**Subcategory Pollutant**  **No. of Sources**  **Industryw ide**  **No. of Sources**  **with Data Determination of** 

**No. of Sources in MA** **CT Floor No. of** 

**Sources in MA** **CT Floor (N)**  **No. Data**

**Points in MA** **CT Floor (n)**  **Av erage** 

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

Equivalent PM 122 71 12% Sources with Data 1 3 0.0020 Small Kilns PM 103 21 12% Sources with Data 1 3 0.0093

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

0.018 lb/ton lognormal 0.0059 gr/dscf @ 7% O2 lognormal 0.029 lb/ton lognormal 0.0072 gr/dscf @ 7% O2 lognormalRuns 514 PM lb/ton 514 PM

conc 599 PM

Small lb/ton 599 PM

Small conc Example

5 Example

6 Example

7 Example

8 Example

9 1 2.99E-03 1.02E-03 8.11E-03 2.03E-03 2 9.53E-03 3.18E-03 1.53E-02 3.77E-03 3 5.61E-03 1.78E-03 4.53E-03 1.12E-03 Average 6.04E-03 2.00E-03 9.30E-03 2.31E-03 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Standard Deviation

3.29E-03 1.09E-03 5.47E-03 1.35E-03 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 1 8.74E-11 8.93E-13 2.01E-12 6.30E-15 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 2 1.48E-10 1.98E-12 1.26E-09 4.61E-12 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 3 3.43E-14 2.10E-15 5.19E-10 1.96E-12 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Testing Normality

Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Sample Size 3.00 3.00 3.00 3.00 0.00 0.00 0.00 0.00 0.00 Kurtosis -2.00 -2.00 -2.00 -2.00 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! SE Kurtosis 2.83 2.83 2.83 2.83 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Result Kurtosis

normal normal normal normal #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Skewness 0.58 0.85 0.93 0.90 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! SE Skewness 1.22 1.22 1.22 1.22 0.00 0.00 0.00 0.00 0.00 Result Skewness

normal normal normal normal #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 0.471984820.690883830.762346740.73496458 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Lognormal Lognormal Lognormal Lognormal #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 1 2 3 4 5 6 7 8 9Example

10 Example

11 Example

12 Example

13 Example

14 Example

15 Example

16 Example

17 Example

18 Example

19

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data Raw Data 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 10 11 12 13 14 15 16 17 18 19Example

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

-5.81E+00 -6.88E+00 -4.81E+00 -6.20E+00 #NUM! #NUM! #NUM! #NUM! -4.65E+00 -5.75E+00 -4.18E+00 -5.58E+00 #NUM! #NUM! #NUM! #NUM! -5.18E+00 -6.33E+00 -5.40E+00 -6.79E+00 #NUM! #NUM! #NUM! #NUM! #DIV/0! -5.22E+00 -6.32E+00 -4.80E+00 -6.19E+00 #NUM! #NUM! #NUM! #NUM! #DIV/0! 5.81E-01 5.67E-01 6.08E-01 6.06E-01 #NUM! #NUM! #NUM! #NUM! #DIV/0! 1.27E-01 1.00E-01 7.46E-08 1.39E-08 #NUM! #NUM! #NUM! #NUM! #DIV/0! 1.01E-01 1.06E-01 1.44E-01 1.40E-01 #NUM! #NUM! #NUM! #NUM! #DIV/0! 1.31E-06 5.57E-09 1.29E-01 1.30E-01 #NUM! #NUM! #NUM! #NUM! Raw Data Log Data Log Data Log Data Log Data Log Data Log Data Log Data Log Data 0.00 3.00 3.00 3.00 3.00 0.00 0.00 0.00 0.00 #DIV/0! -2.00 -2.00 -2.00 -2.00 #NUM! #NUM! #NUM! #NUM! #DIV/0! 2.83 2.83 2.83 2.83 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! normal normal normal normal #NUM! #NUM! #NUM! #NUM! #DIV/0! -0.26 0.07 0.12 0.08 #NUM! #NUM! #NUM! #NUM! 0.00 1.22 1.22 1.22 1.22 0.00 0.00 0.00 0.00 #DIV/0! normal normal normal normal #NUM! #NUM! #NUM! #NUM! #DIV/0! 0.213309410.055985170.099852360.06584724 #NUM! #NUM! #NUM! #NUM! #DIV/0! 20 1 2 3 4 5 6 7 8Example 9 Example 10 Example

11 Example

12 Example

13 Example

14 Example

15 Example

16 Example

17 Example

18 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data Log Data Log Data Log Data Log Data Log Data Log Data 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!
9 10 11 12 13 14 15 16 17 18Example

19 Example

20 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! Log Data Log Data 0.00 0.00 #NUM! #NUM! #DIV/0! #DIV/0! #NUM! #NUM! #NUM! #NUM! 0.00 0.00 #NUM! #NUM! #NUM! #NUM!
19 20Raw Data Runs 514 1 2.99E-03 2 9.53E-03 3 5.61E-03 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 6.04E-03 Result Kurtosis #DIV/0! #DIV/0! Skewness 0.58 -0.26 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 514 1 -5.81E+00 2 -4.65E+00 3 -5.18E+00 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 3

3. Calculate -5.22E+00

0.337445542 4. Calculate

5. m= number future runs = 3 6. Calculate Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1 ˆ n i

i x n     2 2 1 ˆ 1

n i i x x n    

 

### 

222 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 PM-Large-lbton 12/1/201412.2520905
0.483277475

27.35208269
7. Calculate 1.24411698

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

3.484z value from "z-stat"

a- Calculate 0.006424721
b- Calculate 4.1277E-05

c- Calculate 0.401363291

d-calculate 0.140949221

e- Calculate 0.010103

f- Calculate UPL 1.82E-02 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term
b- denominator first term

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

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z



2 ˆ 2 2 2 2 2 4 ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e   2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



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

0.101 27.35208 1.244117

-5=f(a) 463.2035 1 -4.899 421.0576 2 -4.798 381.7066 3 -4.697 345.03 4 -4.596 310.91 5 -4.495 279.2314 6 -4.394 249.8814 7 -4.293 222.7498 8 -4.192 197.7287 9 -4.091 174.7132 10 -3.99 153.6005 11 -3.889 134.2906 12 -3.788 116.6859 13 -3.687 100.6914 14 -3.586 86.21459 15 -3.485 73.16549 16 -3.384 61.4567 17 -3.283 51.00332 18 -3.182 41.723 19 -3.081 33.53592 20 -2.98 26.3648 21 -2.879 20.13487 22 -2.778 14.77392 23 -2.677 10.21226 24 -2.576 6.382752 25 -2.475 3.220766 26 -2.374 0.664222 27 -2.273 -1.34643 28 -2.172 -2.8682 29 -2.071 -3.95557 30 -1.97 -4.66049 31 -1.869 -5.03237 32 -1.768 -5.11808 33 -1.667 -4.96198 34 -1.566 -4.60588 35 -1.465 -4.08904 36 -1.364 -3.44822 37 -1.263 -2.71761 38 -1.162 -1.9289 39 -1.061 -1.11122 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -0.29118

41 -0.859 0.507151 42 -0.758 1.262234 43 -0.657 1.955065 44 -0.556 2.569176 45 -0.455 3.090631 46 -0.354 3.508029 47 -0.253 3.812502 48 -0.152 3.997719 49 -0.051 4.05988 50 0.05 3.99772 51 0.151 3.812508 52 0.252 3.508048 53 0.353 3.090678 54 0.454 2.569268 55 0.555 1.955224 56 0.656 1.262487 57 0.757 0.50753 58 0.858 -0.29064 59 0.959 -1.11048 60 1.06 -1.92791 61 1.161 -2.71633 62 1.262 -3.44659 63 1.363 -4.08701 64 1.464 -4.60338 65 1.565 -4.95895 66 1.666 -5.11444 67 1.767 -5.02804 68 1.868 -4.6554 69 1.969 -3.94963 70 2.07 -2.86133 71 2.171 -1.33853 72 2.272 0.673248 73 2.373 3.23102 74 2.474 6.394343 75 2.575 10.2253 76 2.676 14.78852 77 2.777 20.15115 78 2.878 26.38289 79 2.979 33.55595 80 3.08 41.7451 81 3.181 51.02763 82 3.282 61.48336 83 3.383 73.19465 84 3.484 86.2464 85 3.585 100.726 86 3.686 116.7235 87 3.787 134.3314 88 3.888 153.644689 3.989 174.7607

90 4.09 197.7799 91 4.191 222.8047 92 4.292 249.9404 93 4.393 279.2947 94 4.494 310.9777 95 4.595 345.1022 96 4.696 381.7836 97 4.797 421.1397 98 4.898 463.2908 99 4.999 508.3599distribution of Z Normal distribution

1.48672E-06 0.000688654 2.45094E-06 0.001031987 3.9995E-06 0.001526636 6.46025E-06 0.002228979 1.03291E-05 0.003211417 1.63472E-05 0.004564666 2.56093E-05 0.006399279 3.97118E-05 0.008845786 6.09553E-05 0.012052605 9.26132E-05 0.016180747 0.000139285 0.021394248 0.000207351 0.027845257 0.000305546 0.035652911 0.000445674 0.04487554 0.000643469 0.055476427 0.000919619 0.067284381 0.001300942 0.079951609 0.001821704 0.092912943 0.002525035 0.10535202 0.003464389 0.116181478 0.004704958 0.124045247 0.006324913 0.127351286 0.008416337 0.124342289 0.011085658 0.113209674 0.014453386 0.092252383 0.018652949 0.06007678 0.023828414 0.015827368 0.030130931 -0.040569091 0.03771375 -0.108170447 0.046725789 -0.184827021 0.057303789 -0.267063539 0.069563239 -0.350067665 0.083588399 -0.427812432 0.099421884 -0.493329872 0.117054396 -0.539138381 0.136415346 -0.557808155 0.157365126 -0.542629187 0.179689839 -0.488327241 0.203099244 -0.391758021 0.227228529 -0.252500893 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 -0.073273625 0.275855243 0.139900292 0.299326439 0.377819919 0.321498297 0.628550081 0.341807853 0.878164461 0.359712192 1.11173754 0.37471238 1.314501734 0.386376486 1.473061259 0.394360216 1.576541345 0.398423793 1.617552685 0.398443914 1.592867053 0.394419966 1.503729262 0.386474058 1.355769593 0.374844865 1.158524603 0.359875719 0.924617077 0.34199778 0.668682393 0.32170943 0.406154008 0.299553264 0.152032198 0.276092166 -0.080243366 0.25188591 -0.279714154 0.227469632 -0.438541779 0.203335281 -0.552326028 0.17991684 -0.620099454 0.157579839 -0.64403015 0.136615273 -0.628891587 0.117237788 -0.581376068 0.099587708 -0.509335428 0.083736272 -0.421029404 0.069693339 -0.32445035 0.05741676 -0.226775201 0.046822635 -0.133974896 0.037795734 -0.050590672 0.030199481 0.02033173 0.023885038 0.077173047 0.018699163 0.119568861 0.014490659 0.148171378 0.011115369 0.164379871 0.008439746 0.170070626 0.006343145 0.167350506 0.004718997 0.158350443 0.003475077 0.145067465 0.002533081 0.129257107 0.001827693 0.112372732 0.001305351 0.09554474 0.000922829 0.079590688 0.00064578 0.065046903 0.00044732 0.052212756 0.000306705 0.041200145 0.000208159 0.0319824270.000139842 0.024438844 9.29928E-05 0.018392104 6.12113E-05 0.013638163 3.98826E-05 0.009968273 2.5722E-05 0.007184023 1.64209E-05 0.005106528 1.03767E-05 0.00358101 6.49066E-06 0.002478027 4.01874E-06 0.001692449 2.46298E-06 0.001141074 1.49417E-06 0.000759577Absolute values

0.000688654 3.4777E-05 0.001031987 0.000104231 0.001526636 0.00015419 0.002228979 0.000225127 0.003211417 0.000324353 0.004564666 0.000461031 0.006399279 0.000646327 0.008845786 0.000893424 0.012052605 0.001217313 0.016180747 0.001634255 0.021394248 0.002160819 0.027845257 0.002812371 0.035652911 0.003600944 0.04487554 0.004532429 0.055476427 0.005603119 0.067284381 0.006795723 0.079951609 0.008075112 0.092912943 0.009384207 0.10535202 0.010640554 0.116181478 0.011734329 0.124045247 0.01252857 0.127351286 0.01286248 0.124342289 0.012558571 0.113209674 0.011434177 0.092252383 0.009317491 0.06007678 0.006067755 0.015827368 0.001598564 0.040569091 0.004097478 0.108170447 0.010925215 0.184827021 0.018667529 0.267063539 0.026973417 0.350067665 0.035356834 0.427812432 0.043209056 0.493329872 0.049826317 0.539138381 0.054452976 0.557808155 0.056338624 0.542629187 0.054805548 0.488327241 0.049321051 0.391758021 0.03956756 0.252500893 0.02550259 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.073273625 0.007400636 0.139900292 0.014129929 0.377819919 0.038159812 0.628550081 0.063483558 0.878164461 0.088694611 1.11173754 0.112285492 1.314501734 0.132764675 1.473061259 0.148779187 1.576541345 0.159230676 1.617552685 0.163372821 1.592867053 0.160879572 1.503729262 0.151876655 1.355769593 0.136932729 1.158524603 0.117010985 0.924617077 0.093386325 0.668682393 0.067536922 0.406154008 0.041021555 0.152032198 0.015355252 0.080243366 0.00810458 0.279714154 0.02825113 0.438541779 0.04429272 0.552326028 0.055784929 0.620099454 0.062630045 0.64403015 0.065047045 0.628891587 0.06351805 0.581376068 0.058718983 0.509335428 0.051442878 0.421029404 0.04252397 0.32445035 0.032769485 0.226775201 0.022904295 0.133974896 0.013531464 0.050590672 0.005109658 0.02033173 0.002053505 0.077173047 0.007794478 0.119568861 0.012076455 0.148171378 0.014965309 0.164379871 0.016602367 0.170070626 0.017177133 0.167350506 0.016902401 0.158350443 0.015993395 0.145067465 0.014651814 0.129257107 0.013054968 0.112372732 0.011349646 0.09554474 0.009650019 0.079590688 0.00803866 0.065046903 0.006569737 0.052212756 0.005273488 0.041200145 0.004161215 0.031982427 0.0032302250.024438844 0.002468323 0.018392104 0.001857603 0.013638163 0.001377455 0.009968273 0.001006796 0.007184023 0.000725586 0.005106528 0.000515759 0.00358101 0.000361682 0.002478027 0.000250281 0.001692449 0.000170937 0.001141074 0.000115248 0.000759577 3.83586E-05Cumulative normalization
3.4777E-05 1.13583E-05 0.000139008 4.54005E-05 0.000293198 9.57597E-05 0.000518325 0.000169287 0.000842678 0.000275222 0.001303709 0.000425797 0.001950036 0.00063689 0.002843461 0.000928686 0.004060774 0.001326266 0.005695029 0.001860021 0.007855848 0.002565753 0.010668219 0.003484285 0.014269163 0.004660369 0.018801593 0.006140679 0.024404712 0.007970682 0.031200435 0.010190193 0.039275547 0.012827559 0.048659754 0.01589248 0.059300308 0.01936773 0.071034638 0.023200212 0.083563208 0.027292095 0.096425687 0.031493035 0.108984259 0.035594717 0.120418436 0.039329168 0.129735926 0.042372299 0.135803681 0.044354054 0.137402245 0.044876151 0.141499723 0.046214405 0.152424939 0.049782626 0.171092468 0.05587952 0.198065885 0.064689152 0.233422719 0.076236842 0.276631775 0.090349102 0.326458092 0.106622587 0.380911069 0.124407158 0.437249692 0.142807589 0.49205524 0.160707312 0.541376291 0.176815775 0.580943851 0.189738706 0.606446442 0.19806796 

## ( ) 2 () () f a f x f b        0.613847078 0.200485039 0.627977007 0.205099933 0.666136819 0.217563088 0.729620377 0.238297085 0.818314988 0.267265118 0.930600479 0.303938032 1.063365155 0.347299534 1.212144342 0.395891443 1.371375017 0.447896851 1.534747839 0.501255102 1.695627411 0.553799048 1.847504066 0.603402602 1.984436795 0.648125408 2.10144778 0.686341688 2.194834105 0.716842054 2.262371027 0.738899897 2.303392582 0.75229771 2.318747834 0.7573128 2.326852414 0.759959789 2.355103543 0.769186727 2.399396263 0.783652916 2.455181192 0.801872508 2.517811236 0.822327745 2.582858282 0.843572383 2.646376332 0.864317646 2.705095315 0.883495513 2.756538193 0.900296973 2.799062163 0.914185481 2.831831648 0.924888132 2.854735943 0.932368772 2.868267408 0.936788205 2.873377066 0.938457041 2.87543057 0.939127724 2.883225048 0.941673433 2.895301503 0.945617654 2.910266812 0.950505387 2.926869179 0.95592779 2.944046312 0.961537914 2.960948714 0.967058309 2.976942108 0.972281819 2.991593922 0.977067163 3.00464889 0.981330971 3.015998536 0.985037813 3.025648555 0.98818955 3.033687214 0.99081501 3.040256952 0.992960714 3.04553044 0.994683057 3.049691655 0.996042127 3.05292188 0.9970971323.055390203 0.997903297 3.057247806 0.998509998 3.05862526 0.99895988 3.059632056 0.999288704 3.060357642 0.999525684 3.060873401 0.999694133 3.061235083 0.99981226 3.061485364 0.999894002 3.061656301 0.999949831 3.06177155 0.999987472 3.061809909 1 153.2361334Raw Data ranked by lb/ton, use concentration data Runs 514 1 1.02E-03 2 3.18E-03 3 1.78E-03 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 2.00E-03 Result Kurtosis #DIV/0! #DIV/0! Skewness 0.85 0.07 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 514 1 -6.88E+00 2 -5.75E+00 3 -6.33E+00 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 3

3. Calculate -6.32E+00

0.321232234 4. Calculate

5. m= number future runs = 3 6. Calculate Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1 ˆ n i

i x n     2 2 1 ˆ 1

n i i x x n    

 

### 

222 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 PM-conc 12/1/201411.56063048
0.430526856

28.85228652
7. Calculate 1.200673476

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

3.484z value from "z-stat"

a- Calculate 0.002111346
b- Calculate 4.45778E-06

c- Calculate 0.378825754

d-calculate 0.132874948

e- Calculate 0.003224

f- Calculate UPL 5.86E-03 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term
b- denominator first term

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

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z



2 ˆ 2 2 2 2 2 4 ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e   2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



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

0.101 28.85229 1.200673

-5=f(a) 493.879 1 -4.899 448.9942 2 -4.798 407.0825 3 -4.697 368.0158 4 -4.596 331.6691 5 -4.495 297.9197 6 -4.394 266.6478 7 -4.293 237.7364 8 -4.192 211.0708 9 -4.091 186.5395 10 -3.99 164.0332 11 -3.889 143.4457 12 -3.788 124.6732 13 -3.687 107.6147 14 -3.586 92.17184 15 -3.485 78.24911 16 -3.384 65.75353 17 -3.283 54.59484 18 -3.182 44.6855 19 -3.081 35.94065 20 -2.98 28.27811 21 -2.879 21.61841 22 -2.778 15.88474 23 -2.677 11.00301 24 -2.576 6.901806 25 -2.475 3.512406 26 -2.374 0.768779 27 -2.273 -1.39242 28 -2.172 -3.03183 29 -2.071 -4.20742 30 -1.97 -4.97447 31 -1.869 -5.38554 32 -1.768 -5.49053 33 -1.667 -5.33665 34 -1.566 -4.96839 35 -1.465 -4.42759 36 -1.364 -3.75336 37 -1.263 -2.98215 38 -1.162 -2.1477 39 -1.061 -1.28108 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -0.41066

41 -0.859 0.437898 42 -0.758 1.241592 43 -0.657 1.980129 44 -0.556 2.6359 45 -0.455 3.193989 46 -0.354 3.642168 47 -0.253 3.970899 48 -0.152 4.173336 49 -0.051 4.245323 50 0.05 4.185393 51 0.151 3.994769 52 0.252 3.677366 53 0.353 3.239788 54 0.454 2.691329 55 0.555 2.043973 56 0.656 1.312396 57 0.757 0.513962 58 0.858 -0.33127 59 0.959 -1.20057 60 1.06 -2.06848 61 1.161 -2.90689 62 1.262 -3.68497 63 1.363 -4.36923 64 1.464 -4.92347 65 1.565 -5.3088 66 1.666 -5.48364 67 1.767 -5.40372 68 1.868 -5.0221 69 1.969 -4.28911 70 2.07 -3.15244 71 2.171 -1.55704 72 2.272 0.554806 73 2.373 3.243498 74 2.474 6.572137 75 2.575 10.60651 76 2.676 15.4151 77 2.777 21.06908 78 2.878 27.64229 79 2.979 35.2113 80 3.08 43.85534 81 3.181 53.65635 82 3.282 64.69893 83 3.383 77.0704 84 3.484 90.86077 85 3.585 106.1627 86 3.686 123.0716 87 3.787 141.6856 88 3.888 162.105389 3.989 184.4343

90 4.09 208.7787 91 4.191 235.2473 92 4.292 263.9516 93 4.393 295.0059 94 4.494 328.527 95 4.595 364.6346 96 4.696 403.451 97 4.797 445.101 98 4.898 489.7124 99 4.999 537.4155distribution of Z Normal distribution

1.48672E-06 0.00073426 2.45094E-06 0.001100458 3.9995E-06 0.001628127 6.46025E-06 0.002377473 1.03291E-05 0.003425839 1.63472E-05 0.004870167 2.56093E-05 0.006828654 3.97118E-05 0.009440931 6.09553E-05 0.012865876 9.26132E-05 0.017276016 0.000139285 0.022847365 0.000207351 0.029743563 0.000305546 0.038093374 0.000445674 0.047961043 0.000643469 0.059309735 0.000919619 0.071959376 0.001300942 0.085541534 0.001821704 0.099455631 0.002525035 0.112832439 0.003464389 0.124512395 0.004704958 0.133047326 0.006324913 0.136734543 0.008416337 0.133691335 0.011085658 0.121975616 0.014453386 0.099754472 0.018652949 0.065516732 0.023828414 0.018318787 0.030130931 -0.041954788 0.03771375 -0.114341673 0.046725789 -0.196595176 0.057303789 -0.285055794 0.069563239 -0.374635636 0.083588399 -0.458944983 0.099421884 -0.530579646 0.117054396 -0.581572166 0.136415346 -0.603990604 0.157365126 -0.590647504 0.179689839 -0.535861365 0.203099244 -0.436196588 0.227228529 -0.291098433 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 -0.103339333 0.275855243 0.12079643 0.299326439 0.371641392 0.321498297 0.636608094 0.341807853 0.900971442 0.359712192 1.148916769 0.37471238 1.364765255 0.386376486 1.534261981 0.394360216 1.645797856 0.398423793 1.691437805 0.398443914 1.667644387 0.394419966 1.575616834 0.386474058 1.421206737 0.374844865 1.214417959 0.359875719 0.968543939 0.34199778 0.699034335 0.32170943 0.422210168 0.299553264 0.153958978 0.276092166 -0.091462079 0.25188591 -0.302405574 0.227469632 -0.470516011 0.203335281 -0.591072493 0.17991684 -0.662988673 0.157579839 -0.688502973 0.136615273 -0.672621059 0.117237788 -0.622391438 0.099587708 -0.546102677 0.083736272 -0.45248751 0.069693339 -0.35000673 0.05741676 -0.246267083 0.046822635 -0.147605411 0.037795734 -0.058849316 0.030199481 0.016754864 0.023885038 0.077471072 0.018699163 0.122893462 0.014490659 0.153695357 0.011115369 0.171344555 0.008439746 0.177817661 0.006343145 0.175339083 0.004718997 0.166162026 0.003475077 0.152400712 0.002533081 0.135915856 0.001827693 0.118249807 0.001305351 0.100603957 0.000922829 0.083848961 0.00064578 0.068557805 0.00044732 0.055052392 0.000306705 0.043455725 0.000208159 0.0337436070.000139842 0.025791619 9.29928E-05 0.019414913 6.12113E-05 0.014399788 3.98826E-05 0.010527075 2.5722E-05 0.007588148 1.64209E-05 0.005394705 1.03767E-05 0.003783692 6.49066E-06 0.002618663 4.01874E-06 0.001788744 2.46298E-06 0.001206149 1.49417E-06 0.000802991Absolute values

0.00073426 3.70801E-05 0.001100458 0.000111146 0.001628127 0.000164441 0.002377473 0.000240125 0.003425839 0.00034601 0.004870167 0.000491887 0.006828654 0.000689694 0.009440931 0.000953534 0.012865876 0.001299454 0.017276016 0.001744878 0.022847365 0.002307584 0.029743563 0.0030041 0.038093374 0.003847431 0.047961043 0.004844065 0.059309735 0.005990283 0.071959376 0.007267897 0.085541534 0.008639695 0.099455631 0.010045019 0.112832439 0.011396076 0.124512395 0.012575752 0.133047326 0.01343778 0.136734543 0.013810189 0.133691335 0.013502825 0.121975616 0.012319537 0.099754472 0.010075202 0.065516732 0.00661719 0.018318787 0.001850198 0.041954788 0.004237434 0.114341673 0.011548509 0.196595176 0.019856113 0.285055794 0.028790635 0.374635636 0.037838199 0.458944983 0.046353443 0.530579646 0.053588544 0.581572166 0.058738789 0.603990604 0.061003051 0.590647504 0.059655398 0.535861365 0.054121998 0.436196588 0.044055855 0.291098433 0.029400942 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.103339333 0.010437273 0.12079643 0.012200439 0.371641392 0.037535781 0.636608094 0.064297417 0.900971442 0.090998116 1.148916769 0.116040594 1.364765255 0.137841291 1.534261981 0.15496046 1.645797856 0.166225583 1.691437805 0.170835218 1.667644387 0.168432083 1.575616834 0.1591373 1.421206737 0.14354188 1.214417959 0.122656214 0.968543939 0.097822938 0.699034335 0.070602468 0.422210168 0.042643227 0.153958978 0.015549857 0.091462079 0.00923767 0.302405574 0.030542963 0.470516011 0.047522117 0.591072493 0.059698322 0.662988673 0.066961856 0.688502973 0.0695388 0.672621059 0.067934727 0.622391438 0.062861535 0.546102677 0.05515637 0.45248751 0.045701238 0.35000673 0.03535068 0.246267083 0.024872975 0.147605411 0.014908147 0.058849316 0.005943781 0.016754864 0.001692241 0.077471072 0.007824578 0.122893462 0.01241224 0.153695357 0.015523231 0.171344555 0.0173058 0.177817661 0.017959584 0.175339083 0.017709247 0.166162026 0.016782365 0.152400712 0.015392472 0.135915856 0.013727501 0.118249807 0.011943231 0.100603957 0.010161 0.083848961 0.008468745 0.068557805 0.006924338 0.055052392 0.005560292 0.043455725 0.004389028 0.033743607 0.0034081040.025791619 0.002604953 0.019414913 0.001960906 0.014399788 0.001454379 0.010527075 0.001063235 0.007588148 0.000766403 0.005394705 0.000544865 0.003783692 0.000382153 0.002618663 0.000264485 0.001788744 0.000180663 0.001206149 0.000121821 0.000802991 4.0551E-05Cumulative normalization
3.70801E-05 1.14751E-05 0.000148226 4.58713E-05 0.000312667 9.67604E-05 0.000552792 0.000171071 0.000898802 0.00027815 0.001390689 0.000430373 0.002080383 0.000643811 0.003033917 0.000938899 0.00433337 0.001341038 0.006078248 0.001881021 0.008385832 0.002595143 0.011389931 0.003524815 0.015237362 0.00471547 0.020081428 0.006214552 0.026071711 0.008068351 0.033339608 0.010317529 0.041979303 0.012991235 0.052024321 0.016099844 0.063420398 0.01962656 0.07599615 0.023518348 0.08943393 0.027676906 0.103244119 0.031950713 0.116746943 0.0361294 0.129066481 0.039941898 0.139141682 0.043059847 0.145758872 0.045107653 0.14760907 0.045680229 0.151846503 0.046991578 0.163395012 0.050565467 0.183251125 0.056710292 0.21204176 0.065620061 0.249879959 0.077329759 0.296233403 0.09167465 0.349821947 0.10825857 0.408560736 0.126436323 0.469563787 0.145314793 0.529219185 0.163776208 0.583341183 0.180525214 0.627397038 0.194159076 0.65679798 0.203257715 

## ( ) 2 () () f a f x f b        0.667235252 0.206487713 0.679435692 0.210263355 0.716971472 0.221879464 0.78126889 0.241777433 0.872267006 0.269938405 0.988307599 0.305849212 1.12614889 0.34850663 1.28110935 0.396461877 1.447334933 0.447903314 1.618170152 0.500771285 1.786602235 0.552895563 1.945739535 0.602143406 2.089281415 0.646564972 2.211937629 0.6845231 2.309760567 0.714796133 2.380363035 0.736645312 2.423006262 0.74984201 2.438556119 0.754654187 2.447793789 0.757512947 2.478336752 0.766965006 2.525858869 0.781671563 2.585557191 0.800146261 2.652519047 0.820868788 2.722057847 0.842388796 2.789992574 0.863412395 2.852854109 0.882866005 2.908010479 0.899935117 2.953711718 0.914078171 2.989062398 0.925018062 3.013935373 0.932715443 3.02884352 0.937329031 3.0347873 0.939168439 3.036479542 0.939692133 3.04430412 0.942113587 3.05671636 0.945954773 3.072239591 0.950758711 3.089545391 0.956114296 3.107504975 0.961672206 3.125214222 0.967152645 3.141996587 0.972346243 3.157389059 0.977109715 3.17111656 0.981357932 3.18305979 0.985053976 3.19322079 0.988198476 3.201689535 0.990819279 3.208613873 0.992962137 3.214174165 0.994682867 3.218563193 0.99604113 3.221971298 0.9970958283.224576251 0.997901976 3.226537157 0.998508813 3.227991536 0.998958896 3.229054771 0.999287933 3.229821174 0.99952511 3.230366039 0.999693728 3.230748192 0.999811992 3.231012676 0.999893842 3.23119334 0.999949751 3.231315161 0.999987451 3.231355712 1 161.833731Raw Data Runs 599 1 8.11E-03 2 1.53E-02 3 4.53E-03 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 9.30E-03 Result Kurtosis #DIV/0! #DIV/0! Skewness 0.93 0.12 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 599 1 -4.81E+00 2 -4.18E+00 3 -5.40E+00 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 3

3. Calculate -4.80E+00

0.369349906 4. Calculate

5. m= number future runs = 3 6. Calculate Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1 ˆ n i

i x n     2 2 1 ˆ 1

n i i x x n    

 

### 

222 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 PM-Small-lbton 12/1/201413.71806635
0.598873984

24.90643227
7. Calculate 1.330173276

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

3.484z value from "z-stat"

a- Calculate 0.009916872
b- Calculate 9.83443E-05

c- Calculate 0.446793757

d-calculate 0.157221474

e- Calculate 0.016461

f- Calculate UPL 2.90E-02 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term
b- denominator first term

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

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z



2 ˆ 2 2 2 2 2 4 ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e   2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



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

0.101 24.90643 1.330173

-5=f(a) 412.9166 1 -4.899 375.2538 2 -4.798 340.0945 3 -4.697 307.3309 4 -4.596 276.8569 5 -4.495 248.5693 6 -4.394 222.3667 7 -4.293 198.1502 8 -4.192 175.8232 9 -4.091 155.2912 10 -3.99 136.4622 11 -3.889 119.2462 12 -3.788 103.5559 13 -3.687 89.30591 14 -3.586 76.41324 15 -3.485 64.79719 16 -3.384 54.37934 17 -3.283 45.08354 18 -3.182 36.83593 19 -3.081 29.56492 20 -2.98 23.20119 21 -2.879 17.67773 22 -2.778 12.92977 23 -2.677 8.894862 24 -2.576 5.512804 25 -2.475 2.725688 26 -2.374 0.477885 27 -2.273 -1.28396 28 -2.172 -2.61091 29 -2.071 -3.55177 30 -1.97 -4.15304 31 -1.869 -4.45896 32 -1.768 -4.51148 33 -1.667 -4.35027 34 -1.566 -4.01273 35 -1.465 -3.53397 36 -1.364 -2.94683 37 -1.263 -2.28186 38 -1.162 -1.56734 39 -1.061 -0.82926 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -0.09134

41 -0.859 0.624982 42 -0.758 1.300552 43 -0.657 1.91849 44 -0.556 2.464199 45 -0.455 2.925361 46 -0.354 3.291937 47 -0.253 3.556168 48 -0.152 3.712574 49 -0.051 3.757956 50 0.05 3.691392 51 0.151 3.514242 52 0.252 3.230144 53 0.353 2.845018 54 0.454 2.367062 55 0.555 1.806751 56 0.656 1.176845 57 0.757 0.492379 58 0.858 -0.22933 59 0.959 -0.96869 60 1.06 -1.70381 61 1.161 -2.41056 62 1.262 -3.06248 63 1.363 -3.63088 64 1.464 -4.08474 65 1.565 -4.39081 66 1.666 -4.51352 67 1.767 -4.41505 68 1.868 -4.05528 69 1.969 -3.39182 70 2.07 -2.38 71 2.171 -0.97287 72 2.272 0.878804 73 2.373 3.226532 74 2.474 6.124104 75 2.575 9.627587 76 2.676 13.79533 77 2.777 18.68797 78 2.878 24.3684 79 2.979 30.90182 80 3.08 38.35569 81 3.181 46.79976 82 3.282 56.30606 83 3.383 66.94889 84 3.484 78.80484 85 3.585 91.95278 86 3.686 106.4738 87 3.787 122.4515 88 3.888 139.971389 3.989 159.1214

90 4.09 179.9921 91 4.191 202.6757 92 4.292 227.2673 93 4.393 253.8638 94 4.494 282.5646 95 4.595 313.4715 96 4.696 346.6883 97 4.797 382.3214 98 4.898 420.479 99 4.999 461.2722distribution of Z Normal distribution

1.48672E-06 0.000613891 2.45094E-06 0.000919724 3.9995E-06 0.001360209 6.46025E-06 0.001985433 1.03291E-05 0.00285968 1.63472E-05 0.004063424 2.56093E-05 0.005694647 3.97118E-05 0.007868896 6.09553E-05 0.010717347 9.26132E-05 0.014382012 0.000139285 0.019007128 0.000207351 0.024725792 0.000305546 0.031641088 0.000445674 0.039801316 0.000643469 0.049169559 0.000919619 0.059588731 0.001300942 0.070744376 0.001821704 0.082128862 0.002525035 0.093011996 0.003464389 0.102424363 0.004704958 0.109160612 0.006324913 0.11181007 0.008416337 0.108821326 0.011085658 0.0986054 0.014453386 0.079678686 0.018652949 0.050842128 0.023828414 0.011387237 0.030130931 -0.038686852 0.03771375 -0.098467268 0.046725789 -0.165959148 0.057303789 -0.237984837 0.069563239 -0.310179458 0.083588399 -0.377106983 0.099421884 -0.432511795 0.117054396 -0.469707425 0.136415346 -0.482087746 0.157365126 -0.463728123 0.179689839 -0.410027034 0.203099244 -0.318325173 0.227228529 -0.188431409 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 -0.022985307 0.275855243 0.172404637 0.299326439 0.389289469 0.321498297 0.616791231 0.341807853 0.842282666 0.359712192 1.052288161 0.37471238 1.233529721 0.386376486 1.37401981 0.394360216 1.464091615 0.398423793 1.49725891 0.398443914 1.470812517 0.394419966 1.386087066 0.386474058 1.248367034 0.374844865 1.066440548 0.359875719 0.851847973 0.34199778 0.617904949 0.32170943 0.378602133 0.299553264 0.14749381 0.276092166 -0.0633161 0.25188591 -0.243998237 0.227469632 -0.38756574 0.203335281 -0.490151319 0.17991684 -0.55099227 0.157579839 -0.572152916 0.136615273 -0.558038323 0.117237788 -0.514768897 0.099587708 -0.449491561 0.083736272 -0.369700079 0.069693339 -0.282626253 0.05741676 -0.194747569 0.046822635 -0.111438064 0.037795734 -0.036770395 0.030199481 0.02653942 0.023885038 0.077065847 0.018699163 0.11451561 0.014490659 0.139510086 0.011115369 0.153340208 0.008439746 0.157721699 0.006343145 0.1545723 0.004718997 0.145825584 0.003475077 0.133288992 0.002533081 0.118547577 0.001827693 0.102910217 0.001305351 0.087391828 0.000922829 0.072723399 0.00064578 0.059381302 0.00044732 0.047627868 0.000306705 0.037556517 0.000208159 0.0291362280.000139842 0.022251823 9.29928E-05 0.016737965 6.12113E-05 0.012406041 3.98826E-05 0.009064008 2.5722E-05 0.00652989 1.64209E-05 0.004639961 1.03767E-05 0.003252789 6.49066E-06 0.002250236 4.01874E-06 0.001536449 2.46298E-06 0.001035629 1.49417E-06 0.000689219Absolute values

0.000613891 3.10015E-05 0.000919724 9.28922E-05 0.001360209 0.000137381 0.001985433 0.000200529 0.00285968 0.000288828 0.004063424 0.000410406 0.005694647 0.000575159 0.007868896 0.000794758 0.010717347 0.001082452 0.014382012 0.001452583 0.019007128 0.00191972 0.024725792 0.002497305 0.031641088 0.00319575 0.039801316 0.004019933 0.049169559 0.004966125 0.059588731 0.006018462 0.070744376 0.007145182 0.082128862 0.008295015 0.093011996 0.009394212 0.102424363 0.010344861 0.109160612 0.011025222 0.11181007 0.011292817 0.108821326 0.010990954 0.0986054 0.009959145 0.079678686 0.008047547 0.050842128 0.005135055 0.011387237 0.001150111 0.038686852 0.003907372 0.098467268 0.009945194 0.165959148 0.016761874 0.237984837 0.024036469 0.310179458 0.031328125 0.377106983 0.038087805 0.432511795 0.043683691 0.469707425 0.04744045 0.482087746 0.048690862 0.463728123 0.04683654 0.410027034 0.04141273 0.318325173 0.032150842 0.188431409 0.019031572 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.022985307 0.002321516 0.172404637 0.017412868 0.389289469 0.039318236 0.616791231 0.062295914 0.842282666 0.085070549 1.052288161 0.106281104 1.233529721 0.124586502 1.37401981 0.138776001 1.464091615 0.147873253 1.49725891 0.15122315 1.470812517 0.148552064 1.386087066 0.139994794 1.248367034 0.12608507 1.066440548 0.107710495 0.851847973 0.086036645 0.617904949 0.0624084 0.378602133 0.038238815 0.14749381 0.014896875 0.0633161 0.006394926 0.243998237 0.024643822 0.38756574 0.03914414 0.490151319 0.049505283 0.55099227 0.055650219 0.572152916 0.057787444 0.558038323 0.056361871 0.514768897 0.051991659 0.449491561 0.045398648 0.369700079 0.037339708 0.282626253 0.028545252 0.194747569 0.019669504 0.111438064 0.011255244 0.036770395 0.00371381 0.02653942 0.002680481 0.077065847 0.007783651 0.11451561 0.011566077 0.139510086 0.014090519 0.153340208 0.015487361 0.157721699 0.015929892 0.1545723 0.015611802 0.145825584 0.014728384 0.133288992 0.013462188 0.118547577 0.011973305 0.102910217 0.010393932 0.087391828 0.008826575 0.072723399 0.007345063 0.059381302 0.005997511 0.047627868 0.004810415 0.037556517 0.003793208 0.029136228 0.0029427590.022251823 0.002247434 0.016737965 0.001690534 0.012406041 0.00125301 0.009064008 0.000915465 0.00652989 0.000659519 0.004639961 0.000468636 0.003252789 0.000328532 0.002250236 0.000227274 0.001536449 0.000155181 0.001035629 0.000104599 0.000689219 3.48056E-05Cumulative normalization
3.10015E-05 1.11285E-05 0.000123894 4.44738E-05 0.000261275 9.37891E-05 0.000461804 0.000165772 0.000750631 0.000269452 0.001161037 0.000416774 0.001736196 0.000623238 0.002530955 0.00090853 0.003613407 0.001297095 0.00506599 0.001818525 0.00698571 0.002507642 0.009483015 0.003404093 0.012678765 0.004551262 0.016698698 0.005994287 0.021664823 0.007776964 0.027683285 0.009937395 0.034828467 0.012502282 0.043123482 0.015479921 0.052517694 0.018852136 0.062862554 0.022565603 0.073887776 0.026523298 0.085180593 0.03057705 0.096171547 0.034522444 0.106130693 0.038097451 0.11417824 0.040986258 0.119313295 0.042829575 0.120463406 0.043242427 0.124370778 0.044645046 0.134315972 0.048215046 0.151077846 0.054232011 0.175114314 0.062860318 0.20644244 0.074106091 0.244530245 0.087778369 0.288213936 0.103459387 0.335654386 0.120488959 0.384345248 0.137967388 0.431181789 0.154780176 0.472594519 0.169645993 0.504745362 0.181187095 0.523776934 0.188018808 

## ( ) 2 () () f a f x f b        0.52609845 0.188852156 0.543511318 0.195102807 0.582829555 0.209216769 0.645125469 0.231578967 0.730196018 0.262116515 0.836477123 0.30026796 0.961063625 0.344990444 1.099839625 0.394806494 1.247712878 0.447888161 1.398936028 0.502172332 1.547488093 0.555497669 1.687482886 0.605751227 1.813567957 0.651011648 1.921278452 0.689676197 2.007315097 0.720560542 2.069723497 0.742963119 2.107962313 0.756689604 2.122859187 0.762037095 2.129254113 0.764332665 2.153897935 0.773178992 2.193042075 0.787230459 2.242547358 0.805001239 2.298197577 0.824977849 2.355985022 0.845721654 2.412346892 0.865953724 2.464338551 0.884617031 2.509737199 0.900913662 2.547076907 0.914317398 2.575622158 0.92456421 2.595291663 0.93162492 2.606546907 0.935665185 2.610260717 0.936998321 2.612941198 0.937960527 2.620724849 0.940754603 2.632290926 0.944906447 2.646381444 0.949964482 2.661868805 0.955523939 2.677798697 0.96124225 2.693410499 0.966846377 2.708138883 0.972133385 2.721601071 0.976965871 2.733574377 0.981263897 2.743968309 0.984994979 2.752794883 0.98816343 2.760139947 0.990800068 2.766137458 0.992952979 2.770947873 0.994679761 2.774741081 0.996041399 2.77768384 0.9970977532.779931274 0.997904508 2.781621809 0.998511355 2.782874819 0.998961145 2.783790284 0.999289767 2.784449803 0.999526512 2.784918439 0.999694737 2.78524697 0.99981267 2.785474244 0.999894253 2.785629425 0.999949958 2.785734024 0.999987506 2.78576883 1 139.2201487Raw Data Runs 599 1 2.03E-03 2 3.77E-03 3 1.12E-03 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 3.00 3.00 Kurtosis #DIV/0! #DIV/0! SE Kurtosis #DIV/0! #DIV/0! Average of Raw Data 2.31E-03 Result Kurtosis #DIV/0! #DIV/0! Skewness 0.90 0.08 SE Skewness 1.22 1.22 Result Raw Data Result Log Data Result Skewness normal normal #DIV/0! #DIV/0! 1. Take LN( ) of Raw Data Runs 599 1 -6.20E+00 2 -5.58E+00 3 -6.79E+00 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 3

3. Calculate -6.19E+00

0.367033652 4. Calculate

5. m= number future runs = 3 6. Calculate Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1 ˆ n i

i x n     2 2 1 ˆ 1

n i i x x n    

 

### 

222 ˆˆˆ 4 3 2 233 1 3 1 e e e          

0900006481934216 PM-Small-conc 12/1/201413.60669361
0.589934377

25.06475796
7. Calculate 1.323894336

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

3.484z value from "z-stat"

a- Calculate 0.002459994
b- Calculate 6.05157E-06

c- Calculate 0.443446493

d-calculate 0.156022976

e- Calculate 0.004068

f- Calculate UPL 7.18E-03 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term
b- denominator first term

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

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z 

2 ˆ 2 2 2 2 2 4 ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e   2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



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

0.101 25.06476 1.323894

-5=f(a) 416.185 1 -4.899 378.2311 2 -4.798 342.7997 3 -4.697 309.782 4 -4.596 279.0713 5 -4.495 250.5634 6 -4.394 224.1564 7 -4.293 199.7506 8 -4.192 177.2485 9 -4.091 156.5552 10 -3.99 137.5777 11 -3.889 120.2257 12 -3.788 104.411 13 -3.687 90.04759 14 -3.586 77.05193 15 -3.485 65.34269 16 -3.384 54.84089 17 -3.283 45.46979 18 -3.182 37.15498 19 -3.081 29.82435 20 -2.98 23.40806 21 -2.879 17.83859 22 -2.778 13.0507 23 -2.677 8.981463 24 -2.576 5.570227 25 -2.475 2.758651 26 -2.374 0.490686 27 -2.273 -1.28742 28 -2.172 -2.62713 29 -2.071 -3.57759 30 -1.97 -4.18568 31 -1.869 -4.49597 32 -1.768 -4.55072 33 -1.667 -4.38991 34 -1.566 -4.05123 35 -1.465 -3.57005 36 -1.364 -2.97947 37 -1.263 -2.31028 38 -1.162 -1.59097 39 -1.061 -0.84775 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -0.10451

41 -0.859 0.617125 42 -0.758 1.297854 43 -0.657 1.920659 44 -0.556 2.470819 45 -0.455 2.935911 46 -0.354 3.305807 47 -0.253 3.572675 48 -0.152 3.730981 49 -0.051 3.777483 50 0.05 3.71124 51 0.151 3.533604 52 0.252 3.248223 53 0.353 2.861042 54 0.454 2.380302 55 0.555 1.816539 56 0.656 1.182588 57 0.757 0.493577 58 0.858 -0.23307 59 0.959 -0.97763 60 1.06 -1.71809 61 1.161 -2.43013 62 1.262 -3.08714 63 1.363 -3.66022 64 1.464 -4.11817 65 1.565 -4.42749 66 1.666 -4.55238 67 1.767 -4.45476 68 1.868 -4.09424 69 1.969 -3.42814 70 2.07 -2.41148 71 2.171 -0.99698 72 2.272 0.864917 73 2.373 3.226086 74 2.474 6.140687 75 2.575 9.665179 76 2.676 13.85832 77 2.777 18.78115 78 2.878 24.49702 79 2.979 31.07158 80 3.08 38.57276 81 3.181 47.07079 82 3.282 56.63822 83 3.383 67.34986 84 3.484 79.28284 85 3.585 92.51657 86 3.686 107.1328 87 3.787 123.2155 88 3.888 140.850989 3.989 160.1278

90 4.09 181.137 91 4.191 203.9716 92 4.292 228.7273 93 4.393 255.5017 94 4.494 284.3949 95 4.595 315.5094 96 4.696 348.9498 97 4.797 384.823 98 4.898 423.2384 99 4.999 464.3076distribution of Z Normal distribution

1.48672E-06 0.00061875 2.45094E-06 0.000927022 3.9995E-06 0.001371028 6.46025E-06 0.002001268 1.03291E-05 0.002882552 1.63472E-05 0.004096022 2.56093E-05 0.00574048 3.97118E-05 0.007932448 6.09553E-05 0.010804227 9.26132E-05 0.014499072 0.000139285 0.019162512 0.000207351 0.024928892 0.000305546 0.031902354 0.000445674 0.040131863 0.000643469 0.049580535 0.000919619 0.060090387 0.001300942 0.07134482 0.001821704 0.082832487 0.002525035 0.093817614 0.003464389 0.103323142 0.004704958 0.110133936 0.006324913 0.112827533 0.008416337 0.109839129 0.011085658 0.099565434 0.014453386 0.080508648 0.018652949 0.051456981 0.023828414 0.011692272 0.030130931 -0.038791199 0.03771375 -0.099078831 0.046725789 -0.167165893 0.057303789 -0.239855592 0.069563239 -0.312754163 0.083588399 -0.380387341 0.099421884 -0.436453338 0.117054396 -0.474214065 0.136415346 -0.487009838 0.157365126 -0.468864849 0.179689839 -0.41513362 0.203099244 -0.323124977 0.227228529 -0.192632427 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 -0.026300091 0.275855243 0.170237143 0.299326439 0.388482034 0.321498297 0.61748847 0.341807853 0.844545238 0.359712192 1.056082845 0.37471238 1.238726673 0.386376486 1.380397662 0.394360216 1.471350295 0.398423793 1.505039262 0.398443914 1.478721087 0.394419966 1.393723865 0.386474058 1.255353764 0.374844865 1.072446741 0.359875719 0.856612727 0.34199778 0.621252443 0.32170943 0.380449756 0.299553264 0.147852541 0.276092166 -0.064348664 0.25188591 -0.246251116 0.227469632 -0.390812433 0.203335281 -0.494130352 0.17991684 -0.555428408 0.157579839 -0.576777136 0.136615273 -0.562604983 0.117237788 -0.519068967 0.099587708 -0.45336132 0.083736272 -0.373025154 0.069693339 -0.285341364 0.05741676 -0.19683266 0.046822635 -0.112911738 0.037795734 -0.037681672 0.030199481 0.026120048 0.023885038 0.077055193 0.018699163 0.114825714 0.014490659 0.140054815 0.011115369 0.154040288 0.008439746 0.158508112 0.006343145 0.155388153 0.004718997 0.146626671 0.003475077 0.134043312 0.002533081 0.119234119 0.001827693 0.103517299 0.001305351 0.087915229 0.000922829 0.073164505 0.00064578 0.059745389 0.00044732 0.047922624 0.000306705 0.037790843 0.000208159 0.0293193280.000139842 0.022392556 9.29928E-05 0.016844434 6.12113E-05 0.012485365 3.98826E-05 0.009122237 2.5722E-05 0.006572019 1.64209E-05 0.004670016 1.03767E-05 0.003273935 6.49066E-06 0.002264914 4.01874E-06 0.001546502 2.46298E-06 0.001042426 1.49417E-06 0.000693755Absolute values

0.00061875 3.12469E-05 0.000927022 9.36292E-05 0.001371028 0.000138474 0.002001268 0.000202128 0.002882552 0.000291138 0.004096022 0.000413698 0.00574048 0.000579788 0.007932448 0.000801177 0.010804227 0.001091227 0.014499072 0.001464406 0.019162512 0.001935414 0.024928892 0.002517818 0.031902354 0.003222138 0.040131863 0.004053318 0.049580535 0.005007634 0.060090387 0.006069129 0.07134482 0.007205827 0.082832487 0.008366081 0.093817614 0.009475579 0.103323142 0.010435637 0.110133936 0.011123528 0.112827533 0.011395581 0.109839129 0.011093752 0.099565434 0.010056109 0.080508648 0.008131373 0.051456981 0.005197155 0.011692272 0.001180919 0.038791199 0.003917911 0.099078831 0.010006962 0.167165893 0.016883755 0.239855592 0.024225415 0.312754163 0.03158817 0.380387341 0.038419121 0.436453338 0.044081787 0.474214065 0.047895621 0.487009838 0.049187994 0.468864849 0.04735535 0.41513362 0.041928496 0.323124977 0.032635623 0.192632427 0.019455875 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.026300091 0.002656309 0.170237143 0.017193951 0.388482034 0.039236685 0.61748847 0.062366335 0.844545238 0.085299069 1.056082845 0.106664367 1.238726673 0.125111394 1.380397662 0.139420164 1.471350295 0.14860638 1.505039262 0.152008965 1.478721087 0.14935083 1.393723865 0.14076611 1.255353764 0.12679073 1.072446741 0.108317121 0.856612727 0.086517885 0.621252443 0.062746497 0.380449756 0.038425425 0.147852541 0.014933107 0.064348664 0.006499215 0.246251116 0.024871363 0.390812433 0.039472056 0.494130352 0.049907166 0.555428408 0.056098269 0.576777136 0.058254491 0.562604983 0.056823103 0.519068967 0.052425966 0.45336132 0.045789493 0.373025154 0.037675541 0.285341364 0.028819478 0.19683266 0.019880099 0.112911738 0.011404086 0.037681672 0.003805849 0.026120048 0.002638125 0.077055193 0.007782575 0.114825714 0.011597397 0.140054815 0.014145536 0.154040288 0.015558069 0.158508112 0.016009319 0.155388153 0.015694203 0.146626671 0.014809294 0.134043312 0.013538375 0.119234119 0.012042646 0.103517299 0.010455247 0.087915229 0.008879438 0.073164505 0.007389615 0.059745389 0.006034284 0.047922624 0.004840185 0.037790843 0.003816875 0.029319328 0.0029612520.022392556 0.002261648 0.016844434 0.001701288 0.012485365 0.001261022 0.009122237 0.000921346 0.006572019 0.000663774 0.004670016 0.000471672 0.003273935 0.000330667 0.002264914 0.000228756 0.001546502 0.000156197 0.001042426 0.000105285 0.000693755 3.50346E-05Cumulative normalization
3.12469E-05 1.11452E-05 0.000124876 4.4541E-05 0.00026335 9.3932E-05 0.000465478 0.000166027 0.000756616 0.000269871 0.001170314 0.000417429 0.001750102 0.000624229 0.00255128 0.000909994 0.003642507 0.001299214 0.005106913 0.001821541 0.007042327 0.002511867 0.009560145 0.003409926 0.012782283 0.004559202 0.016835601 0.006004945 0.021843235 0.007791075 0.027912364 0.009955821 0.035118191 0.012526005 0.043484272 0.01551003 0.052959851 0.018889793 0.063395488 0.022611991 0.074519016 0.026579546 0.085914597 0.030644137 0.097008349 0.034601072 0.107064457 0.038187899 0.115195831 0.041088209 0.120392986 0.042941937 0.121573905 0.043363149 0.125491816 0.044760595 0.135498778 0.048329892 0.152382533 0.054352013 0.176607948 0.062992768 0.208196119 0.07425968 0.24661524 0.087963066 0.290697027 0.103686219 0.338592648 0.120769695 0.387780642 0.138314137 0.435135991 0.155204909 0.477064487 0.170160023 0.50970011 0.181800542 0.529155985 0.18874009 

## ( ) 2 () () f a f x f b        0.531812294 0.189687546 0.549006245 0.195820309 0.588242931 0.209815304 0.650609266 0.232060215 0.735908335 0.262484805 0.842572703 0.30053 0.967684096 0.345154905 1.10710426 0.394883483 1.25571064 0.447888612 1.407719606 0.502107381 1.557070435 0.555378042 1.697836546 0.605586694 1.824627276 0.650810587 1.932944397 0.689445288 2.019462282 0.720304607 2.082208779 0.742685115 2.120634204 0.756390749 2.135567311 0.76171711 2.142066526 0.764035259 2.166937889 0.772906411 2.206409944 0.786985359 2.25631711 0.804786316 2.312415379 0.824795524 2.37066987 0.845573817 2.427492973 0.86584156 2.479918939 0.884540926 2.525708432 0.900873186 2.563383973 0.91431135 2.59220345 0.924590721 2.612083549 0.931681583 2.623487635 0.935749208 2.627293483 0.937106683 2.629931608 0.938047653 2.637714183 0.940823553 2.64931158 0.944960128 2.663457116 0.950005578 2.679015185 0.955554851 2.695024505 0.961265078 2.710718708 0.966862908 2.725528002 0.972145107 2.739066376 0.976973993 2.751109022 0.981269381 2.76156427 0.984998573 2.770443708 0.988165704 2.777833323 0.990801442 2.783867607 0.992953759 2.788707792 0.994680162 2.792524667 0.996041571 2.795485919 0.9970977942.797747567 0.997904482 2.799448855 0.9985113 2.800709877 0.998961083 2.801631223 0.99928971 2.802294997 0.999526466 2.802766669 0.999694702 2.803097336 0.999812645 2.803326093 0.999894238 2.803482289 0.999949951 2.803587574 0.999987504 2.803622609 1 140.1275143Raw 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 11 n i i s x X n   

 1 1 1 i

n N ij i j X x n   

1 1 n m 

21 1 s n m        1 1 N i i df n

         

df t 2 1 1

s n m        1 1 i n i i

k k i x x n  1

N i in

  N

2
, 1 1 dfp X t s n m    

   

0900006481934216 UPL Pooled Template 12/1/2014Raw 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 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 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 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 ˆˆˆ 4 3 2 233 1 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

3.484z value from "z-stat"

a- Calculate #DIV/0!
b- Calculate #DIV/0!

c- Calculate #DIV/0!

d-calculate #DIV/0!

e- Calculate #DIV/0!

f- Calculate UPL #DIV/0! 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula b- denominator first term a- Numerator first term

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

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 ˆˆˆ 4 3 2 233 e e e   

###   2 2 ˆ 1 m e



2z    2 2 ˆ ˆ 1 1 2
z e e

m      

2z

 1z



2 ˆ 2 2 2 2 2 4 ˆ ˆ ˆ ˆ ˆ ˆ 2 2 2 .99 ˆ ˆ

( 1) 2( 1) z UPL e me e m e

m n n                    

  

2 ˆ 2 ˆ e   2 ˆ ˆ 2 e  

2 ˆ ( 1) e



2 4 ˆ ˆ 2( 1) n n       

  

2 2 2 2 4
ˆ ˆ ˆ ˆ ˆ 2 2 2 ˆ ˆ ( 1) 2( 1) me e me n n             

 

     2 2 2 ˆ 233 1 3 1 1 z e e e m

m e           



0900006481934216 Lognormal Template 12/1/2014∆z z

0.101 24.12133 1.363077

-5=f(a) 396.6768 1 -4.899 360.4593 2 -4.798 326.6518 3 -4.697 295.1499 4 -4.596 265.852 5 -4.495 238.6581 6 -4.394 213.4709 7 -4.293 190.195 8 -4.192 168.7373 9 -4.091 149.0069 10 -3.99 130.9151 11 -3.889 114.3752 12 -3.788 99.30312 13 -3.687 85.61659 14 -3.586 73.23572 15 -3.485 62.08277 16 -3.384 52.08221 17 -3.283 43.16072 18 -3.182 35.24714 19 -3.081 28.27255 20 -2.98 22.1702 21 -2.879 16.87554 22 -2.778 12.32624 23 -2.677 8.46214 24 -2.576 5.225296 25 -2.475 2.559955 26 -2.374 0.412563 27 -2.273 -1.26824 28 -2.172 -2.5316 29 -2.071 -3.42449 30 -1.97 -3.99166 31 -1.869 -4.27569 32 -1.768 -4.31694 33 -1.667 -4.15357 34 -1.566 -3.82156 35 -1.465 -3.35468 36 -1.364 -2.78451 37 -1.263 -2.14041 38 -1.162 -1.44958 39 -1.061 -0.737 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -0.02543

41 -0.859 0.664518 42 -0.758 1.314469 43 -0.657 1.908233 44 -0.556 2.431816 45 -0.455 2.873425 46 -0.354 3.223463 47 -0.253 3.474533 48 -0.152 3.621435 49 -0.051 3.661166 50 0.05 3.592923 51 0.151 3.418097 52 0.252 3.140281 53 0.353 2.765264 54 0.454 2.301034 55 0.555 1.757774 56 0.656 1.147869 57 0.757 0.4859 58 0.858 -0.21136 59 0.959 -0.92492 60 1.06 -1.63362 61 1.161 -2.31408 62 1.262 -2.94074 63 1.363 -3.48582 64 1.464 -3.91936 65 1.565 -4.20919 66 1.666 -4.32095 67 1.767 -4.21808 68 1.868 -3.86183 69 1.969 -3.21124 70 2.07 -2.22315 71 2.171 -0.85222 72 2.272 0.949112 73 2.373 3.230583 74 2.474 6.044142 75 2.575 9.443932 76 2.676 13.48629 77 2.777 18.22976 78 2.878 23.73508 79 2.979 30.06518 80 3.08 37.2852 81 3.181 45.46245 82 3.282 54.66648 83 3.383 64.96901 84 3.484 76.44396 85 3.585 89.16746 86 3.686 103.2178 87 3.787 118.6756 88 3.888 135.623489 3.989 154.1462

90 4.09 174.3312 91 4.191 196.2676 92 4.292 220.047 93 4.393 245.7629 94 4.494 273.5115 95 4.595 303.3907 96 4.696 335.5008 97 4.797 369.9444 98 4.898 406.8261 99 4.999 446.2528distribution of Z Normal distribution

1.48672E-06 0.000589747 2.45094E-06 0.000883464 3.9995E-06 0.001306445 6.46025E-06 0.001906741 1.03291E-05 0.002746009 1.63472E-05 0.003901404 2.56093E-05 0.005466833 3.97118E-05 0.007552981 6.09553E-05 0.010285427 9.26132E-05 0.013800006 0.000139285 0.018234501 0.000207351 0.023715784 0.000305546 0.030341661 0.000445674 0.038157079 0.000643469 0.047124921 0.000919619 0.057092498 0.001300942 0.067755949 0.001821704 0.078626046 0.002525035 0.089000254 0.003464389 0.097947106 0.004704958 0.104309835 0.006324913 0.10673633 0.008416337 0.103741783 0.011085658 0.093808392 0.014453386 0.075523218 0.018652949 0.047750703 0.023828414 0.009830716 0.030130931 -0.03821313 0.03771375 -0.095476112 0.046725789 -0.160011912 0.057303789 -0.228737477 0.069563239 -0.297430965 0.083588399 -0.360845903 0.099421884 -0.412955775 0.117054396 -0.447330438 0.136415346 -0.457629943 0.157365126 -0.43818421 0.179689839 -0.384610712 0.203099244 -0.29440934 0.227228529 -0.167466401 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

### f z z z z             

## ( ) z 0.251644341 -0.006400143 0.275855243 0.183310648 0.299326439 0.393455457 0.321498297 0.613493584 0.341807853 0.831213719 0.359712192 1.033605832 0.37471238 1.207871534 0.386376486 1.342478021 0.394360216 1.42815002 0.398423793 1.45869584 0.398443914 1.431578111 0.394419966 1.348165671 0.386474058 1.213637172 0.374844865 1.036545118 0.359875719 0.828086167 0.34199778 0.601154969 0.32170943 0.369280442 0.299553264 0.145552829 0.276092166 -0.058353815 0.25188591 -0.23297474 0.227469632 -0.371599393 0.203335281 -0.470535074 0.17991684 -0.529088588 0.157579839 -0.549294713 0.136615273 -0.535443759 0.117237788 -0.493475585 0.099587708 -0.43031327 0.083736272 -0.353206452 0.069693339 -0.26914386 0.05741676 -0.184378871 0.046822635 -0.104093763 0.037795734 -0.032210166 0.030199481 0.028662687 0.023885038 0.077162607 0.018699163 0.113020402 0.014490659 0.1368488 0.011115369 0.149905125 0.008439746 0.153854586 0.006343145 0.150555076 0.004718997 0.141877499 0.003475077 0.129568938 0.002533081 0.115160063 0.001827693 0.09991357 0.001305351 0.084807386 0.000922829 0.070544709 0.00064578 0.057582596 0.00044732 0.046171384 0.000306705 0.036398428 0.000208159 0.0282311680.000139842 0.021556082 9.29928E-05 0.016211547 6.12113E-05 0.012013793 3.98826E-05 0.008776044 2.5722E-05 0.00632152 1.64209E-05 0.004491301 1.03767E-05 0.003148183 6.49066E-06 0.002177621 4.01874E-06 0.001486709 2.46298E-06 0.001002002 1.49417E-06 0.000666778Absolute values

0.000589747 2.97822E-05 0.000883464 8.92299E-05 0.001306445 0.000131951 0.001906741 0.000192581 0.002746009 0.000277347 0.003901404 0.000394042 0.005466833 0.00055215 0.007552981 0.000762851 0.010285427 0.001038828 0.013800006 0.001393801 0.018234501 0.001841685 0.023715784 0.002395294 0.030341661 0.003064508 0.038157079 0.003853865 0.047124921 0.004759617 0.057092498 0.005766342 0.067755949 0.006843351 0.078626046 0.007941231 0.089000254 0.008989026 0.097947106 0.009892658 0.104309835 0.010535293 0.10673633 0.010780369 0.103741783 0.01047792 0.093808392 0.009474648 0.075523218 0.007627845 0.047750703 0.004822821 0.009830716 0.000992902 0.03821313 0.003859526 0.095476112 0.009643087 0.160011912 0.016161203 0.228737477 0.023102485 0.297430965 0.030040527 0.360845903 0.036445436 0.412955775 0.041708533 0.447330438 0.045180374 0.457629943 0.046220624 0.43818421 0.044256605 0.384610712 0.038845682 0.29440934 0.029735343 0.167466401 0.016914107 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

## ( ) 2 () () 2 M i

i 

## xf a f x f b

 

##  

      

# 0.006400143 0.000646414 0.183310648 0.018514375 0.393455457 0.039739001 0.613493584 0.061962852 0.831213719 0.083952586 1.033605832 0.104394189 1.207871534 0.121995025 1.342478021 0.13559028 1.42815002 0.144243152 1.45869584 0.14732828 1.431578111 0.144589389 1.348165671 0.136164733 1.213637172 0.122577354 1.036545118 0.104691057 0.828086167 0.083636703 0.601154969 0.060716652 0.369280442 0.037297325 0.145552829 0.014700836 0.058353815 0.005893735 0.23297474 0.023530449 0.371599393 0.037531539 0.470535074 0.047524042 0.529088588 0.053437947 0.549294713 0.055478766 0.535443759 0.05407982 0.493475585 0.049841034 0.43031327 0.04346164 0.353206452 0.035673852 0.26914386 0.02718353 0.184378871 0.018622266 0.104093763 0.01051347 0.032210166 0.003253227 0.028662687 0.002894931 0.077162607 0.007793423 0.113020402 0.011415061 0.1368488 0.013821729 0.149905125 0.015140418 0.153854586 0.015539313 0.150555076 0.015206063 0.141877499 0.014329627 0.129568938 0.013086463 0.115160063 0.011631166 0.09991357 0.010091271 0.084807386 0.008565546 0.070544709 0.007125016 0.057582596 0.005815842 0.046171384 0.00466331 0.036398428 0.003676241 0.028231168 0.0028513480.021556082 0.002177164 0.016211547 0.001637366 0.012013793 0.001213393 0.008776044 0.00088638 0.00632152 0.000638474 0.004491301 0.000453621 0.003148183 0.000317966 0.002177621 0.00021994 0.001486709 0.000150158 0.001002002 0.000101202 0.000666778 3.36723E-05Cumulative normalization
2.97822E-05 1.10416E-05 0.000119012 4.41231E-05 0.000250963 9.30431E-05 0.000443544 0.000164441 0.000720891 0.000267266 0.001114933 0.000413355 0.001667083 0.000618061 0.002429934 0.000900884 0.003468762 0.001286024 0.004862562 0.001802767 0.006704247 0.002485561 0.009099541 0.003373603 0.012164049 0.004509751 0.016017914 0.00593855 0.020777531 0.00770315 0.026543873 0.009840989 0.033387224 0.012378122 0.041328455 0.015322287 0.05031748 0.018654917 0.060210138 0.022322563 0.070745431 0.026228463 0.081525801 0.030225223 0.092003721 0.034109851 0.101478368 0.037622523 0.109106214 0.040450502 0.113929034 0.042238536 0.114921937 0.042606648 0.118781463 0.044037546 0.12842455 0.047612665 0.144585753 0.053604338 0.167688239 0.062169452 0.197728766 0.073306805 0.234174202 0.086818741 0.275882735 0.102281941 0.32106311 0.119032305 0.367283734 0.136168336 0.411540339 0.152576218 0.450386021 0.166978032 0.480121364 0.17800224 0.497035471 0.184273048 

## ( ) 2 () () f a f x f b        0.497681885 0.184512702 0.516196261 0.191376801 0.555935262 0.206109807 0.617898114 0.229082179 0.701850699 0.260207118 0.806244889 0.298910664 0.928239914 0.34413962 1.063830194 0.394408938 1.208073346 0.447886258 1.355401625 0.502507372 1.499991015 0.556113058 1.636155747 0.606595351 1.758733102 0.652040202 1.863424159 0.69085381 1.947060862 0.721861638 2.007777513 0.744371988 2.045074838 0.758199757 2.059775674 0.763650007 2.065669409 0.765835076 2.089199858 0.774558855 2.126731397 0.788473457 2.174255439 0.806092723 2.227693386 0.825904535 2.283172152 0.846472969 2.337251972 0.866522752 2.387093006 0.885001029 2.430554646 0.901114183 2.466228498 0.914340059 2.493412028 0.924418196 2.512034294 0.931322295 2.522547764 0.935220103 2.525800991 0.936426219 2.528695922 0.937499498 2.536489345 0.940388865 2.547904406 0.944620933 2.561726135 0.949745259 2.576866552 0.955358482 2.592405866 0.961119594 2.607611928 0.966757154 2.621941556 0.972069781 2.635028018 0.976921512 2.646659185 0.9812337 2.656750455 0.984974981 2.665316001 0.988150608 2.672441017 0.990792166 2.678256859 0.992948357 2.682920169 0.994677252 2.68659641 0.996040197 2.689447758 0.9970973182.691624922 0.997904489 2.693262289 0.998511533 2.694475682 0.998961392 2.695362062 0.999290012 2.696000536 0.999526723 2.696454157 0.9996949 2.696772123 0.999812785 2.696992063 0.999894326 2.697142221 0.999949996 2.697243423 0.999987516 2.697277095 1 134.7207183**Raw Data** Runs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 n for each source 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Sample size 0 KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY Testing Normality Raw Data Logtransformed Sample Size 0.00 0.00 Kurtosis #DIV/0! #DIV/0! http://www.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics SE Kurtosis 0.00 0.00 note: cannot use this equation when n=3 because the denominator in second component is not defined (the term n-3 in denominator of second component is zero). 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 23 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 0.00 Kurtosis -3.00 SE Kurtosis #DIV/0! Result Kurtosis #DIV/0! Skewness #DIV/0! SE Skewness 0.00 Result Raw Data Result Skewness #DIV/0! #DIV/0! ### The kurtosis and skewness are pulled into "Calculations for Template skew", in cells H12 and H14 respectively kurtosis= #DIV/0! depending on sample size skewness= #DIV/0! same for n>3 and n=3 ni = number test runs = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 0
n =Total # test runs = 0

N   1 N i in

  N

1 N i in

 -1

This step is for QC only:
means #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Mean = #DIV/0!

Pooled Variance = #DIV/0!

m= number future runs = 3
Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0! NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)
t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = #NUM! UPL based on t-statistic with df at 99% confidence level #DIV/0!

0 #NUM! Adjustment for non-normality (skewness and kurtosis) for any sample size in particular for small samples a) n = number of test-runs = b) Calculate u0 2 1 0 1 1 u t n 



##  2 2 1 11n i i s xX n   



###  1 1

1 i n N ij i j X x n   

### 

1 1 n m 

21 1 s n m        1 1 N i i df n

         

### 

df t 21 1

s n m        1 1 i n i ik

k i x x n   

### #NUM!

Conclusion interval of differences for t-values used

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

Current Confidence Level is #NUM!

new t-statistic from tab Recalculate t-statisticLogtransformed

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

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 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 

### 1x w b x     

   

## , x 

### I ab a a w a a

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn  

#      

1 2 4 1 1 11 11 31 ( ) , , ,

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 2 4 1 1 11 11 31 ( ) , , , 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 11

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

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 SkewnessP t KurtosisP 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a        

2

### 3 1 2 6 w a  

    #NUM! Result3:



# 1 2 4 1 1 11 11 31 ( ) , , , n n n n P t I I I    
               #NUM! 0 0 0



# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I I    
                     

1 1 1 1 72 2 2 24 2 2

3 1 5 1 24 2 2 72 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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   

2 

### 3 11 1 2 6 3 w a a         3 1 2 6 w a  

    #NUM!#NUM!

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 4) If you don't get a YES in row 11, play around with the cell G3, the interval for differences until you get one p-value closed to 0.01 5) Once you get a YES, use the t-statistic in the calculations of the UPL. Go back to Template_nonnormal tab and paste the value in row 5 that has a Yes in row 11. Copy the new t-value in cell E164 of**Template_nonnormal n<3** 1) Start with initial t-statistic from Template 2) Determine interval of differences for t-values 2) Candidate values for t-statistic 3) Corrected p-values Acutal significance level with t-value 4) Is p-value closed enough to 0.01?t-statistic=

Incomplete Beta Function approximation

where Inputs kurtosis= #DIV/0! x=u0= #NUM! skewness= #DIV/0!
n= 0 t0=t= #NUM!

a= -0.5 a= b= 0.5 b=
w= #NUM! w=

c.1.1. #NUM! d.1.1. c.1.2. #NUM! d.1.2. c.1.3 #NUM! d.1.3
c.1.4. #NUM! d.1.4. c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6 Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

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

Calculations Calculations 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 11

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

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 SkewnessP t KurtosisP 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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   

2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 11

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

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 SkewnessP t KurtosisP 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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   

2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 11

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

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 SkewnessP t KurtosisP 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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   

2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 11

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

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 SkewnessP t KurtosisP 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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   

2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 11

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

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 SkewnessP t KurtosisP 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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   

2 

### 3 11 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 

### 1x w b x     

   

##   , x 

### I ab a a w a a         

0 

### 1 1 , 2 2 u 

### n I       



## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 11 ( ) , 2 2 2 u 

### n P t I 

      

3 0 0 

### 2 1 1 1 1 ( ) ,1 ,1

2 2 6 2 3 2

u u 

### n n n n P t I I

n n  

###  

         

        2 2 01 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 nn 

2

0 

## 2 1 1 1 n t n 

 1 2 2 0 

## 1 1 n 

## t n  

##     

  

1 3 2 nn  

## 2 1 6 2 nn 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 2 4 1 1 11 11 31 ( ) , , , 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 11

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

72 2 2 24 2 2 1 2 5 12 1 2 5 12 3 1 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 SkewnessP t KurtosisP 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 3 2 326 3 2 3 1 1 2 6 2 w a 

### a w e w a w a I ab 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 0 0 

# 1 2 4 1 1 11 11 31 ( ) , , , 24 2 2 12 2 2 24 2 2 u u u 

# n n n n P t I I 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 1 2 6 3 w a a         

## ( ,) ( ) a  ( ) w a e w a  1 2 a w b  

3 2 5 3 1 2 3 2 3 a a a   

2 

### 3 11 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   

2 

### 3 11 1 2 6 3 w a a        #NUM!

#NUM! 

### 2 6 3     3 1 2 6 w a  

    