 =f(a) 6377891

1 -4.899 5804260

2 -4.798 5268199

3 -4.697 4768108

4 -4.596 4302421

5 -4.495 3869604

6 -4.394 3468158

7 -4.293 3096616

8 -4.192 2753546

9 -4.091 2437548

10 -3.99 2147255

11 -3.889 1881334

12 -3.788 1638485

13 -3.687 1417443

14 -3.586 1216973

15 -3.485 1035876

16 -3.384 872986

17 -3.283 727169.3

18 -3.182 597326.2

19 -3.081 482390.2

20 -2.98 381328

21 -2.879 293140

22 -2.778 216859.4

23 -2.677 151553.1

24 -2.576 96321.29

25 -2.475 50297.27

26 -2.374 12647.86

27 -2.273 -17426.9

28 -2.172 -40693.5

29 -2.071 -57885.4

30 -1.97 -69702.4

31 -1.869 -76811.1

32 -1.768 -79845

33 -1.667 -79403.9

34 -1.566 -76054.5

35 -1.465 -70330.2

36 -1.364 -62730.9

37 -1.263 -53723.4

38 -1.162 -43741

39 -1.061 -33183.7

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 -22418.4

41 -0.859 -11778.3

42 -0.758 -1563.57

43 -0.657 7959.066

44 -0.556 16556.17

45 -0.455 24027.64

46 -0.354 30206.71

47 -0.253 34959.9

48 -0.152 38187.09

49 -0.051 39821.47

50 0.05 39829.55

51 0.151 38211.15

52 0.252 34999.45

53 0.353 30260.93

54 0.454 24095.38

55 0.555 16635.93

56 0.656 8049.039

57 0.757 -1465.52

58 0.858 -11674.7

59 0.959 -22311.9

60 1.06 -33077.6

61 1.161 -43638.7

62 1.262 -53628.7

63 1.363 -62647.9

64 1.464 -70263.3

65 1.565 -76008.6

66 1.666 -79384.1

67 1.767 -79856.7

68 1.868 -76860.1

69 1.969 -69794.7

70 2.07 -58027.5

71 2.171 -40892.3

72 2.272 -17689.3

73 2.373 12314.41

74 2.474 49884.99

75 2.575 95822.1

76 2.676 150958.6

77 2.777 216160.8

78 2.878 292328.3

79 2.979 380393.8

80 3.08 481323.7

81 3.181 596117.4

82 3.282 725807.8

83 3.383 871461.1

84 3.484 1034177

85 3.585 1215088

86 3.686 1415360

87 3.787 1636193

88 3.888 1878820
89 3.989 2144506

90 4.09 2434551

91 4.191 2750289

92 4.292 3093084

93 4.393 3464337

94 4.494 3865480

95 4.595 4297979

96 4.696 4763334

97 4.797 5263077

98 4.898 5798774

99 4.999 6372025
distribution of Z

Normal 

distribution

1.48672E-06 9.482135231

2.45094E-06 14.22589281

3.9995E-06 21.07017796

6.46025E-06 30.80315723

1.03291E-05 44.44008603

1.63472E-05 63.25737464

2.56093E-05 88.8169625

3.97118E-05 122.9721103

6.09553E-05 167.8431131

9.26132E-05 225.7490785

0.000139285 299.0803286

0.000207351 390.0958602

0.000305546 500.6324415

0.000445674 631.7171745

0.000643469 783.0844331

0.000919619 952.6114436

0.001300942 1135.704304

0.001821704 1324.687107

0.002525035 1508.269244

0.003464389 1671.187086

0.004704958 1794.132213

0.006324913 1854.084705

0.008416337 1825.161903

0.011085658 1680.066356

0.014453386 1392.168772

0.018652949 938.1924676

0.023828414 301.3784363

0.030130931 -525.0880094

0.03771375 -1534.705674

0.046725789 -2704.740072

0.057303789 -3994.210046

0.069563239 -5343.231447

0.083588399 -6674.1139

0.099421884 -7894.483006

0.117054396 -8902.512616

0.136415346 -9594.112976

0.157365126 -9871.654295

0.179689839 -9653.544101

0.203099244 -8883.758627

0.227228529 -7540.291364

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 -5641.457864

0.275855243 -3249.104849

0.299326439 -468.0164235

0.321498297 2558.8262

0.341807853 5659.029172

0.359712192 8643.036797

0.37471238 11318.82738

0.386376486 13507.68414

0.394360216 15059.47049

0.398423793 15865.82151

0.398443914 15869.84028

0.394419966 15071.24222

0.386474058 13526.38094

0.374844865 11343.15265

0.359875719 8671.340423

0.34199778 5689.45084

0.32170943 2589.451717

0.299553264 -439.0018051

0.276092166 -3223.280932

0.25188591 -5620.06389

0.227469632 -7524.158318

0.203335281 -8873.282863

0.17991684 -9648.701009

0.157579839 -9872.046658

0.136615273 -9599.045599

0.117237788 -8911.083046

0.099587708 -7905.677738

0.083736272 -6686.900326

0.069693339 -5356.637602

0.05741676 -4007.386645

0.046822635 -2717.001999

0.037795734 -1545.55264

0.030199481 -534.2062408

0.023885038 294.1300802

0.018699163 932.807623

0.014490659 1388.525396

0.011115369 1677.960975

0.008439746 1824.342699

0.006343145 1854.280771

0.004718997 1795.0773

0.003475077 1672.637104

0.002533081 1510.013548

0.001827693 1326.554202

0.001305351 1137.562967

0.000922829 954.3684071

0.00064578 784.6798087

0.00044732 633.1186513

0.000306705 501.829117

0.000208159 391.0923349
0.000139842 299.8914797

9.29928E-05 226.395735

6.12113E-05 168.3487008

3.98826E-05 123.3602238

2.5722E-05 89.10975263

1.64209E-05 63.47460261

1.03767E-05 44.59868829

6.49066E-06 30.91717381

4.01874E-06 21.15091801

2.46298E-06 14.28223616

1.49417E-06 9.520894664
Absolute values

9.482135231 0.478847829

14.22589281 1.436815174

21.07017796 2.128087974

30.80315723 3.11111888

44.44008603 4.488448689

63.25737464 6.388994839

88.8169625 8.970513213

122.9721103 12.42018314

167.8431131 16.95215442

225.7490785 22.80065693

299.0803286 30.20711319

390.0958602 39.39968188

500.6324415 50.56387659

631.7171745 63.80343462

783.0844331 79.09152774

952.6114436 96.21375581

1135.704304 114.7061347

1324.687107 133.7933978

1508.269244 152.3351937

1671.187086 168.7898957

1794.132213 181.2073535

1854.084705 187.2625552

1825.161903 184.3413522

1680.066356 169.6867019

1392.168772 140.609046

938.1924676 94.75743923

301.3784363 30.43922207

525.0880094 53.03388895

1534.705674 155.005273

2704.740072 273.1787473

3994.210046 403.4152146

5343.231447 539.6663761

6674.1139 674.0855039

7894.483006 797.3427836

8902.512616 899.1537742

9594.112976 969.0054106

9871.654295 997.0370838

9653.544101 975.0079542

8883.758627 897.2596213

7540.291364 761.5694277

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
5641.457864 569.7872442

3249.104849 328.1595898

468.0164235 47.26965877

2558.8262 258.4414462

5659.029172 571.5619464

8643.036797 872.9467164

11318.82738 1143.201565

13507.68414 1364.276098

15059.47049 1521.006519

15865.82151 1602.447973

15869.84028 1602.853868

15071.24222 1522.195464

13526.38094 1366.164475

11343.15265 1145.658418

8671.340423 875.8053828

5689.45084 574.6345348

2589.451717 261.5346234

439.0018051 44.33918231

3223.280932 325.5513742

5620.06389 567.6264529

7524.158318 759.9399901

8873.282863 896.2015692

9648.701009 974.5188019

9872.046658 997.0767124

9599.045599 969.5036055

8911.083046 900.0193876

7905.677738 798.4734515

6686.900326 675.3769329

5356.637602 541.0203978

4007.386645 404.7460512

2717.001999 274.4172019

1545.55264 156.1008167

534.2062408 53.95483033

294.1300802 29.7071381

932.807623 94.21356992

1388.525396 140.241065

1677.960975 169.4740585

1824.342699 184.2586126

1854.280771 187.2823578

1795.0773 181.3028073

1672.637104 168.9363475

1510.013548 152.5113683

1326.554202 133.9819744

1137.562967 114.8938597

954.3684071 96.39120912

784.6798087 79.25266068

633.1186513 63.94498378

501.829117 50.68474082

391.0923349 39.50032582
299.8914797 30.28903945

226.395735 22.86596924

168.3487008 17.00321878

123.3602238 12.4593826

89.10975263 9.000085016

63.47460261 6.410934864

44.59868829 4.504467517

30.91717381 3.122634555

21.15091801 2.136242719

14.28223616 1.442505852

9.520894664 0.480805181
Cumulative normalization

0.478847829 1.28191E-05

1.915663003 5.12837E-05

4.043750977 0.000108254

7.154869857 0.000191541

11.64331855 0.0003117

18.03231338 0.000482738

27.0028266 0.000722885

39.42300974 0.001055382

56.37516416 0.001509204

79.17582108 0.002119593

109.3829343 0.002928259

148.7826162 0.003983017

199.3464927 0.005336648

263.1499274 0.007044712

342.2414551 0.009162049

438.4552109 0.01173776

553.1613456 0.014808525

686.9547434 0.01839027

839.2899371 0.022468393

1008.079833 0.026987019

1189.287186 0.03183807

1376.549741 0.036851222

1560.891094 0.041786172

1730.577796 0.046328807

1871.186842 0.050093012

1965.944281 0.052629736

1996.383503 0.053444616

2049.417392 0.054864371

2204.422665 0.059013973

2477.601412 0.066327164

2881.016627 0.077126878

3420.683003 0.091574133

4094.768507 0.109619885

4892.11129 0.130965322

5791.265065 0.155036313

6760.270475 0.180977282

7757.307559 0.207668678

8732.315513 0.233770339

9629.575134 0.257790622

10391.14456 0.278178381

( f x ) ( f b)



 




10960.93181 0.293431993

11289.0914 0.302217061

11336.36105 0.303482503

11594.8025 0.310401166

12166.36445 0.325702288

13039.31116 0.349071697

14182.51273 0.379676022

15546.78883 0.41619867

17067.79535 0.456917103

18670.24332 0.499815783

20273.09719 0.542725329

21795.29265 0.583475591

23161.45713 0.620048793

24307.11554 0.650718889

25182.92093 0.674164826

25757.55546 0.689548204

26019.09009 0.696549673

26063.42927 0.697736665

26388.98064 0.706451908

26956.6071 0.721647675

27716.54709 0.741991813

28612.74865 0.765983771

29587.26746 0.792072337

30584.34417 0.818764794

31553.84777 0.8447191

32453.86716 0.868813264

33252.34061 0.89018897

33927.71755 0.908269294

34468.73794 0.922752797

34873.484 0.933588139

35147.9012 0.940934484

35304.00201 0.945113414

35357.95684 0.946557824

35387.66398 0.947353105

35481.87755 0.94987527

35622.11862 0.953629624

35791.59268 0.958166566

35975.85129 0.963099301

36163.13365 0.968112983

36344.43645 0.972966589

36513.3728 0.977489136

36665.88417 0.981571975

36799.86614 0.985158768

36914.76 0.988234559

37011.15121 0.990815021

37090.40387 0.992936671

37154.34886 0.994648524

37205.0336 0.996005391

37244.53392 0.997062843
37274.82296 0.997873703

37297.68893 0.998485841

37314.69215 0.998941029

37327.15153 0.999274576

37336.15162 0.999515515

37342.56255 0.99968714

37347.06702 0.999807728

37350.18966 0.999891323

37352.3259 0.999948512

37353.7684 0.999987129

37354.24921 1

1886224.019
Raw Data

Runs

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 0.00

Kurtosis #DIV/0!

SE Kurtosis 0.00

Result Kurtosis #DIV/0!

Skewness #DIV/0!

SE Skewness 0.00 Result Raw Data

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

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

Total Number of sources = 0

n =Total # test runs = 0

-1

This step is for QC only:

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

#DIV/0!

Mean =

Pooled Variance = #DIV/0!

m= number future runs = 3

Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0!

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

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

UPL POOLED VARIANCE = #DIV/0!

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

0900006481934216 UPL Pooled Template 12/1/2014
Raw Data

Runs 123456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 0.00 0.00

Kurtosis #DIV/0! #DIV/0!

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

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

Skewness #DIV/0! #DIV/0!

SE Skewness 0.00 0.00 Result Raw Data Result Log Data

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

1. Take LN( ) of Raw Data

Runs 123456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

0

3. Calculate #DIV/0!

#DIV/0!

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

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

#DIV/0!

#DIV/0!

7. Calculate

#DIV/0!

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

3.484 z value from "z-stat "

a- Calculate #DIV/0!

b- Calculate #DIV/0!

c- Calculate #DIV/0!

d-calculate #DIV/0!

e- Calculate #DIV/0!

f- Calculate UPL #DIV/0!

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

12. calculate the UPL using the formula 

a- Numerator first term

b- denominator first term

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

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

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

0900006481934216 Lognormal Template 12/1/2014
∆z z

0.101 14.78551 3.387135

-5 =f(a) 173.6307

1 -4.899 156.5385

2 -4.798 140.6598

3 -4.697 125.9391

4 -4.596 112.3223

5 -4.495 99.75638

6 -4.394 88.18975

7 -4.293 77.57188

8 -4.192 67.85352

9 -4.091 58.98665

10 -3.99 50.92447

11 -3.889 43.62141

12 -3.788 37.03312

13 -3.687 31.11647

14 -3.586 25.82959

15 -3.485 21.13179

16 -3.384 16.98365

17 -3.283 13.34694

18 -3.182 10.18468

19 -3.081 7.461103

20 -2.98 5.141681

21 -2.879 3.193102

22 -2.778 1.583284

23 -2.677 0.281372

24 -2.576 -0.74226

25 -2.475 -1.51602

26 -2.374 -2.06709

27 -2.273 -2.42141

28 -2.172 -2.6037

29 -2.071 -2.63747

30 -1.97 -2.54498

31 -1.869 -2.34728

32 -1.768 -2.06419

33 -1.667 -1.71429

34 -1.566 -1.31496

35 -1.465 -0.88233

36 -1.364 -0.43132

37 -1.263 0.024392

38 -1.162 0.472344

39 -1.061 0.901303

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 1.301261

41 -0.859 1.663439

42 -0.758 1.980282

43 -0.657 2.245462

44 -0.556 2.45388

45 -0.455 2.601659

46 -0.354 2.686152

47 -0.253 2.705937

48 -0.152 2.660818

49 -0.051 2.551826

50 0.05 2.381219

51 0.151 2.152479

52 0.252 1.870318

53 0.353 1.540671

54 0.454 1.170702

55 0.555 0.7688

56 0.656 0.344581

57 0.757 -0.09111

58 0.858 -0.52622

59 0.959 -0.94743

60 1.06 -1.34023

61 1.161 -1.68888

62 1.262 -1.97639

63 1.363 -2.18458

64 1.464 -2.294

65 1.565 -2.28401

66 1.666 -2.13273

67 1.767 -1.81705

68 1.868 -1.31264

69 1.969 -0.59394

70 2.07 0.365836

71 2.171 1.594699

72 2.272 3.121887

73 2.373 4.977865

74 2.474 7.194324

75 2.575 9.804181

76 2.676 12.84158

77 2.777 16.34189

78 2.878 20.34171

79 2.979 24.87887

80 3.08 29.9924

81 3.181 35.7226

82 3.282 42.11095

83 3.383 49.20019

84 3.484 57.03427

85 3.585 65.65838

86 3.686 75.11893
87 3.787 85.46354

88 3.888 96.74107

89 3.989 109.0016

90 4.09 122.2965

91 4.191 136.6783

92 4.292 152.2006

93 4.393 168.9187

94 4.494 186.8885

95 4.595 206.1677

96 4.696 226.8148

97 4.797 248.8898

98 4.898 272.4537

99 4.999 297.569
distribution of Z

Normal 

distribution

1.48672E-06 0.00025814

2.45094E-06 0.000383666

3.9995E-06 0.000562569

6.46025E-06 0.000813598

1.03291E-05 0.001160187

1.63472E-05 0.001630742

2.56093E-05 0.002258474

3.97118E-05 0.003080516

6.09553E-05 0.004136028

9.26132E-05 0.005462942

0.000139285 0.007093014

0.000207351 0.009044929

0.000305546 0.011315317

0.000445674 0.0138678

0.000643469 0.016620542

0.000919619 0.0194332

0.001300942 0.022094744

0.001821704 0.02431417

0.002525035 0.025716663

0.003464389 0.025848163

0.004704958 0.024191389

0.006324913 0.02019609

0.008416337 0.013325453

0.011085658 0.003119197

0.014453386 -0.010728213

0.018652949 -0.028278328

0.023828414 -0.049255451

0.030130931 -0.072959218

0.03771375 -0.098195275

0.046725789 -0.123237716

0.057303789 -0.145836905

0.069563239 -0.163284321

0.083588399 -0.172542023

0.099421884 -0.170438144

0.117054396 -0.153922026

0.136415346 -0.120363821

0.157365126 -0.067874755

0.179689839 0.004383042

0.203099244 0.095932777

0.227228529 0.204801723

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 0.327454986

0.275855243 0.458868267

0.299326439 0.592750629

0.321498297 0.72191232

0.341807853 0.838755395

0.359712192 0.93584856

0.37471238 1.006534541

0.386376486 1.045510485

0.394360216 1.049320814

0.398423793 1.016708275

0.398443914 0.948782077

0.394419966 0.848980757

0.386474058 0.722829327

0.374844865 0.577512693

0.359875719 0.421307289

0.34199778 0.262927915

0.32170943 0.110854823

0.299553264 -0.027293533

0.276092166 -0.145283984

0.25188591 -0.238643935

0.227469632 -0.304862135

0.203335281 -0.343408575

0.17991684 -0.355586415

0.157579839 -0.344245108

0.136615273 -0.31339536

0.117237788 -0.267772357

0.099587708 -0.212393663

0.083736272 -0.152153024

0.069693339 -0.091482269

0.05741676 -0.034102101

0.046822635 0.017129395

0.037795734 0.060272801

0.030199481 0.094279357

0.023885038 0.118896489

0.018699163 0.134527832

0.014490659 0.142069051

0.011115369 0.142738913

0.008439746 0.137921435

0.006343145 0.129030453

0.004718997 0.117403303

0.003475077 0.104225924

0.002533081 0.090488222

0.001827693 0.076965905

0.001305351 0.064223535

0.000922829 0.052632887

0.00064578 0.042400898

0.00044732 0.033602192
0.000306705 0.026212124

0.000208159 0.020137481

0.000139842 0.01524298

9.29928E-05 0.011372694

6.12113E-05 0.008366251

3.98826E-05 0.006070157

2.5722E-05 0.004344929

1.64209E-05 0.003068875

1.03767E-05 0.002139333

6.49066E-06 0.001472177

4.01874E-06 0.001000222

2.46298E-06 0.000671047

1.49417E-06 0.000444619
Absolute values

0.00025814 1.30361E-05

0.000383666 3.87503E-05

0.000562569 5.68195E-05

0.000813598 8.21734E-05

0.001160187 0.000117179

0.001630742 0.000164705

0.002258474 0.000228106

0.003080516 0.000311132

0.004136028 0.000417739

0.005462942 0.000551757

0.007093014 0.000716394

0.009044929 0.000913538

0.011315317 0.001142847

0.0138678 0.001400648

0.016620542 0.001678675

0.0194332 0.001962753

0.022094744 0.002231569

0.02431417 0.002455731

0.025716663 0.002597383

0.025848163 0.002610664

0.024191389 0.00244333

0.02019609 0.002039805

0.013325453 0.001345871

0.003119197 0.000315039

0.010728213 0.00108355

0.028278328 0.002856111

0.049255451 0.004974801

0.072959218 0.007368881

0.098195275 0.009917723

0.123237716 0.012447009

0.145836905 0.014729527

0.163284321 0.016491716

0.172542023 0.017426744

0.170438144 0.017214253

0.153922026 0.015546125

0.120363821 0.012156746

0.067874755 0.00685535

0.004383042 0.000442687

0.095932777 0.009689211

0.204801723 0.020684974

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.327454986 0.033072954

0.458868267 0.046345695

0.592750629 0.059867814

0.72191232 0.072913144

0.838755395 0.084714295

0.93584856 0.094520705

1.006534541 0.101659989

1.045510485 0.105596559

1.049320814 0.105981402

1.016708275 0.102687536

0.948782077 0.09582699

0.848980757 0.085747056

0.722829327 0.073005762

0.577512693 0.058328782

0.421307289 0.042552036

0.262927915 0.026555719

0.110854823 0.011196337

0.027293533 0.002756647

0.145283984 0.014673682

0.238643935 0.024103037

0.304862135 0.030791076

0.343408575 0.034684266

0.355586415 0.035914228

0.344245108 0.034768756

0.31339536 0.031652931

0.267772357 0.027045008

0.212393663 0.02145176

0.152153024 0.015367455

0.091482269 0.009239709

0.034102101 0.003444312

0.017129395 0.001730069

0.060272801 0.006087553

0.094279357 0.009522215

0.118896489 0.012008545

0.134527832 0.013587311

0.142069051 0.014348974

0.142738913 0.01441663

0.137921435 0.013930065

0.129030453 0.013032076

0.117403303 0.011857734

0.104225924 0.010526818

0.090488222 0.00913931

0.076965905 0.007773556

0.064223535 0.006486577

0.052632887 0.005315922

0.042400898 0.004282491

0.033602192 0.003393821
0.026212124 0.002647425

0.020137481 0.002033886

0.01524298 0.001539541

0.011372694 0.001148642

0.008366251 0.000844991

0.006070157 0.000613086

0.004344929 0.000438838

0.003068875 0.000309956

0.002139333 0.000216073

0.001472177 0.00014869

0.001000222 0.000101022

0.000671047 6.77757E-05

0.000444619 2.24533E-05
Cumulative normalization

1.30361E-05 7.04745E-06

5.17864E-05 2.79963E-05

0.000108606 5.87135E-05

0.000190779 0.000103137

0.000307958 0.000166486

0.000472663 0.000255527

0.000700769 0.000378843

0.001011901 0.000547045

0.00142964 0.000772879

0.001981397 0.001071165

0.002697791 0.001458456

0.003611329 0.001952325

0.004754176 0.00257016

0.006154824 0.003327366

0.007833499 0.004234876

0.009796252 0.005295962

0.012027821 0.006502373

0.014483552 0.007829968

0.017080935 0.009234142

0.0196916 0.010645496

0.02213493 0.011966388

0.024174735 0.013069129

0.025520606 0.013796721

0.025835645 0.013967035

0.026919195 0.014552814

0.029775306 0.016096859

0.034750106 0.018786291

0.042118987 0.02276999

0.05203671 0.02813162

0.064483719 0.034860611

0.079213247 0.042823556

0.095704963 0.05173916

0.113131707 0.06116025

0.13034596 0.070466465

0.145892085 0.078870872

0.15804883 0.085442943

0.164904181 0.089149021

0.165346868 0.089388343

0.175036079 0.094626437

0.195721052 0.105808962

( f x ) ( f b)



 




0.228794006 0.123688566

0.275139701 0.14874356

0.335007515 0.181108761

0.407920659 0.220526412

0.492634954 0.266323895

0.587155658 0.317422832

0.688815647 0.372381344

0.794412206 0.42946801

0.900393608 0.486762726

1.003081144 0.542276742

1.098908134 0.594081871

1.18465519 0.640437676

1.257660952 0.679905397

1.315989734 0.711438581

1.35854177 0.734442682

1.38509749 0.748798997

1.396293827 0.75485186

1.399050474 0.756342134

1.413724156 0.764274889

1.437827193 0.777305257

1.468618269 0.793951252

1.503302535 0.812701949

1.539216763 0.832117578

1.573985519 0.850913951

1.60563845 0.868025875

1.632683458 0.882646705

1.654135218 0.894243763

1.669502674 0.90255158

1.678742383 0.907546668

1.682186695 0.909408702

1.683916764 0.910343996

1.690004317 0.913634994

1.699526532 0.918782809

1.711535077 0.925274761

1.725122388 0.932620211

1.739471362 0.940377425

1.753887993 0.948171215

1.767818058 0.955701962

1.780850133 0.962747246

1.792707867 0.969157667

1.803234685 0.974848582

1.812373996 0.979789394

1.820147552 0.983991864

1.826634129 0.987498579

1.831950051 0.990372425

1.836232541 0.992687587

1.839626363 0.994522324
1.842273787 0.995953551

1.844307673 0.997053092

1.845847214 0.997885385

1.846995856 0.998506354

1.847840847 0.998963165

1.848453933 0.999294606

1.848892771 0.999531847

1.849202727 0.999699413

1.8494188 0.999816224

1.84956749 0.999896607

1.849668512 0.999951221

1.849736288 0.999987862

1.849758741 1

90.98161566
Raw Data

Runs

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

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

Sample size 0

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY

Testing Normality Raw Data Logtransformed

Sample Size 0.00 0.00

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

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

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

Skewness #DIV/0! #DIV/0!

SE Skewness 0.00 0.00 Result Raw Data Result Log Data

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

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n = 3 ONLY

Average #DIV/0!

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

1

2

3
4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 0.00

Kurtosis -3.00

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness #DIV/0!

SE Skewness 0.00 Result Raw Data

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

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

kurtosis= #DIV/0! depending on sample size

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

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

Total Number of sources = 0

n =Total # test runs = 0

1

N

i df n

 

    

1

N

i

i

n





N

1

N

i

i

n




-1

This step is for QC only:

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

Mean = #DIV/0!

Pooled Variance = #DIV/0!

m= number future runs = 3

Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0!

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

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

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

0

#NUM!

#NUM!

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

a) n = number of test-runs =

b) Calculate u0

Current Confidence Level is

2

1

0

1

1

u

t

n







 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1 i

i

df n



      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 
Conclusion

interval of differences for t-values used

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

new t-statistic from tab Recalculate t-statistic

#NUM!
Logtransformed

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

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






Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

c.1.7 #NUM! d.1.7

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

1

x

w b

x

 

     

 ,

 

x

I a b 

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )



( )

w a

e w





a w  
Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

Result #DIV/0!

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



1 1 1 1  1 2   1 3  4 1   1

( ) , , ,

n n n n n n n n

P t I I I

             

        
4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

Corrected probability is::

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

3 4 3/2

2

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

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

#NUM! e.1.7

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

Calculations Calculations



 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
#NUM! Result3:

3 4 1 1

,

n

I

 

   

 
#NUM!

0

3 4 1 1

,

24 2 2

u

n

I

 

   

  

 

1 1

2 2

5 1

2 2



 








a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

#NUM! f.1.7 #NUM!

f.1.) Calculate 

Calculations

0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
#NUM! Result4: #NUM!

a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.7 #NUM! h.1.7

h.1.) Calculate 

Calculations Calculations

g.1.) Calculate 0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
Result5: #NUM! Result6:

#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

3 1

2 6

w a

    

 
#NUM!
#NUM!

0.236

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

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

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

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

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

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

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

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

1) Start with initial t-statistic from Template

2) Determine interval of differences for t-values

2) Candidate values for t-statistic

3) Corrected p-values

Acutal significance level with t-value

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

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

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

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

e.1.) Calculate 

Calculations

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 