**Subcategory Pollutant**  **No. of Sources**  **Industry wide**  **No. of Sources**  **with Data Determination of No. of** 

**Sources in MA CT Floor**  Floor tile roller kilns Hg 27 7 best performing 5 sources PM 27 10 best performing 5 sources D/F TEQ 27 6 best performing 5 sources Floor tile press dryers DF/TEQ 5 4 best performing 5 sources Floor tile spray dryers DF/TEQ 11 5 best performing 5 sources Wall tile roller kilns Hg 14 1 best performing 5 sources PM 14 1 best performing 5 sources D/F TEQ 14 1 best performing 5 sources Wall tile spray dryers D/F TEQ 5 1 best performing 5 sources Tile glaze lines Hg 24 3 best performing 5 sources PM 24 3 best performing 5 sources First-fire sanitaryware tunnel kilns Hg 11 5 best performing 5 sources PM 11 6 best performing 5 sources D/F TEQ 11 5 best performing 5 sources First-fire sanitaryware manual glaze application PM 14 3 best performing 5 sources First-fire sanitaryware spray machine glaze application PM 4 3 best performing 5 sources First-fire sanitaryware robot glaze application PM 10 4 best performing 5 sources**No. of**

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

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

**Top Sources**  **UPL Unit of Measure**  **Data Distribution** 5 15 7.8E-05 1.3E-04 lb/ton Normal 5 15 0.054 0.18 lb/ton Skewed/Unknown 5 18 2.9 4.6 ng/dscm @ 7% O2 Normal 4 12 0.078 0.17 ng/dscm @ 7% O2 Normal 5 15 0.96 44 ng/dscm @ 7% O2 Lognormal 1 6 5.0E-05 2.0E-04 lb/ton Normal 1 9 0.071 0.20 lb/ton Normal 1 6 0.065 0.17 ng/dscm @ 7% O2 Normal 1 3 0.053 0.12 ng/dscm @ 7% O2 Normal 3 9 2.0E-05 4.9E-05 lb/ton Lognormal 3 9 0.67 1.9 lb/ton Normal 5 15 1.6E-04 2.6E-04 lb/ton Normal 5 15 0.12 0.33 lb/ton Skewed/Unknown 5 15 0.81 1.5 ng/dscm @ 7% O2 Normal 3 9 14 33 lb/ton Normal 3 9 5.9 12 lb/ton Normal 4 12 4.4 8.8 lb/ton NormalRuns Wall Tile Spray Dryer TEQ (ng/dscm @7%O2) SD-105 Example 2 Example 3 Example 4

1 5.90E-02 2 5.95E-02 3 3.96E-02 Average 5.27E-02 #DIV/0! #DIV/0! #DIV/0! Standard Deviation

1.14E-02 #DIV/0! #DIV/0! #DIV/0! 1 1.59E-09 #DIV/0! #DIV/0! #DIV/0! 2 2.14E-09 #DIV/0! #DIV/0! #DIV/0! 3 2.96E-08 #DIV/0! #DIV/0! #DIV/0! Testing Normality

Raw Data Raw Data Raw Data Raw Data Sample Size 3.00 0.00 0.00 0.00 Kurtosis -2.00 #DIV/0! #DIV/0! #DIV/0! SE Kurtosis 2.83 #DIV/0! #DIV/0! #DIV/0! Result Kurtosis normal #DIV/0! #DIV/0! #DIV/0! Skewness -1.73 #DIV/0! #DIV/0! #DIV/0! SE Skewness 1.22 0.00 0.00 0.00 Result Skewnessnormal #DIV/0! #DIV/0! #DIV/0!

1.411216784 #DIV/0! #DIV/0! #DIV/0! Normal #DIV/0! #DIV/0! #DIV/0! 1 2 3 4Example 5 Example 6 Example 7 Example 8 Example 9

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! Raw Data Raw Data Raw Data Raw Data Raw Data 0.00 0.00 0.00 0.00 0.00 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 0.00 0.00 0.00 0.00 0.00 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! 5 6 7 8 9Example 10 Example 11 Example 12 Example 13 Example 14

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

#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #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 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! 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! 15 16 17 18Example 1 Example 2 Example 3 Example 4 Example 5

-2.83E+00 #NUM! #NUM! #NUM! #NUM! -2.82E+00 #NUM! #NUM! #NUM! #NUM! -3.23E+00 #NUM! #NUM! #NUM! #NUM! -2.96E+00 #NUM! #NUM! #NUM! #NUM! 2.33E-01 #NUM! #NUM! #NUM! #NUM! 2.88E-04 #NUM! #NUM! #NUM! #NUM! 3.69E-04 #NUM! #NUM! #NUM! #NUM! 5.23E-03 #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data Log Data 3.00 0.00 0.00 0.00 0.00 -2.00 #NUM! #NUM! #NUM! #NUM! 2.83 #DIV/0! #DIV/0! #DIV/0! #DIV/0! normal #NUM! #NUM! #NUM! #NUM! -1.73 #NUM! #NUM! #NUM! #NUM! 1.22 0.00 0.00 0.00 0.00 normal #NUM! #NUM! #NUM! #NUM! 1.412183555 #NUM! #NUM! #NUM! #NUM!Example 6 Example 7 Example 8 Example 9 Example 10

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data Log Data 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!Example 11 Example 12 Example 13 Example 14 Example 15

#NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! Log Data Log Data Log Data Log Data Log Data 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 0.00 0.00 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!Example 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! Log Data Log Data Log Data 0.00 0.00 0.00 #NUM! #NUM! #NUM! #DIV/0! #DIV/0! #DIV/0! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 0.00 0.00 0.00 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!Raw Data: Floor Tile - Roller Kiln - Hg lb/ton Runs 36 84 77 60 61 1 3.35E-05 5.26E-05 1.12E-04 1.07E-04 1.07E-04 2 3.61E-05 9.16E-05 1.02E-04 9.68E-05 9.68E-05 3 3.39E-05 6.82E-05 5.26E-05 8.88E-05 8.88E-05 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 15.00 Kurtosis -1.38 SE Kurtosis 1.20 Result Kurtosis normal Skewness -0.54 SE Skewness 0.58 Result Raw Data Result Skewness normal normal ni = number test runs = 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 5 n =Total # test runs = 15

14
This step is for QC only: means 3.45E-05 7.08E-05 8.88E-05 9.74E-05 9.74E-05 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

7.78E-05 Mean =
Pooled Variance = 8.25E-10

m= number future runs = 3 Term1 4.00E-01
Term2 3.30E-10
Squared root Term2 1.82E-05 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.624494068 UPL POOLED VARIANCE = 1.25466E-04 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    

   

09000064819365f6 FT-RK-Hg 12/1/2014Raw Data: Floor Tile - Roller Kiln - PM lb/ton Runs **77 84**  **36 76**  **41** 1 1.85E-02 2.42E-02 3.62E-02 4.78E-02 2.01E-01 2 2.11E-02 3.83E-02 3.32E-02 3.89E-02 9.62E-02 3 1.99E-02 3.17E-02 4.24E-02 2.83E-02 1.31E-01 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 n for each source 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 Sample size 15 KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY Testing Normality Raw Data Logtransformed Sample Size 15.00 15.00 Kurtosis 4.62 0.79 http://www.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics SE Kurtosis 1.20 1.20 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 non normal normal Skewness 2.20 1.18 SE Skewness 0.58 0.58 Result Raw Data Result Log Data Result Skewness non normal non normal non normal non normal 3.791282.027281 KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n = 3 ONLY Average 5.40E-02 Standard Deviation 5.10E-02 http://www.spiderfinancial.com/support/documentation/numxl/reference-manual/statistical-tests/test_xkurt 77 84 36 76 41 1 2.35E-01 1.16E-01 1.46E-02 2.14E-04 6.97E+01 2 1.73E-01 8.98E-03 2.73E-02 7.53E-03 4.73E-01 3 2.00E-01 3.62E-02 2.59E-03 6.40E-02 5.29E+004 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 15.00 Kurtosis 2.45 SE Kurtosis 1.26 Result Kurtosis normal Skewness 2.20 SE Skewness 0.58 Result Raw Data Result Skewness non normal non normal ### The kurtosis and skewness are pulled into "Calculations for Template skew", in cells H12 and H14 respectively kurtosis= 4.62 depending on sample size skewness= 2.20 same for n>3 and n=3 ni = number test runs = 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 5 n =Total # test runs = 15

1 N i df n       

###  1

N i in

  N

1 N i in

 14

This step is for QC only: means 0.01981 0.03141 0.03731 0.03835 0.14288 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Mean = 5.40E-02

Pooled Variance = 2.5971E-03

m= number future runs = 3 Term1 0.4000
Term2 1.0388E-03

Squared root Term2 0.03223077 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.624494 UPL based on t-statistic with df at 99% confidence level 0.13854
15 0.670242

65.29% Adjustment for non-normality (skewness and kurtosis) for any sample size in particular for small samples a) n = number of test-runs = b) Calculate u0

Current Confidence Level is 2 1 0 1 1 u t n 



###  2 2 1 11n i i s xX n   



###  1 1

1 i n N ij i j X x n   

### 

1 1 n m 

2 1 1 s n m        1 1 i i df n

       

### 

df t 21 1

s n m        1 1 i n i ik

k i x x n   

### Conclusion interval of differences for t-values used 0.216 3.920494068 Adjusted UPL based on recalculated t-statistic 0.18

new t-statistic from tab Recalculate t-statistic Confidence Level is not 99%, Go to Recalculate t-stat n=3 tab to fix the Confidence LevelLogtransformed **77 84**  **36 76**  **41** -3.99E+00 -3.72E+00-3.32E+00-3.04E+00-1.60E+00 -3.86E+00 -3.26E+00-3.40E+00-3.25E+00-2.34E+00 -3.92E+00 -3.45E+00-3.16E+00-3.56E+00-2.03E+00

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!Raw Data: Floor Tile - Roller Kiln - TEQ ng/dscm @ 7% O2 Runs 84 60 61 62 41 1 1.04E+00 2.47E+00 2.47E+00 2.47E+00 3.91E+00 2 1.26E+00 2.75E+00 2.75E+00 2.75E+00 4.91E+00 3 1.07E+00 3.29E+00 3.29E+00 3.29E+00 4.53E+00 4 3.05E+00 5 3.33E+00 6 3.60E+00 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 18.00 Kurtosis 0.10 SE Kurtosis 1.10 Result Kurtosis normal Skewness -0.18 SE Skewness 0.54 Result Raw Data Result Skewness normal normal ni = number test runs = 3 3 3 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 5 n =Total # test runs = 18

17
This step is for QC only: means 1.12E+00 2.84E+00 2.84E+00 2.84E+00 3.89E+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!

2.90E+00 Mean =
Pooled Variance = 1.12E+00

m= number future runs = 3 Term1 3.89E-01
Term2 4.35E-01
Squared root Term2 6.60E-01 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.566933984 UPL POOLED VARIANCE = 4.59E+00 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    

   

09000064819365f6 FT-RK-TEQ 12/1/2014Raw Data: Floor Tile - Press Dryer - TEQ ng/dscm @ 7% O2 Runs PD-203 PD-201 PD-202 PD-200 1 7.55E-03 7.53E-02 1.25E-01 1.15E-01 2 7.57E-03 5.85E-02 5.59E-02 1.42E-01 3 5.89E-03 1.20E-01 8.44E-02 1.38E-01 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 12.00 Kurtosis -1.41 SE Kurtosis 1.34 Result Kurtosis normal Skewness -0.30 SE Skewness 0.64 Result Raw Data Result Skewness normal normal ni = number test runs = 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 4 n =Total # test runs = 12

11
This step is for QC only: means 7.00E-03 8.47E-02 8.85E-02 1.32E-01 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

7.79E-02 Mean =
Pooled Variance = 2.64E-03

m= number future runs = 3 Term1 4.17E-01
Term2 1.10E-03
Squared root Term2 3.32E-02 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.718079184 UPL POOLED VARIANCE = 1.68E-01 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    

   

09000064819365f6 FT-PD-TEQ 12/1/2014Raw Data: Floor Tile - Spray Dryer - TEQ ng/dscm @ 7% O2 Runs SD-104 SD-101 SD-102 SD-100 SD-103 1 9.11E-03 3.00E-02 1.31E-01 9.22E-01 4.34E+00 2 1.03E-02 5.82E-02 2.00E-01 9.52E-01 4.02E+00 3 1.09E-02 5.76E-02 1.22E-01 7.68E-01 2.72E+00 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Logtransformed Data Sample Size 15.00 15.00 Kurtosis 1.46 -1.32 SE Kurtosis 1.20 1.20 Average of Raw Data 9.57E-01 Result Kurtosis normal normal Variance 2.22 Skewness 1.66 0.06 SE Skewness 0.58 0.58 Result Raw Data Result Log Data Result Skewness non normal normal non normal normal 1. Take LN( ) of Raw Data Runs SD-104 SD-101 SD-102 SD-100 SD-103 1 -4.70E+00 -3.51E+00 -2.03E+00 -8.08E-02 1.47E+00 2 -4.58E+00 -2.84E+00 -1.61E+00 -4.96E-02 1.39E+00 3 -4.52E+00 -2.85E+00 -2.10E+00 -2.64E-01 1.00E+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 15

3. Calculate -1.69E+00

4.693184628 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          

09000064819365f6 FT-SD-TEQ 12/1/2014144839207.2 35121.97272

4125.891569 7. Calculate 667.8210312

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

3.585z value from "lognormal distribution 99%"

a- Calculate 1.937380601 b- Calculate 3.753443593
c- Calculate 108.2003893
d-calculate 1.099521188
e- Calculate 35.43325
f- Calculate UPL 4.43E+01 11. In column B, go down the the row number from Step 10. above, and copy the z value. 12. calculate the UPL using the formula a- Numerator first term b- denominator first term

8. Go to tab distribution for 99 percentile "lognormal distribution 99%"
10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 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           



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

0.101 4125.892 667.821

-5=f(a) 69871.87 1 -4.899 63278.96 2 -4.798 57136.37 3 -4.697 51424.21 4 -4.596 46122.98 5 -4.495 41213.65 6 -4.394 36677.6 7 -4.293 32496.62 8 -4.192 28652.96 9 -4.091 25129.29 10 -3.99 21908.7 11 -3.889 18974.72 12 -3.788 16311.29 13 -3.687 13902.81 14 -3.586 11734.09 15 -3.485 9790.353 16 -3.384 8057.287 17 -3.283 6520.984 18 -3.182 5167.973 19 -3.081 3985.212 20 -2.98 2960.086 21 -2.879 2080.409 22 -2.778 1334.427 23 -2.677 710.8123 24 -2.576 198.6668 25 -2.475 -212.478 26 -2.374 -532.662 27 -2.273 -771.497 28 -2.172 -938.163 29 -2.071 -1041.41 30 -1.97 -1089.57 31 -1.869 -1090.53 32 -1.768 -1051.76 33 -1.667 -980.299 34 -1.566 -882.746 35 -1.465 -765.286 36 -1.364 -633.665 37 -1.263 -493.206 38 -1.162 -348.8 39 -1.061 -204.909 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 -65.5661

41 -0.859 65.62382 42 -0.758 185.4855 43 -0.657 291.2727 44 -0.556 380.6682 45 -0.455 451.7836 46 -0.354 503.1598 47 -0.253 533.7666 48 -0.152 543.0028 49 -0.051 530.6964 50 0.05 497.1041 51 0.151 442.912 52 0.252 369.2349 53 0.353 277.617 54 0.454 170.0312 55 0.555 48.87952 56 0.656 -83.0069 57 0.757 -222.368 58 0.858 -365.514 59 0.959 -508.328 60 1.06 -646.262 61 1.161 -774.34 62 1.262 -887.157 63 1.363 -978.878 64 1.464 -1043.24 65 1.565 -1073.55 66 1.666 -1062.69 67 1.767 -1003.1 68 1.868 -886.815 69 1.969 -705.416 70 2.07 -450.069 71 2.171 -111.507 72 2.272 319.9662 73 2.373 854.4743 74 2.474 1502.571 75 2.575 2275.24 76 2.676 3183.891 77 2.777 4240.366 78 2.878 5456.935 79 2.979 6846.296 80 3.08 8421.578 81 3.181 10196.34 82 3.282 12184.56 83 3.383 14400.66 84 3.484 16859.48 85 3.585 19576.29 86 3.686 22566.8 87 3.787 25847.14 88 3.888 29433.8789 3.989 33343.97

90 4.09 37594.87 91 4.191 42204.41 92 4.292 47190.87 93 4.393 52572.95 94 4.494 58369.79 95 4.595 64600.95 96 4.696 71286.42 97 4.797 78446.62 98 4.898 86102.41 99 4.999 94275.07distribution of Z Normal distribution

1.48672E-06 0.103879875 2.45094E-06 0.155092911 3.9995E-06 0.228517069 6.46025E-06 0.332213062 1.03291E-05 0.476408366 1.63472E-05 0.673729786 2.56093E-05 0.939286119 3.97118E-05 1.2904982 6.09553E-05 1.746548621 9.26132E-05 2.327303805 0.000139285 3.051553065 0.000207351 3.934420937 0.000305546 4.983849036 0.000445674 6.196120611 0.000643469 7.550521316 0.000919619 9.003395486 0.001300942 10.48206382 0.001821704 11.87930163 0.002525035 13.04931156 0.003464389 13.80632414 0.004704958 13.92707704 0.006324913 13.15840735 0.008416337 11.23098915 0.011085658 7.879822157 0.014453386 2.871407996 0.018652949 -3.963343645 0.023828414 -12.69249817 0.030130931 -23.24590849 0.03771375 -35.38162716 0.046725789 -48.66084685 0.057303789 -62.43659083 0.069563239 -75.86103201 0.083588399 -87.91522222 0.099421884 -97.46313523 0.117054396 -103.329355 0.136415346 -104.3966924 0.157365126 -99.71682664 0.179689839 -88.62417104 0.203099244 -70.84102435 0.227228529 -46.56112792 

# 

#  

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 -16.49933062 0.275855243 18.10267489 0.299326439 55.52072175 0.321498297 93.64368967 0.341807853 130.1153758 0.359712192 162.5120774 0.37471238 188.5402177 0.386376486 206.2348721 0.394360216 214.1387194 0.398423793 211.4420607 0.398443914 198.0681032 0.394419966 174.6933234 0.386474058 142.6997243 0.374844865 104.0633061 0.359875719 61.19009124 0.34199778 16.71668674 0.32170943 -26.70409926 0.299553264 -66.61104196 0.276092166 -100.9156866 0.25188591 -128.0407524 0.227469632 -147.0050636 0.203335281 -157.4506967 0.17991684 -159.6144507 0.157579839 -154.251391 0.136615273 -142.522465 0.117237788 -125.8606525 0.099587708 -105.8306739 0.083736272 -83.99609801 0.069693339 -61.80510854 0.05741676 -40.50272253 0.046822635 -21.07342068 0.037795734 -4.214479829 0.030199481 9.662812245 0.023885038 20.40915077 0.018699163 28.09682316 0.014490659 32.96972092 0.011115369 35.39012386 0.008439746 35.78761626 0.006343145 34.61413312 0.004718997 32.30765156 0.003475077 29.26563463 0.002533081 25.82814272 0.001827693 22.26963515 0.001305351 18.797916 0.000922829 15.55841498 0.00064578 12.64198556 0.00044732 10.09458032 0.000306705 7.927456819 0.000208159 6.1269112650.000139842 4.662879835 9.29928E-05 3.496052047 6.12113E-05 2.583386032 3.98826E-05 1.882094273 2.5722E-05 1.352282551 1.64209E-05 0.958483578 1.03767E-05 0.670342365 6.49066E-06 0.462695809 4.01874E-06 0.315256302 2.46298E-06 0.212068106 1.49417E-06 0.140863072Absolute values
0.103879875 0.005245934 0.155092911 0.015664384 0.228517069 0.023080224 0.332213062 0.033553519 0.476408366 0.048117245 0.673729786 0.068046708 0.939286119 0.094867898 1.2904982 0.130340318 1.746548621 0.176401411 2.327303805 0.235057684 3.051553065 0.30820686 3.934420937 0.397376515 4.983849036 0.503368753 6.196120611 0.625808182 7.550521316 0.762602653 9.003395486 0.909342944 10.48206382 1.058688446 11.87930163 1.199809464 13.04931156 1.317980468 13.80632414 1.394438738 13.92707704 1.406634781 13.15840735 1.328999142 11.23098915 1.134329905 7.879822157 0.795862038 2.871407996 0.290012208 3.963343645 0.400297708 12.69249817 1.281942316 23.24590849 2.347836758 35.38162716 3.573544343 48.66084685 4.914745532 62.43659083 6.306095674 75.86103201 7.661964233 87.91522222 8.879437445 97.46313523 9.843776659 103.329355 10.43626485 104.3966924 10.54406593 99.71682664 10.07139949 88.62417104 8.951041275 70.84102435 7.154943459 46.56112792 4.70267392 

## 

#  

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

 

##  

      

# 16.49933062 1.666432393 18.10267489 1.828370164 55.52072175 5.607592896 93.64368967 9.458012656 130.1153758 13.14165296 162.5120774 16.41371982 188.5402177 19.04256198 206.2348721 20.82972208 214.1387194 21.62801066 211.4420607 21.35564813 198.0681032 20.00487842 174.6933234 17.64402566 142.6997243 14.41267216 104.0633061 10.51039391 61.19009124 6.180199216 16.71668674 1.688385361 26.70409926 2.697114025 66.61104196 6.727715238 100.9156866 10.19248434 128.0407524 12.932116 147.0050636 14.84751143 157.4506967 15.90252036 159.6144507 16.12105952 154.251391 15.57939049 142.522465 14.39476897 125.8606525 12.7119259 105.8306739 10.68889807 83.99609801 8.483605899 61.80510854 6.242315963 40.50272253 4.090774976 21.07342068 2.128415488 4.214479829 0.425662463 9.662812245 0.975944037 20.40915077 2.061324228 28.09682316 2.837779139 32.96972092 3.329941813 35.39012386 3.57440251 35.78761626 3.614549243 34.61413312 3.496027445 32.30765156 3.263072808 29.26563463 2.955829098 25.82814272 2.608642414 22.26963515 2.24923315 18.797916 1.898589516 15.55841498 1.571399913 12.64198556 1.276840541 10.09458032 1.019552612 7.927456819 0.800673139 6.126911265 0.6188180384.662879835 0.470950863 3.496052047 0.353101257 2.583386032 0.260921989 1.882094273 0.190091522 1.352282551 0.136580538 0.958483578 0.096806841 0.670342365 0.067704579 0.462695809 0.046732277 0.315256302 0.031840887 0.212068106 0.021418879 0.140863072 0.007113585Cumulative normalization 0.005245934 1.03521E-05 0.020910318 4.12637E-05 0.043990542 8.68094E-05 0.077544061 0.000153023 0.125661306 0.000247976 0.193708014 0.000382257 0.288575912 0.000569466 0.418916231 0.000826675 0.595317641 0.00117478 0.830375326 0.001638635 1.138582185 0.00224684 1.5359587 0.003031009 2.039327452 0.00402434 2.665135634 0.005259289 3.427738287 0.006764183 4.337081231 0.00855865 5.395769677 0.01064783 6.595579141 0.013015493 7.913559609 0.015616351 9.307998346 0.01836809 10.71463313 0.021143895 12.04363227 0.023766497 13.17796217 0.026004946 13.97382421 0.027575473 14.26383642 0.028147773 14.66413413 0.028937707 15.94607644 0.031467449 18.2939132 0.036100591 21.86745754 0.043152502 26.78220308 0.052851095 33.08829875 0.065295332 40.75026298 0.080415193 49.62970043 0.097937575 59.47347709 0.117362952 69.90974194 0.137957525 80.45380787 0.158764828 90.52520736 0.178639388 99.47624863 0.196303071 106.6311921 0.210422395 111.333866 0.219702493 

## ( ) 2 ( ) () f a f x f b        113.0002984 0.222990974 114.8286686 0.226599018 120.4362615 0.237664852 129.8942741 0.256328975 143.0359271 0.28226227 159.4496469 0.31465255 178.4922089 0.352230499 199.321931 0.393335169 220.9499416 0.436015155 242.3055898 0.478157671 262.3104682 0.517634622 279.9544938 0.552452747 294.367166 0.580894227 304.8775599 0.601635084 311.0577591 0.61383088 312.7461445 0.617162683 315.4432585 0.622485077 322.1709737 0.635761323 332.3634581 0.655874827 345.2955741 0.681394629 360.1430855 0.710694206 376.0456059 0.742075703 392.1666654 0.773888458 407.7460559 0.804632301 422.1408249 0.833038452 434.8527508 0.858123738 445.5416488 0.879216848 454.0252547 0.89595811 460.2675707 0.908276485 464.3583457 0.916349082 466.4867611 0.920549226 466.9124236 0.921389214 467.8883676 0.923315109 469.9496919 0.927382856 472.787471 0.932982834 476.1174128 0.939554029 479.6918153 0.946607634 483.3063646 0.953740463 486.802392 0.960639406 490.0654648 0.967078644 493.0212939 0.972911577 495.6299363 0.978059384 497.8791695 0.982497945 499.777759 0.986244557 501.3491589 0.989345505 502.6259995 0.991865179 503.6455521 0.993877129 504.4462252 0.995457151 505.0650432 0.996678305505.5359941 0.997607664 505.8890954 0.998304462 506.1500174 0.998819356 506.3401089 0.999194477 506.4766894 0.999464 506.5734963 0.999655035 506.6412008 0.999788641 506.6879331 0.999880861 506.719774 0.999943695 506.7411929 0.999985962 506.7483065 1 24703.49797Raw Data: Wall Tile - Roller Kiln - TEQ ng/dscm @ 7% O2 Runs 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75 1 1.03E-04 2 4.77E-05 3 1.49E-04 4 1.14E-06 5 9.87E-07 6 1.12E-06 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 6.00 Kurtosis -0.92 SE Kurtosis 2.06 Result Kurtosis normal Skewness 0.89 SE Skewness 0.85 Result Raw Data Result Skewness normal normal ni = number test runs = 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 1 n =Total # test runs = 6

5
This step is for QC only: means 5.05E-05 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

5.05E-05 Mean =
Pooled Variance = 3.95E-09

m= number future runs = 3 Term1 5.00E-01
Term2 1.98E-09
Squared root Term2 4.44E-05 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 3.364929999 UPL POOLED VARIANCE = 2.00E-04 Use for large sample sets and small sample sets that are Normal.

 2 2 1 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    

   

09000064819365f6 WT-RK-Hg 12/1/2014Raw Data: Wall Tile - Roller Kiln - TEQ ng/dscm @ 7% O2 Runs 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75 1 1.93E-01 2 1.21E-01 3 1.59E-01 4 2.29E-02 5 2.35E-02 6 5.01E-02 7 2.45E-02 8 1.64E-02 9 2.82E-02 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 9.00 Kurtosis -0.64 SE Kurtosis 1.56 Result Kurtosis normal Skewness 1.05 SE Skewness 0.72 Result Raw Data Result Skewness normal normal ni = number test runs = 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 1 n =Total # test runs = 9

8
This step is for QC only: means 7.10E-02 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

7.10E-02 Mean =
Pooled Variance = 4.65E-03

m= number future runs = 3 Term1 4.44E-01
Term2 2.07E-03
Squared root Term2 4.55E-02 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.896459448 UPL POOLED VARIANCE = 0.2027 Use for large sample sets and small sample sets that are Normal.

 2 2 1 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    

   

09000064819365f6 WT-RK-PM 12/1/2014Raw Data: Wall Tile - Roller Kiln - TEQ ng/dscm @ 7% O2 Runs 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75 1 1.19E-01 2 9.98E-02 3 9.89E-02 4 3.06E-02 5 2.96E-02 6 1.07E-02 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 6.00 Kurtosis -2.73 SE Kurtosis 2.06 Result Kurtosis normal Skewness 0.00 SE Skewness 0.85 Result Raw Data Result Skewness normal normal ni = number test runs = 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 1 n =Total # test runs = 6

5
This step is for QC only: means 6.47E-02 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

6.47E-02 Mean =
Pooled Variance = 2.13E-03

m= number future runs = 3 Term1 5.00E-01
Term2 1.06E-03
Squared root Term2 3.26E-02 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 3.364929999 UPL POOLED VARIANCE = 1.74E-01 Use for large sample sets and small sample sets that are Normal.

 2 2 1 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    

   

09000064819365f6 WT-RK-TEQ 12/1/2014Raw Data: Wall Tile - Spray Dryer - TEQ ng/dscm @ 7% O2 Runs SD-105 1 5.90E-02 2 5.95E-02 3 3.96E-02 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 3.00 Kurtosis #DIV/0! SE Kurtosis #DIV/0! Result Kurtosis #DIV/0! Skewness -1.73 SE Skewness 1.22 Result Raw Data Result Skewness normal #DIV/0! ni = number test runs = 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 1 n =Total # test runs = 3

2
This step is for QC only: means 5.27E-02 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

5.27E-02 Mean =
Pooled Variance = 1.29E-04

m= number future runs = 3 Term1 6.67E-01
Term2 8.60E-05
Squared root Term2 9.27E-03 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 6.964556734 UPL POOLED VARIANCE = 1.17E-01 Use for large sample sets and small sample sets that are Normal.

 2 2 1 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    

   

09000064819365f6 WT-SD-TEQ 12/1/2014Raw Data: Tile - Glazing - Hg lb/ton Runs 114; 116; 117; 119 71 73, 74, 75 1 4.45E-05 2.33E-05 2.87E-06 2 9.73E-06 4.28E-05 4.08E-06 3 1.53E-05 2.06E-05 1.52E-05 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 9.00 Kurtosis -0.44 SE Kurtosis 1.56 Result Kurtosis normal Skewness 0.80 SE Skewness 0.72 Result Raw Data Result Skewness normal normal ni = number test runs = 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 3 n =Total # test runs = 9

8
This step is for QC only: means 2.32E-05 2.89E-05 7.37E-06 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1.98E-05 Mean =
Pooled Variance = 2.29E-10

m= number future runs = 3 Term1 4.44E-01
Term2 1.02E-10
Squared root Term2 1.01E-05 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.896459448 UPL POOLED VARIANCE = 4.90E-05 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    

   

09000064819365f6 T-Glaze-Hg 12/1/2014Raw Data: Tile - Glazing - PM lb/ton Runs 71 114; 116; 117; 119 73, 74, 75 1 1.52E-01 5.46E-01 1.24E+00 2 2.33E-01 5.41E-01 1.69E+00 3 7.21E-02 1.38E-01 1.40E+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 Testing Normality Raw Data Sample Size 9.00 Kurtosis -1.20 SE Kurtosis 1.56 Result Kurtosis normal Skewness 0.74 SE Skewness 0.72 Result Raw Data Result Skewness normal normal ni = number test runs = 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 3 n =Total # test runs = 9

8
This step is for QC only: means 1.52E-01 4.09E-01 1.44E+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!

6.68E-01 Mean =
Pooled Variance = 3.80E-01

m= number future runs = 3 Term1 4.44E-01
Term2 1.69E-01
Squared root Term2 4.11E-01 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.896459448 UPL POOLED VARIANCE = 1.86E+00 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    

   

09000064819365f6 T-Glaze-PM 12/1/2014Raw Data: Sanitaryware - Tunnel Kiln - Hg lb/ton Runs TK #2 POC stack TK #19 POC stack TK #6 POC stack TK #12 POC stack TK #8 POC stack 1 7.38E-05 1.34E-04 1.92E-04 1.92E-04 2.16E-04 2 6.53E-05 1.51E-04 1.98E-04 1.98E-04 2.18E-04 3 5.41E-05 1.42E-04 1.97E-04 1.97E-04 2.24E-04 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 15.00 Kurtosis -0.43 SE Kurtosis 1.20 Result Kurtosis normal Skewness -0.98 SE Skewness 0.58 Result Raw Data Result Skewness normal normal ni = number test runs = 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 5 n =Total # test runs = 15

14
This step is for QC only: means 6.44E-05 1.43E-04 1.96E-04 1.96E-04 2.19E-04 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1.64E-04 Mean =
Pooled Variance = 3.35E-09

m= number future runs = 3 Term1 4.00E-01
Term2 1.34E-09
Squared root Term2 3.66E-05 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.624494068 UPL POOLED VARIANCE = 2.60E-04 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    

   

09000064819365f6 S-TK-Hg 12/1/2014Raw Data: Sanitaryware - Tunnel Kiln - PM lb/ton Runs **TK #12 POC stack TK #6 POC stack**  **TK #19 POC stack TK #2 POC + EE + FE stack**  **TK #8 POC stack** 1 9.21E-02 9.21E-02 1.05E-01 1.58E-01 6.53E-02 2 9.33E-02 9.33E-02 1.06E-01 1.02E-01 3.98E-01 3 9.15E-02 9.15E-02 8.84E-02 1.30E-01 7.33E-02 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 n for each source 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 Sample size 15 KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY Testing Normality Raw Data Logtransformed Sample Size 15.00 15.00 Kurtosis 12.36 7.38 http://www.spcforexcel.com/are-skewness-and-kurtosis-useful-statistics SE Kurtosis 1.20 1.20 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 non normal non normal Skewness 3.41 2.44 SE Skewness 0.58 0.58 Result Raw Data Result Log Data Result Skewness non normal non normal non normal non normal KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n = 3 ONLY Average 1.19E-01 Standard Deviation 8.03E-02 http://www.spiderfinancial.com/support/documentation/numxl/reference-manual/statistical-tests/test_xkurt TK #12 POC stack TK #6 POC stack TK #19 POC stackTK #2 POC + EE + FE stack TK #8 POC stack 1 1.19E-02 1.19E-02 8.98E-04 5.86E-02 1.94E-012 9.91E-03 9.91E-03 6.47E-04 1.82E-03 1.46E+02 3 1.29E-02 1.29E-02 1.99E-02 3.79E-04 1.01E-01 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 15.00 Kurtosis 7.48 SE Kurtosis 1.26 Result Kurtosis non normal Skewness 3.41 SE Skewness 0.58 Result Raw Data Result Skewness non normal non normal ### The kurtosis and skewness are pulled into "Calculations for Template skew", in cells H12 and H14 respectively kurtosis= 12.36 depending on sample size skewness= 3.41 same for n>3 and n=3 ni = number test runs = 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 5 n =Total # test runs = 15 1 N i in

  N

1 N i in

 14

This step is for QC only: means 0.09228 0.09228 0.09964 0.12996 0.17881 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

Mean = 1.19E-01

Pooled Variance = 6.4463E-03

m= number future runs = 3 Term1 0.4000
Term2 2.5785E-03

Squared root Term2 0.050779199 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.624494 UPL based on t-statistic with df at 99% confidence level 0.251864
15 0.670242 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 

2 1 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   

### 13.81%

Conclusion interval of differences for t-values used 0.2582 4.173694068 Adjusted UPL based on recalculated t-statistic 0.33

new t-statistic from tab Recalculate t-statistic Current Confidence Level is Confidence Level is not 99%, Go to Recalculate t-stat n=3 tab to fix the Confidence LevelLogtransformed **TK #12 POC stack**  **TK #6 POC**

**stack TK #19 POC** 

**stack** -2.39E+00 -2.39E+00 -2.26E+00 -2.37E+00 -2.37E+00 -2.25E+00 -2.39E+00 -2.39E+00 -2.43E+00

0 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 0#DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!**TK #2 POC +**

**EE + FE stack TK #8** 

**POC stack**  -1.84E+00 -2.73E+00 -2.28E+00 -9.22E-01 -2.04E+00 -2.61E+00Raw Data: Sanitaryware - Tunnel Kiln - TEQ ng/dscm @ 7% O2 Runs TK #10 POC + RCE stacks TK #19 POC stack TK #8 POC stack TK #6 POC stack TK #12 POC stack 1 2.87E-01 4.93E-01 8.26E-01 1.24E+00 1.24E+00 2 2.19E-01 4.50E-01 9.13E-01 1.09E+00 1.09E+00 3 1.99E-01 5.50E-01 9.39E-01 1.28E+00 1.28E+00 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 15.00 Kurtosis -1.54 SE Kurtosis 1.20 Result Kurtosis normal Skewness -0.30 SE Skewness 0.58 Result Raw Data Result Skewness normal normal ni = number test runs = 3 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 5 n =Total # test runs = 15

14
This step is for QC only: means 2.35E-01 4.98E-01 8.93E-01 1.20E+00 1.20E+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!

8.06E-01 Mean =
Pooled Variance = 1.63E-01

m= number future runs = 3 Term1 4.00E-01
Term2 6.53E-02
Squared root Term2 2.56E-01 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.624494068 UPL POOLED VARIANCE = 1.48E+00 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    

   

09000064819365f6 S-TK-TEQ 12/1/2014Raw Data: Sanitaryware - Manual Glaze Application - PM lb/ton Runs SO3/SO4 LBM EO13 (EPN013-Manual SB) 1 3.52E+00 1.53E+01 1.75E+01 2 3.17E+00 1.77E+01 1.91E+01 3 3.29E+00 1.68E+01 3.19E+01 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 9.00 Kurtosis 0.01 SE Kurtosis 1.56 Result Kurtosis normal Skewness 0.32 SE Skewness 0.72 Result Raw Data Result Skewness normal normal ni = number test runs = 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 3 n =Total # test runs = 9

8
This step is for QC only: means 3.33E+00 1.66E+01 2.28E+01 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

1.43E+01 Mean =
Pooled Variance = 9.03E+01

m= number future runs = 3 Term1 4.44E-01
Term2 4.01E+01
Squared root Term2 6.33E+00 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.896459448 UPL POOLED VARIANCE = 3.26E+01 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    

   

09000064819365f6 S-Manual-PM 12/1/2014Raw Data: Sanitaryware - Spray Machine Glaze Application - PM lb/ton Runs EO14 (EPN014-Bowl SM 1) BBM NTM 1 2.14E+00 6.01E+00 1.04E+01 2 1.72E+00 6.90E+00 9.93E+00 3 2.01E+00 6.68E+00 7.68E+00 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 9.00 Kurtosis -1.43 SE Kurtosis 1.56 Result Kurtosis normal Skewness -0.15 SE Skewness 0.72 Result Raw Data Result Skewness normal normal ni = number test runs = 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 3 n =Total # test runs = 9

8
This step is for QC only: means 1.95E+00 6.53E+00 9.32E+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!

5.93E+00 Mean =
Pooled Variance = 1.10E+01

m= number future runs = 3 Term1 4.44E-01
Term2 4.87E+00
Squared root Term2 2.21E+00 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.896459448 UPL POOLED VARIANCE = 1.23E+01 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    

   

09000064819365f6 S-SprayMachine-PM 12/1/2014Raw Data: Sanitaryware - Robot Glaze Application - PM lb/ton Runs SR1 (EPN 104-1P) SR5 (EPN251-Robot) Robot 3 P18-Q24/S18 1 1.68E+00 2.51E+00 6.31E+00 7.14E+00 2 1.17E+00 2.80E+00 4.62E+00 7.26E+00 3 1.10E+00 4.48E+00 7.90E+00 6.11E+00 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 12.00 Kurtosis -1.67 SE Kurtosis 1.34 Result Kurtosis normal Skewness -0.04 SE Skewness 0.64 Result Raw Data Result Skewness normal normal ni = number test runs = 3 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 4 n =Total # test runs = 12

11
This step is for QC only: means 1.32E+00 3.26E+00 6.28E+00 6.84E+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!

4.42E+00 Mean =
Pooled Variance = 6.31E+00

m= number future runs = 3 Term1 4.17E-01
Term2 2.63E+00
Squared root Term2 1.62E+00 NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01) t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 2.718079184 UPL POOLED VARIANCE = 8.83E+00 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    

   

09000064819365f6 S-Robot-PM 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 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    

   

09000064819365f6 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          

09000064819365f6 Lognormal Template 12/1/2014#DIV/0! #DIV/0! #DIV/0! 7. Calculate #DIV/0!

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

3.484z value from "lognormal distribution 99%"

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

8. Go to tab distribution for 99 percentile "lognormal distribution 99%"
10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is. 222 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           



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

0.101 23.80249 1.377354

-5=f(a) 390.0648 1 -4.899 354.4355 2 -4.798 321.1779 3 -4.697 290.1895 4 -4.596 261.37 5 -4.495 234.6213 6 -4.394 209.8473 7 -4.293 186.9542 8 -4.192 165.8504 9 -4.091 146.4462 10 -3.99 128.6545 11 -3.889 112.3899 12 -3.788 97.56945 13 -3.687 84.11235 14 -3.586 71.93989 15 -3.485 60.97555 16 -3.384 51.14496 17 -3.283 42.37594 18 -3.182 34.59846 19 -3.081 27.74465 20 -2.98 21.74883 21 -2.879 16.54744 22 -2.778 12.07914 23 -2.677 8.284705 24 -2.576 5.107107 25 -2.475 2.491468 26 -2.374 0.38508 27 -2.273 -1.2626 28 -2.172 -2.49996 29 -2.071 -3.37321 30 -1.97 -3.92639 31 -1.869 -4.20141 32 -1.768 -4.23797 33 -1.667 -4.07363 34 -1.566 -3.74379 35 -1.465 -3.28168 36 -1.364 -2.71835 37 -1.263 -2.0827 38 -1.162 -1.40147 39 -1.061 -0.69922 

#   1 3 

## 1 3 6 24 z 

## z z      

     2z  1z



## 40 -0.96 0.001637

41 -0.859 0.68087 42 -0.758 1.320402 43 -0.657 1.904325 44 -0.556 2.418893 45 -0.455 2.852526 46 -0.354 3.19581 47 -0.253 3.441494 48 -0.152 3.584492 49 -0.051 3.621883 50 0.05 3.55291 51 0.151 3.378983 52 0.252 3.103674 53 0.353 2.73272 54 0.454 2.274026 55 0.555 1.737657 56 0.656 1.135845 57 0.757 0.482988 58 0.858 -0.20435 59 0.959 -0.90745 60 1.06 -1.60542 61 1.161 -2.2752 62 1.262 -2.89157 63 1.363 -3.42715 64 1.464 -3.85238 65 1.565 -4.13556 66 1.666 -4.2428 67 1.767 -4.13806 68 1.868 -3.78313 69 1.969 -3.13764 70 2.07 -2.15904 71 2.171 -0.80265 72 2.272 0.978414 73 2.373 3.233181 74 2.474 6.012847 75 2.575 9.370774 76 2.676 13.36249 77 2.777 18.04568 78 2.878 23.4802 79 2.979 29.72808 80 3.08 36.8535 81 3.181 44.92281 82 3.282 54.00452 83 3.383 64.16932 84 3.484 75.49004 85 3.585 88.04169 86 3.686 101.9015 87 3.787 117.1487 88 3.888 133.864889 3.989 152.1336

90 4.09 172.0408 91 4.191 193.6745 92 4.292 217.1248 93 4.393 242.484 94 4.494 269.8467 95 4.595 299.3094 96 4.696 330.971 97 4.797 364.9326 98 4.898 401.2971 99 4.999 440.17distribution of Z Normal distribution

1.48672E-06 0.000579917 2.45094E-06 0.0008687 3.9995E-06 0.001284552 6.46025E-06 0.001874696 1.03291E-05 0.002699714 1.63472E-05 0.003835413 2.56093E-05 0.005374035 3.97118E-05 0.007424283 6.09553E-05 0.010109452 9.26132E-05 0.013562852 0.000139285 0.017919633 0.000207351 0.023304116 0.000305546 0.029811945 0.000445674 0.037486681 0.000643469 0.0462911 0.000919619 0.056074278 0.001300942 0.066536639 0.001821704 0.077196417 0.002525035 0.087362307 0.003464389 0.09611827 0.004704958 0.102327303 0.006324913 0.104661137 0.008416337 0.101662098 0.011085658 0.091841411 0.014453386 0.073814986 0.018652949 0.046473228 0.023828414 0.00917585 0.030130931 -0.03804338 0.03771375 -0.094282831 0.046725789 -0.1576157 0.057303789 -0.224997187 0.069563239 -0.292263436 0.083588399 -0.354244929 0.099421884 -0.405008262 0.117054396 -0.438227575 0.136415346 -0.447671321 0.157365126 -0.427773224 0.179689839 -0.374240219 0.203099244 -0.284637455 0.227228529 -0.158883344 

# 

#  

2 4 2 

## 336 6 24 z 

## z z   

    

     

#   2 1 3 

### ( ) 1 3 ( )

6 24 z G 

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

## ( ) z 0.251644341 0.000411873 0.275855243 0.18782147 0.299326439 0.395231257 0.321498297 0.612237125 0.341807853 0.8267965 0.359712192 1.026088455 0.37471238 1.197509618 0.386376486 1.329712319 0.394360216 1.413580947 0.398423793 1.443044227 0.398443914 1.415635461 0.394419966 1.33273831 0.386474058 1.199489353 0.374844865 1.024346204 0.359875719 0.818366595 0.34199778 0.594274668 0.32170943 0.365412073 0.299553264 0.144680625 0.276092166 -0.056420378 0.25188591 -0.228574626 0.227469632 -0.365184292 0.203335281 -0.462628185 0.17991684 -0.520241996 0.157579839 -0.540049159 0.136615273 -0.526293985 0.117237788 -0.484843542 0.099587708 -0.422530427 0.083736272 -0.346505441 0.069693339 -0.263658786 0.05741676 -0.180152901 0.046822635 -0.101092161 0.037795734 -0.030336761 0.030199481 0.029547598 0.023885038 0.07722465 0.018699163 0.112435203 0.014490659 0.135788687 0.011115369 0.148528978 0.008439746 0.152300952 0.006343145 0.148938347 0.004718997 0.140286733 0.003475077 0.128068772 0.002533081 0.113793105 0.001827693 0.098703708 0.001305351 0.083763503 0.000922829 0.0696644 0.00064578 0.0568556 0.00044732 0.045582551 0.000306705 0.035930124 0.000208159 0.0278651090.000139842 0.021274633 9.29928E-05 0.015998559 6.12113E-05 0.011855066 3.98826E-05 0.008659501 2.5722E-05 0.00623718 1.64209E-05 0.004431121 1.03767E-05 0.003105833 6.49066E-06 0.00214822 4.01874E-06 0.001466568 2.46298E-06 0.000988385 1.49417E-06 0.000657689Absolute values
0.000579917 2.92858E-05 0.0008687 8.77387E-05 0.001284552 0.00012974 0.001874696 0.000189344 0.002699714 0.000272671 0.003835413 0.000387377 0.005374035 0.000542778 0.007424283 0.000749853 0.010109452 0.001021055 0.013562852 0.001369848 0.017919633 0.001809883 0.023304116 0.002353716 0.029811945 0.003011006 0.037486681 0.003786155 0.0462911 0.004675401 0.056074278 0.005663502 0.066536639 0.006720201 0.077196417 0.007796838 0.087362307 0.008823593 0.09611827 0.009707945 0.102327303 0.010335058 0.104661137 0.010570775 0.101662098 0.010267872 0.091841411 0.009275983 0.073814986 0.007455314 0.046473228 0.004693796 0.00917585 0.000926761 0.03804338 0.003842381 0.094282831 0.009522566 0.1576157 0.015919186 0.224997187 0.022724716 0.292263436 0.029518607 0.354244929 0.035778738 0.405008262 0.040905835 0.438227575 0.044260985 0.447671321 0.045214803 0.427773224 0.043205096 0.374240219 0.037798262 0.284637455 0.028748383 0.158883344 0.016047218 

## 

#  

2 4 2 

### 336 ( ) 1 3 ( )

6 24 z 

### z z f z z z z    

       1 

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

i 

## xf a f x f b

 

##  

      

# 0.000411873 4.15991E-05 0.18782147 0.018969969 0.395231257 0.039918357 0.612237125 0.06183595 0.8267965 0.083506447 1.026088455 0.103634934 1.197509618 0.120948471 1.329712319 0.134300944 1.413580947 0.142771676 1.443044227 0.145747467 1.415635461 0.142979182 1.33273831 0.134606569 1.199489353 0.121148425 1.024346204 0.103458967 0.818366595 0.082655026 0.594274668 0.060021741 0.365412073 0.036906619 0.144680625 0.014612743 0.056420378 0.005698458 0.228574626 0.023086037 0.365184292 0.036883613 0.462628185 0.046725447 0.520241996 0.052544442 0.540049159 0.054544965 0.526293985 0.053155692 0.484843542 0.048969198 0.422530427 0.042675573 0.346505441 0.03499705 0.263658786 0.026629537 0.180152901 0.018195443 0.101092161 0.010210308 0.030336761 0.003064013 0.029547598 0.002984307 0.07722465 0.00779969 0.112435203 0.011355955 0.135788687 0.013714657 0.148528978 0.015001427 0.152300952 0.015382396 0.148938347 0.015042773 0.140286733 0.01416896 0.128068772 0.012934946 0.113793105 0.011493104 0.098703708 0.009969074 0.083763503 0.008460114 0.0696644 0.007036104 0.0568556 0.005742416 0.045582551 0.004603838 0.035930124 0.003628943 0.027865109 0.0028143760.021274633 0.002148738 0.015998559 0.001615854 0.011855066 0.001197362 0.008659501 0.00087461 0.00623718 0.000629955 0.004431121 0.000447543 0.003105833 0.000313689 0.00214822 0.00021697 0.001466568 0.000148123 0.000988385 9.98269E-05 0.000657689 3.32133E-05Cumulative normalization 2.92858E-05 1.10037E-05 0.000117025 4.39703E-05 0.000246764 9.27182E-05 0.000436108 0.000163862 0.00070878 0.000266314 0.001096156 0.000411865 0.001638934 0.000615806 0.002388786 0.000897553 0.003409841 0.0012812 0.004779689 0.001795901 0.006589572 0.002475939 0.008943288 0.003360314 0.011954294 0.004491657 0.015740449 0.005914251 0.02041585 0.007670967 0.026079352 0.009798948 0.032799553 0.012323968 0.040596391 0.01525352 0.049419984 0.01856886 0.059127929 0.022216484 0.069462986 0.026099735 0.080033761 0.030071555 0.090301633 0.033929562 0.099577616 0.037414882 0.107032929 0.04021611 0.111726725 0.041979738 0.112653486 0.042327955 0.116495868 0.043771676 0.126018434 0.047349645 0.141937619 0.053331054 0.164662335 0.061869545 0.194180942 0.072960745 0.22995968 0.0864041 0.270865514 0.101773889 0.3151265 0.118404328 0.360341303 0.135393152 0.403546399 0.151626856 0.441344661 0.165829018 0.470093044 0.176630817 0.486140262 0.182660332 

## ( ) 2 ( ) () f a f x f b        0.486181861 0.182675963 0.505151829 0.18980366 0.545070186 0.204802419 0.606906136 0.228036403 0.690412582 0.25941277 0.794047516 0.298352131 0.914995988 0.34379681 1.049296932 0.394258492 1.192068608 0.447902931 1.337816075 0.502665482 1.480795256 0.556387888 1.615401825 0.606964404 1.73655025 0.652484212 1.840009217 0.691357457 1.922664243 0.722413915 1.982685984 0.744966236 2.019592604 0.758833376 2.034205347 0.764323908 2.039903805 0.766465023 2.062989842 0.775139275 2.099873456 0.788997772 2.146598902 0.806554198 2.199143344 0.826297029 2.253688309 0.846791529 2.306844002 0.866764029 2.355813199 0.885163513 2.398488772 0.901198257 2.433485822 0.914347903 2.460115359 0.924353576 2.478310802 0.931190256 2.488521111 0.935026635 2.491585124 0.936177895 2.494569431 0.937299206 2.502369121 0.940229829 2.513725076 0.944496669 2.527439734 0.949649758 2.54244116 0.955286332 2.557823556 0.96106605 2.572866329 0.966718159 2.58703529 0.972041945 2.599970236 0.976902069 2.611463339 0.98122044 2.621432414 0.984966179 2.629892527 0.988144947 2.636928632 0.990788664 2.642671047 0.992946295 2.647274885 0.994676122 2.650903828 0.996039646 2.653718204 0.9970971082.655866942 0.997904466 2.657482796 0.9985116 2.658680158 0.998961492 2.659554767 0.999290114 2.660184722 0.999526811 2.660632266 0.999694969 2.660945955 0.999812833 2.661162925 0.999894357 2.661311048 0.999950012 2.661410875 0.999987521 2.661444089 1 132.8984354Raw 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 2 34 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Testing Normality Raw Data Sample Size 0.00 Kurtosis -3.00 SE Kurtosis #DIV/0! Result Kurtosis #DIV/0! Skewness #DIV/0! SE Skewness 0.00 Result Raw Data Result Skewness #DIV/0! #DIV/0! ### The kurtosis and skewness are pulled into "Calculations for Template skew", in cells H12 and H14 respectively kurtosis= #DIV/0! depending on sample size skewness= #DIV/0! same for n>3 and n=3 ni = number test runs = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Total Number of sources = 0 n =Total # test runs = 0

1 N i df n       

###  1

N i 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!

#NUM! Current Confidence Level is 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 

2 1 1 s n m        1 1 i i df n

       

### 

df t 21 1

s n m        1 1 i n i ik

k i x x n   

### Conclusion interval of differences for t-values used 0.0172 2.56522136 Adjusted UPL based on recalculated t-statistic #DIV/0!

#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 Calculations Calculations Calculate the Incomplete Beta Functions needed for adjustment

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

### 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 Skewness P t Kurtosis P t P t    

###       0.5 a= 1.5

0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM! #NUM! e.1.4. #NUM! 0.0625 e.1.5.
#NUM! e.1.6 #NUM! e.1.7 Calculations Calculations d.1.) Calculate e.1.) Calculate 

##    3 2 2 2
2 3 

### ( , ) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 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!

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        
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! 0.0172 #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! #NUM! 4) If you don't get a YES in row 11, play around with the cell G3, the interval for differences until you get one p-value closed to 0.01 5) Once you get a YES, use the t-statistic in the calculations of the UPL. Go back to Template_nonnormal tab and paste the value in row 5 that has a Yes in row 11. Copy the new t-value in cell E164 of**Template_nonnormal n<3** 1) Start with initial t-statistic from Template 2) Determine interval of differences for t-values 2) Candidate values for t-statistic 3) Corrected p-values Acutal significance level with t-value 4) Is p-value closed enough to 0.01?t-statistic=

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

a= -0.5 a= b= 0.5 b= w= #NUM! w=
c.1.1. #NUM! d.1.1. c.1.2. #NUM! d.1.2. c.1.3 #NUM! d.1.3 c.1.4. #NUM! d.1.4. c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 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 Skewness P t Kurtosis P t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM! #NUM! e.1.4. #NUM! 0.0625 e.1.5.
#NUM! e.1.6 Calculations e.1.) Calculate 

##   3 2 2 2
2 3 

### ( , ) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 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 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 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 Skewness P t Kurtosis P t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM! #NUM! e.1.4. #NUM! 0.0625 e.1.5.
#NUM! e.1.6 Calculations e.1.) Calculate 

##   3 2 2 2
2 3 

### ( , ) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 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 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 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 Skewness P t Kurtosis P t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM! #NUM! e.1.4. #NUM! 0.0625 e.1.5.
#NUM! e.1.6 Calculations e.1.) Calculate 

##   3 2 2 2
2 3 

### ( , ) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 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 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 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 Skewness P t Kurtosis P t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM! #NUM! e.1.4. #NUM! 0.0625 e.1.5.
#NUM! e.1.6 Calculations e.1.) Calculate 

##   3 2 2 2
2 3 

### ( , ) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 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 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 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 Skewness P t Kurtosis P t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM! #NUM! e.1.4. #NUM! 0.0625 e.1.5.
#NUM! e.1.6 Calculations e.1.) Calculate 

##   3 2 2 2
2 3 

### ( , ) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 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 Calculations Calculations Calculate u0. Sustitute each candidate t-value in u0 equation
Calculate the Incomplete Beta Functions needed for adjustment

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

### 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 Skewness P t Kurtosis P t P t    

###       #NUM! u0= #NUM!

0.5 a= 1.5 0.5 b= 0.5 #NUM! w= #NUM!

#NUM! e.1.1. #NUM! 1.772453851 e.1.2. 0.886226925 #NUM! e.1.3 #NUM! #NUM! e.1.4. #NUM! 0.0625 e.1.5.
#NUM! e.1.6 Calculations e.1.) Calculate 

##   3 2 2 2
2 3 

### ( , ) 1 1 5 3 1 3 11 1 , ( ) ( ) 2 2 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  

    