Subcategory Pollutant

No. of 

Sources 

Industrywide

No. of 

Sources 

with Data

Determination of No. of 

Sources in MACT Floor

Floor tile roller kilns Hg 27 7 best performing 5 sources

PM 27 10 best performing 5 sources

D/F TEQ 27 6 best performing 5 sources

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

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

Wall tile roller kilns Hg 14 1 best performing 5 sources

PM 14 1 best performing 5 sources

D/F TEQ 14 1 best performing 5 sources

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

Tile glaze lines Hg 24 3 best performing 5 sources

PM 24 3 best performing 5 sources

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

PM 11 6 best performing 5 sources

D/F TEQ 11 5 best performing 5 sources

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

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

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

Sources in 

MACT Floor 

(N)

No. Data 

Points in 

MACT Floor 

(n)

Average of 

Top 

Sources

UPL Unit of Measure Data Distribution

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 Normal
Runs

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 Deviation1.14E-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 NormalityRaw 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 Skewness normal #DIV/0! #DIV/0! #DIV/0!

1.411216784 #DIV/0! #DIV/0! #DIV/0!

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

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

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

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

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

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

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

Raw Data Raw Data Raw Data Raw Data Raw Data

0.00 0.00 0.00 0.00 0.00

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

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

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

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

0.00 0.00 0.00 0.00 0.00

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

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

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

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

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

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

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

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

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

Raw Data Raw Data Raw Data Raw Data Raw Data

0.00 0.00 0.00 0.00 0.00

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

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

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

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

0.00 0.00 0.00 0.00 0.00

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

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

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

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

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

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

#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 18
Example 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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

09000064819365f6 FT-RK-Hg 12/1/2014
Raw 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 compone

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 3.79128 2.027281

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+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 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

i

n





N

1

N

i

i

n




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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1 i

i

df n



      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 
Conclusion

interval of differences for t-values used 0.216

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 Level
Logtransformed

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

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






Raw Data: 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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs SD-104 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 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 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 456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

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 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

09000064819365f6 FT-SD-TEQ 12/1/2014
144839207.2

35121.97272

4125.891569

7. Calculate

667.8210312

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

3.585 z 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 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

09000064819365f6 FT-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 29433.87
89 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 94275.07
distribution 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

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f 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 6.126911265
0.000139842 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 0.140863072
Absolute 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

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f 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 0.618818038
4.662879835 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 0.007113585
Cumulative 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

( 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 0.996678305
505.5359941 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

24703.49797
Raw Data: Wall Tile - Roller Kiln - TEQ ng/dscm @ 7% O2

Runs

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

71, 72, 73, 74, 75

1 1.03E-04

2 4.77E-05

3 1.49E-04

4 1.14E-06

5 9.87E-07

6 1.12E-06

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 6.00

Kurtosis -0.92

SE Kurtosis 2.06

Result Kurtosis normal

Skewness 0.89

SE Skewness 0.85 Result Raw Data

Result Skewness normal normal

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

Total Number of sources = 1

n =Total # test runs = 6

5

This step is for QC only:

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

5.05E-05

Mean =

Pooled Variance = 3.95E-09

m= number future runs = 3

Term1 5.00E-01

Term2 1.98E-09

Squared root Term2 4.44E-05

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

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 3.364929999

UPL POOLED VARIANCE = 2.00E-04

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

 

    

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs

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

71, 72, 73, 74, 75

1 1.93E-01

2 1.21E-01

3 1.59E-01

4 2.29E-02

5 2.35E-02

6 5.01E-02

7 2.45E-02

8 1.64E-02

9 2.82E-02

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 9.00

Kurtosis -0.64

SE Kurtosis 1.56

Result Kurtosis normal

Skewness 1.05

SE Skewness 0.72 Result Raw Data

Result Skewness normal normal

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

Total Number of sources = 1

n =Total # test runs = 9

8

This step is for QC only:

means 7.10E-02 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

7.10E-02

Mean =

Pooled Variance = 4.65E-03

m= number future runs = 3

Term1 4.44E-01

Term2 2.07E-03

Squared root Term2 4.55E-02

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

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

UPL POOLED VARIANCE = 0.2027

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

 

    

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs

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

71, 72, 73, 74, 75

1 1.19E-01

2 9.98E-02

3 9.89E-02

4 3.06E-02

5 2.96E-02

6 1.07E-02

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 6.00

Kurtosis -2.73

SE Kurtosis 2.06

Result Kurtosis normal

Skewness 0.00

SE Skewness 0.85 Result Raw Data

Result Skewness normal normal

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

Total Number of sources = 1

n =Total # test runs = 6

5

This step is for QC only:

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

6.47E-02

Mean =

Pooled Variance = 2.13E-03

m= number future runs = 3

Term1 5.00E-01

Term2 1.06E-03

Squared root Term2 3.26E-02

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

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 3.364929999

UPL POOLED VARIANCE = 1.74E-01

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

 

    

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs SD-105

1 5.90E-02

2 5.95E-02

3 3.96E-02

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 3.00

Kurtosis #DIV/0!

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness -1.73

SE Skewness 1.22 Result Raw Data

Result Skewness normal #DIV/0!

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

Total Number of sources = 1

n =Total # test runs = 3

2

This step is for QC only:

means 5.27E-02 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!

5.27E-02

Mean =

Pooled Variance = 1.29E-04

m= number future runs = 3

Term1 6.67E-01

Term2 8.60E-05

Squared root Term2 9.27E-03

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

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

UPL POOLED VARIANCE = 1.17E-01

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs 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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs TK #2 POC 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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs TK #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 o

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 stack TK #2 POC + EE + FE stack TK #8 POC stack

1 1.19E-02 1.19E-02 8.98E-04 5.86E-02 1.94E-01
2 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

i

n





N

1

N

i

i

n




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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 
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 Level
Logtransformed

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

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+00




Raw 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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

 

    

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs SR1 (EPN 104-1P) 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

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

09000064819365f6 S-Robot-PM 12/1/2014
Raw Data

Runs

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 0.00

Kurtosis #DIV/0!

SE Kurtosis 0.00

Result Kurtosis #DIV/0!

Skewness #DIV/0!

SE Skewness 0.00 Result Raw Data

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

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

Total Number of sources = 0

n =Total # test runs = 0

-1

This step is for QC only:

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

#DIV/0!

Mean =

Pooled Variance = #DIV/0!

m= number future runs = 3

Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0!

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

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

UPL POOLED VARIANCE = #DIV/0!

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

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

    

 

1

1

i n

i ik

i k

x x

n 

 

1

N

i

i

n





N

2

,

1 1 X tdf p

s

n m

 

     

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

Runs 123456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Logtransformed Data

Sample Size 0.00 0.00

Kurtosis #DIV/0! #DIV/0!

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

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

Skewness #DIV/0! #DIV/0!

SE Skewness 0.00 0.00 Result Raw Data Result Log Data

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

1. Take LN( ) of Raw Data

Runs 123456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

0

3. Calculate #DIV/0!

#DIV/0!

4. Calculate

5. m= number future runs =3

6. Calculate

Use for small sample sets that are Lognormal.

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n







2

2

1

ˆ

1

n

i

i

x x n













222 4 3 ˆˆˆ 2

2331

3 1 e e e





  

    

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

#DIV/0!

#DIV/0!

7. Calculate

#DIV/0!

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

3.484 z value from "lognormal distribution 99% "

a- Calculate #DIV/0!

b- Calculate #DIV/0!

c- Calculate #DIV/0!

d-calculate #DIV/0!

e- Calculate #DIV/0!

f- Calculate UPL #DIV/0!

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

12. calculate the UPL using the formula 

b- denominator first term

a- Numerator first term

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

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

222 4 3 ˆˆˆ 2

e e233 e

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )

 

2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

ˆ

233 1

3 1

1

z

e e e

m m e





  

    

  

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

0.101 23.80249 1.377354

-5 =f(a) 390.0648

1 -4.899 354.4355

2 -4.798 321.1779

3 -4.697 290.1895

4 -4.596 261.37

5 -4.495 234.6213

6 -4.394 209.8473

7 -4.293 186.9542

8 -4.192 165.8504

9 -4.091 146.4462

10 -3.99 128.6545

11 -3.889 112.3899

12 -3.788 97.56945

13 -3.687 84.11235

14 -3.586 71.93989

15 -3.485 60.97555

16 -3.384 51.14496

17 -3.283 42.37594

18 -3.182 34.59846

19 -3.081 27.74465

20 -2.98 21.74883

21 -2.879 16.54744

22 -2.778 12.07914

23 -2.677 8.284705

24 -2.576 5.107107

25 -2.475 2.491468

26 -2.374 0.38508

27 -2.273 -1.2626

28 -2.172 -2.49996

29 -2.071 -3.37321

30 -1.97 -3.92639

31 -1.869 -4.20141

32 -1.768 -4.23797

33 -1.667 -4.07363

34 -1.566 -3.74379

35 -1.465 -3.28168

36 -1.364 -2.71835

37 -1.263 -2.0827

38 -1.162 -1.40147

39 -1.061 -0.69922

 

 1 3

1 3

6 24

z

z z

 

 

 

  

 

2z

1z
40 -0.96 0.001637

41 -0.859 0.68087

42 -0.758 1.320402

43 -0.657 1.904325

44 -0.556 2.418893

45 -0.455 2.852526

46 -0.354 3.19581

47 -0.253 3.441494

48 -0.152 3.584492

49 -0.051 3.621883

50 0.05 3.55291

51 0.151 3.378983

52 0.252 3.103674

53 0.353 2.73272

54 0.454 2.274026

55 0.555 1.737657

56 0.656 1.135845

57 0.757 0.482988

58 0.858 -0.20435

59 0.959 -0.90745

60 1.06 -1.60542

61 1.161 -2.2752

62 1.262 -2.89157

63 1.363 -3.42715

64 1.464 -3.85238

65 1.565 -4.13556

66 1.666 -4.2428

67 1.767 -4.13806

68 1.868 -3.78313

69 1.969 -3.13764

70 2.07 -2.15904

71 2.171 -0.80265

72 2.272 0.978414

73 2.373 3.233181

74 2.474 6.012847

75 2.575 9.370774

76 2.676 13.36249

77 2.777 18.04568

78 2.878 23.4802

79 2.979 29.72808

80 3.08 36.8535

81 3.181 44.92281

82 3.282 54.00452

83 3.383 64.16932

84 3.484 75.49004

85 3.585 88.04169

86 3.686 101.9015

87 3.787 117.1487

88 3.888 133.8648
89 3.989 152.1336

90 4.09 172.0408

91 4.191 193.6745

92 4.292 217.1248

93 4.393 242.484

94 4.494 269.8467

95 4.595 299.3094

96 4.696 330.971

97 4.797 364.9326

98 4.898 401.2971

99 4.999 440.17
distribution of Z

Normal 

distribution

1.48672E-06 0.000579917

2.45094E-06 0.0008687

3.9995E-06 0.001284552

6.46025E-06 0.001874696

1.03291E-05 0.002699714

1.63472E-05 0.003835413

2.56093E-05 0.005374035

3.97118E-05 0.007424283

6.09553E-05 0.010109452

9.26132E-05 0.013562852

0.000139285 0.017919633

0.000207351 0.023304116

0.000305546 0.029811945

0.000445674 0.037486681

0.000643469 0.0462911

0.000919619 0.056074278

0.001300942 0.066536639

0.001821704 0.077196417

0.002525035 0.087362307

0.003464389 0.09611827

0.004704958 0.102327303

0.006324913 0.104661137

0.008416337 0.101662098

0.011085658 0.091841411

0.014453386 0.073814986

0.018652949 0.046473228

0.023828414 0.00917585

0.030130931 -0.03804338

0.03771375 -0.094282831

0.046725789 -0.1576157

0.057303789 -0.224997187

0.069563239 -0.292263436

0.083588399 -0.354244929

0.099421884 -0.405008262

0.117054396 -0.438227575

0.136415346 -0.447671321

0.157365126 -0.427773224

0.179689839 -0.374240219

0.203099244 -0.284637455

0.227228529 -0.158883344

 

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z

G

f z z z 

 

 

    

( )z  
0.251644341 0.000411873

0.275855243 0.18782147

0.299326439 0.395231257

0.321498297 0.612237125

0.341807853 0.8267965

0.359712192 1.026088455

0.37471238 1.197509618

0.386376486 1.329712319

0.394360216 1.413580947

0.398423793 1.443044227

0.398443914 1.415635461

0.394419966 1.33273831

0.386474058 1.199489353

0.374844865 1.024346204

0.359875719 0.818366595

0.34199778 0.594274668

0.32170943 0.365412073

0.299553264 0.144680625

0.276092166 -0.056420378

0.25188591 -0.228574626

0.227469632 -0.365184292

0.203335281 -0.462628185

0.17991684 -0.520241996

0.157579839 -0.540049159

0.136615273 -0.526293985

0.117237788 -0.484843542

0.099587708 -0.422530427

0.083736272 -0.346505441

0.069693339 -0.263658786

0.05741676 -0.180152901

0.046822635 -0.101092161

0.037795734 -0.030336761

0.030199481 0.029547598

0.023885038 0.07722465

0.018699163 0.112435203

0.014490659 0.135788687

0.011115369 0.148528978

0.008439746 0.152300952

0.006343145 0.148938347

0.004718997 0.140286733

0.003475077 0.128068772

0.002533081 0.113793105

0.001827693 0.098703708

0.001305351 0.083763503

0.000922829 0.0696644

0.00064578 0.0568556

0.00044732 0.045582551

0.000306705 0.035930124

0.000208159 0.027865109
0.000139842 0.021274633

9.29928E-05 0.015998559

6.12113E-05 0.011855066

3.98826E-05 0.008659501

2.5722E-05 0.00623718

1.64209E-05 0.004431121

1.03767E-05 0.003105833

6.49066E-06 0.00214822

4.01874E-06 0.001466568

2.46298E-06 0.000988385

1.49417E-06 0.000657689
Absolute values

0.000579917 2.92858E-05

0.0008687 8.77387E-05

0.001284552 0.00012974

0.001874696 0.000189344

0.002699714 0.000272671

0.003835413 0.000387377

0.005374035 0.000542778

0.007424283 0.000749853

0.010109452 0.001021055

0.013562852 0.001369848

0.017919633 0.001809883

0.023304116 0.002353716

0.029811945 0.003011006

0.037486681 0.003786155

0.0462911 0.004675401

0.056074278 0.005663502

0.066536639 0.006720201

0.077196417 0.007796838

0.087362307 0.008823593

0.09611827 0.009707945

0.102327303 0.010335058

0.104661137 0.010570775

0.101662098 0.010267872

0.091841411 0.009275983

0.073814986 0.007455314

0.046473228 0.004693796

0.00917585 0.000926761

0.03804338 0.003842381

0.094282831 0.009522566

0.1576157 0.015919186

0.224997187 0.022724716

0.292263436 0.029518607

0.354244929 0.035778738

0.405008262 0.040905835

0.438227575 0.044260985

0.447671321 0.045214803

0.427773224 0.043205096

0.374240219 0.037798262

0.284637455 0.028748383

0.158883344 0.016047218

 

2 4

2

336

( )

24

z

z z

z z





   





1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
0.000411873 4.15991E-05

0.18782147 0.018969969

0.395231257 0.039918357

0.612237125 0.06183595

0.8267965 0.083506447

1.026088455 0.103634934

1.197509618 0.120948471

1.329712319 0.134300944

1.413580947 0.142771676

1.443044227 0.145747467

1.415635461 0.142979182

1.33273831 0.134606569

1.199489353 0.121148425

1.024346204 0.103458967

0.818366595 0.082655026

0.594274668 0.060021741

0.365412073 0.036906619

0.144680625 0.014612743

0.056420378 0.005698458

0.228574626 0.023086037

0.365184292 0.036883613

0.462628185 0.046725447

0.520241996 0.052544442

0.540049159 0.054544965

0.526293985 0.053155692

0.484843542 0.048969198

0.422530427 0.042675573

0.346505441 0.03499705

0.263658786 0.026629537

0.180152901 0.018195443

0.101092161 0.010210308

0.030336761 0.003064013

0.029547598 0.002984307

0.07722465 0.00779969

0.112435203 0.011355955

0.135788687 0.013714657

0.148528978 0.015001427

0.152300952 0.015382396

0.148938347 0.015042773

0.140286733 0.01416896

0.128068772 0.012934946

0.113793105 0.011493104

0.098703708 0.009969074

0.083763503 0.008460114

0.0696644 0.007036104

0.0568556 0.005742416

0.045582551 0.004603838

0.035930124 0.003628943

0.027865109 0.002814376
0.021274633 0.002148738

0.015998559 0.001615854

0.011855066 0.001197362

0.008659501 0.00087461

0.00623718 0.000629955

0.004431121 0.000447543

0.003105833 0.000313689

0.00214822 0.00021697

0.001466568 0.000148123

0.000988385 9.98269E-05

0.000657689 3.32133E-05
Cumulative normalization

2.92858E-05 1.10037E-05

0.000117025 4.39703E-05

0.000246764 9.27182E-05

0.000436108 0.000163862

0.00070878 0.000266314

0.001096156 0.000411865

0.001638934 0.000615806

0.002388786 0.000897553

0.003409841 0.0012812

0.004779689 0.001795901

0.006589572 0.002475939

0.008943288 0.003360314

0.011954294 0.004491657

0.015740449 0.005914251

0.02041585 0.007670967

0.026079352 0.009798948

0.032799553 0.012323968

0.040596391 0.01525352

0.049419984 0.01856886

0.059127929 0.022216484

0.069462986 0.026099735

0.080033761 0.030071555

0.090301633 0.033929562

0.099577616 0.037414882

0.107032929 0.04021611

0.111726725 0.041979738

0.112653486 0.042327955

0.116495868 0.043771676

0.126018434 0.047349645

0.141937619 0.053331054

0.164662335 0.061869545

0.194180942 0.072960745

0.22995968 0.0864041

0.270865514 0.101773889

0.3151265 0.118404328

0.360341303 0.135393152

0.403546399 0.151626856

0.441344661 0.165829018

0.470093044 0.176630817

0.486140262 0.182660332

( f x ) ( f b)



 




0.486181861 0.182675963

0.505151829 0.18980366

0.545070186 0.204802419

0.606906136 0.228036403

0.690412582 0.25941277

0.794047516 0.298352131

0.914995988 0.34379681

1.049296932 0.394258492

1.192068608 0.447902931

1.337816075 0.502665482

1.480795256 0.556387888

1.615401825 0.606964404

1.73655025 0.652484212

1.840009217 0.691357457

1.922664243 0.722413915

1.982685984 0.744966236

2.019592604 0.758833376

2.034205347 0.764323908

2.039903805 0.766465023

2.062989842 0.775139275

2.099873456 0.788997772

2.146598902 0.806554198

2.199143344 0.826297029

2.253688309 0.846791529

2.306844002 0.866764029

2.355813199 0.885163513

2.398488772 0.901198257

2.433485822 0.914347903

2.460115359 0.924353576

2.478310802 0.931190256

2.488521111 0.935026635

2.491585124 0.936177895

2.494569431 0.937299206

2.502369121 0.940229829

2.513725076 0.944496669

2.527439734 0.949649758

2.54244116 0.955286332

2.557823556 0.96106605

2.572866329 0.966718159

2.58703529 0.972041945

2.599970236 0.976902069

2.611463339 0.98122044

2.621432414 0.984966179

2.629892527 0.988144947

2.636928632 0.990788664

2.642671047 0.992946295

2.647274885 0.994676122

2.650903828 0.996039646

2.653718204 0.997097108
2.655866942 0.997904466

2.657482796 0.9985116

2.658680158 0.998961492

2.659554767 0.999290114

2.660184722 0.999526811

2.660632266 0.999694969

2.660945955 0.999812833

2.661162925 0.999894357

2.661311048 0.999950012

2.661410875 0.999987521

2.661444089 1

132.8984354
Raw Data

Runs

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

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

Sample size 0

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n > 3 ONLY

Testing Normality Raw Data Logtransformed

Sample Size 0.00 0.00

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

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

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

Skewness #DIV/0! #DIV/0!

SE Skewness 0.00 0.00 Result Raw Data Result Log Data

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

KURTOSIS CALCULATIONS FOR SAMPLE SETS WITH n = 3 ONLY

Average #DIV/0!

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

1

2

3
4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 0.00

Kurtosis -3.00

SE Kurtosis #DIV/0!

Result Kurtosis #DIV/0!

Skewness #DIV/0!

SE Skewness 0.00 Result Raw Data

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

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

kurtosis= #DIV/0! depending on sample size

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

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

Total Number of sources = 0

n =Total # test runs = 0

1

N

i df n

 

    

1

N

i

i

n





N

1

N

i

i

n




-1

This step is for QC only:

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

Mean = #DIV/0!

Pooled Variance = #DIV/0!

m= number future runs = 3

Term1 #DIV/0!

Term2 #DIV/0!

Squared root Term2 #DIV/0!

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

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

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

0

#NUM!

Current Confidence Level is #NUM!

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

a) n = number of test-runs =

b) Calculate u0

2

1

0

1

1

u

t

n







 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

    

 

1

1 i

i

df n



      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 
Conclusion

interval of differences for t-values used 0.0172

2.56522136

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

#NUM!

new t-statistic from tab Recalculate t-statistic
Logtransformed

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

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






Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

c.1.7 #NUM! d.1.7

Calculations Calculations

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 ,

 

x

I a b 

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )



( )

w a

e w





a w  
Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

Result #DIV/0!

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



1 1 1 1  1 2   1 3  4 1   1

( ) , , ,

n n n n n n n n

P t I I I

             

        
4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

Corrected probability is::

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

3 4 3/2

2

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

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

#NUM! e.1.7

Calculations Calculations

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



 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
#NUM! Result3:

3 4 1 1

,

n

I

 

   

 
#NUM!

0

3 4 1 1

,

24 2 2

u

n

I

 

   

  

 

1 1

2 2

5 1

2 2



 








a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

#NUM! f.1.7 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
#NUM! Result4: #NUM!

a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

g.1.7 #NUM! h.1.7

h.1.) Calculate 

Calculations Calculations

g.1.) Calculate 0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
Result5: #NUM! Result6:

#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

3 1

2 6

w a

    

 
#NUM!
#NUM!

0.0172

#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 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs kurtosis= #DIV/0!

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

n= 0

t0=t= #NUM!

a= -0.5 a=

b= 0.5 b=

w= #NUM! w=

c.1.1. #NUM! d.1.1. 

c.1.2. #NUM! d.1.2. 

c.1.3 #NUM! d.1.3

c.1.4. #NUM! d.1.4.

c.1.5. -1.5625 d.1.5.

c.1.6 #NUM! d.1.6

Calculations Calculations

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

Calculate the Incomplete Beta Functions needed for adjustment

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

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 #NUM! d.1.7

Result1: #NUM! Result2:

1- Calculate #NUM!

2- Calculate

2.1. Calculate #DIV/0!

2.2. Calculate #NUM!

2.3. Calculate #NUM!

Result: #DIV/0!

3-Calculate

3.1. Calculate #DIV/0!

3.2. Calculate #DIV/0!

2 6 3

   

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result #DIV/0!

4-Calculate

4.1. Calculate #NUM!

4.2. Calculate #DIV/0!

4.3. Calculate #DIV/0!

Result: #NUM!

5- Calculate

#NUM!

5.1. Calculate

#DIV/0!

5.2. Calculate

5.3. Calculate #DIV/0!

5.4. Calculate #DIV/0!

Result #NUM!

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

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

n n n n n n n n

P t I I I

n n



             

              

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

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

72 2 2 24 2 2

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

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

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

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

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

0.5 a= 1.5

0.5 b= 0.5

#NUM! w= #NUM!

#NUM! e.1.1. #NUM!

1.772453851 e.1.2. 0.886226925

#NUM! e.1.3 #NUM!

#NUM! e.1.4. #NUM!

0.0625 e.1.5.

#NUM! e.1.6

Calculations

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

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

  

     

  

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

 a

1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! e.1.7

#NUM! Result3:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 
0

3 

1 1

,

2 2

u

n

I

   

  

 
#NUM!
a= 1.5

b= 0.5

w= #NUM!

f.1.1. #NUM!

f.1.2. 0.886226925

f.1.3 #NUM!

f.1.4. #NUM!

-0.14583 f.1.5. -0.14583

#NUM! f.1.6 #NUM!

Calculations

f.1.) Calculate 0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM! f.1.7 #NUM!

#NUM! Result4: #NUM!

2 6 3

   

2 3 1

2 6

w a

    

 


a= -0.5 a= 0.5

b= 1 b= 1

w= #NUM! w= #NUM!

g.1.1. #NUM! h.1.1. 

g.1.2. #NUM! h.1.2. 

g.1.3 #NUM! h.1.3

g.1.4. #NUM! h.1.4.

g.1.5. -1.5625 h.1.5.

g.1.6 #NUM! h.1.6

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

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a

e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 #NUM! h.1.7

Result5: #NUM! Result6:

2 6 3

   

2 3 1

2 6

w a

    

 

2 6 3

   

2 3 1

2 6

w a

    

 


#NUM!

1.772454

#NUM!

#NUM!

0.0625

#NUM!

0

1

,1

2

u

n

I

  

   

5 3 1

2 3 2 3

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
#NUM!

#NUM!

2 6 3

   

3 1

2 6

w a

    

 