Ejemplo Grubbs y Cochran

8/12/2019 Ejemplo Grubbs y Cochran

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Challenge 9.2-1

Method p erformanc e stud y acc ordin g to DIN EN ISO 10304-1 (2009)

Method Determinat ion of b romide in industr ia l water

Laboratoriesn i 8 Replicates n k 4

Data set The number of replicates of each laboratory must be equal for the test!

n k \ ni 2 3 4 5 6

1 15.10 15.20 15.50 14.80 15.20

2 15.00 15.00 15.20 14.90 15.30

3 14.90 15.10 15.40 14.80 15.00

4 15.10 15.10 15.40 14.90 15.20

4 4 4 4 4

16 16 16 16 16

Stage 1: Preliminary data

15.025 15.1 15.375 14.85 15.175

0.0957 0.0816 0.1258 0.0577 0.1258

Stage 2: Test for outliers of type 1 by Grubbs rm (P = 95%, n k)

xmax 15.1 15.2 15.5 14.9 15.3

0.783 1.225 0.993 0.866 0.993

Result no OL no OL no OL no OL no OL

xmin 14.90 15.00 15.20 14.80 15.00

1.306 1.225 1.391 0.866 1.391


If outliers within the laboratories are checked remove them from the data se

Type 1 OL-free data set

n k \ ni 2 3 4 5 6

1 15.1 15.2 15.5 14.8 15.2

2 15 15 15.2 14.9 15.3

3 14.9 15.1 15.4 14.8 15

4 15.1 15.1 15.4 14.9 15.2

Stage 3: Recalculation of the laboratory mean values and the standard deviations

15.025 15.100 15.375 14.850 15.175

0.0957 0.0816 0.1258 0.0577 0.1258

Stage 4: Test for outliers of type 2 by Grubbs rm (P = 95%, n k)

*i

x

.*i

s

max,mr

min,mr

*L

x*

Lxs


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15.143 sxL 0.169934482

0.6927 0.2513 1.3669 1.7225 0.1900


If oultliers of the mean value of the laboratories are checked remove them from the

Type 2 OL-free data

n k \ ni 2 3 4 5 61 15.10 15.20 15.50 14.80 15.20

2 15.00 15.00 15.20 14.90 15.30

3 14.90 15.10 15.40 14.80 15.00

4 15.10 15.10 15.40 14.90 15.20

15.03 15.10 15.38 14.85 15.18

Stage 7: Test for reintegration of outliers of type 1

Copy the outlier within the laboratory checked in stage 2 into G74!xmax 15.375 xmin 14.850 14.50

Result Reintegration of the outlier of type 1 is not possible!

Stage 8 : Test for outliers of type 3 by Cochran

0.009166667 0.00666667 0.015833333 0.00333333 0.01583333

0.015833333 0.0725

0.2184 C(P= 99%,n k, n i) 0.5209

Result The laboratory with the maximal variance must not be rejected!

Stage 9 : Calculation of parameters on the basis of the data set free of outliers of type

15.025 15.100 15.375 14.850 15.175

s i 0.095742711 0.08164966 0.125830574 0.05773503 0.12583057

15.143 Laboratories n L 8 n total 31

SSxx 0.027500 0.020000 0.047500 0.010000 0.047500

SSSxx 0.214167

Repeatability standard deviation, s r

sr 0.0964966

CVr% 0.64 nmean 3.870967742

Reproducibilitx standard deviation, s R

0.055421007 0.00729601 0.215837674 0.34271267 0.00417101

0.8027941 0.1146849 0.02722143

sR 0.1911361

CVR% 1.26

Stage 10 : Recovery rate

99.62

*

ikOLx

C

2

is

2i

s

2maxs

ix

x

2b

s

2xxnii

2L

s

x

iLmr ,

ix

2i

s

2xxn ii


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Stage 11 : Est imat ion of bias by an t- test

1.669 t(P= 99%, df =n - 1) 2.750

Result The result is valid!

t


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Grubbs Table

3

15.2 mg/L 4

5

7 8 9 6

15.40 15.10 15.30 7

15.40 15.00 15.20 8

15.20 15.10 15.30 9

15.30 15.10 10

11

12

rm(95%, nk)

4 3 4 rm(95%, ni)

16 9 16

31 Cochran Tabl121

15.325 15.0666667 15.225 k

0.0957 0.0577 0.0957

2

1.463 3

4

15.4 15.1 15.3 5

0.783 0.577 0.783 6

no OL no OL no OL 7

8

15.20 15.00 15.10 9

1.306 1.155 1.306 10

no OL no OL no OL

table in B10:M17!

7 8 9

15.4 15.1 15.3

15.4 15 15.2

15.2 15.1 15.3

15.3 15.1

15.325 15.067 15.225

0.0957 0.0577 0.0957

2.032


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1.0727 0.4475 0.4843

no OL no OL no OL

data set table in B10:M17!

7 8 915.40 15.10 15.30

15.40 15.00 15.20

15.20 15.10 15.30

15.30 15.10

15.33 15.07 15.23

0.00916667 0.00333333 0.00916667

1 - 3

15.325 15.067 15.225

0.09574271 0.05773503 0.09574271

0.027500 0.006667 0.027500

0.13292101 0.01734701 0.02708767


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P = 95%

P=95%

1.153

1.463

1.672

1.822

1.938

2.032

2.11

2.176

2.234

2.285

1.463

2.032

P = 99%

Degrees of freedom

1 2 3 4 5 6 7

0.9999 0.995 0.9794 0.9582 0.9373 0.9172 0.8988

0.9933 0.9423 0.8831 0.8335 0.7933 0.7606 0.7335

0.9676 0.8643 0.7814 0.7212 0.6761 0.641 0.6129

0.9279 0.7885 0.6957 0.6329 0.5875 0.5531 0.5259

0.8828 0.7218 0.6258 0.5635 0.5195 0.4866 0.4608

0.8376 0.6644 0.5685 0.508 0.4659 0.4347 0.4105

0.7945 0.6152 0.5209 0.4627 0.4226 0.3932 0.3704

0.7544 0.5727 0.481 0.4251 0.387 0.3592 0.3378

0.7175 0.5358 0.4469 0.3934 0.3572 0.3308 0.3106

0.5209


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8

0.8823

0.7105

0.5897

0.5037

0.4401

0.3911

0.3522

0.3207

0.2945


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Data set for Challenge 9.2-1

Method Determination of brom ide in indus tr ial water

Data set

n k \ i 1 2 3 4 5 6

1 13.80 15.10 15.20 15.50 14.80 15.20

2 13.90 15.00 15.00 15.20 14.90 15.30

3 14.00 14.90 15.10 15.40 14.80 15.00

4 13.70 15.10 15.10 15.40 14.90 15.20


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15.2 mg/L

7 8 9

15.40 15.10 15.30

15.40 15.00 15.20

15.20 15.10 15.30

15.30 14.50 15.10

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Ejemplo Grubbs y Cochran