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Determination of reference distribution using Approach 1: all methods

Table 2 shows the results obtained from all SRM 114p data without sets R and L. It also shows the mean and the two-sided 95 % confidence limits as calculated using the bootstrap method (detailed on this method in Appendix C). The next step is to determine any outliers. The bootstrap method does not give the criterion needed to determine the outlier. Therefore, we arbitrarily selected the following criterion: if an organization's set of data contained more than four data points that are more than 5 % absolute value outside the confidence limits of all data sets, then it is considered an outlier. The absolute value 5 % is defined as the difference between the measured value and the confidence limits, high or low.

Table 2: Data from all methods except sets L and R. The results from the Bootstrap calculation are also shown. Data that lie outside the 95 % confidence limits are shown in bold, and data that are greater than 5 % outside the 95 % confidence limits are indicated as yellow cells.

Size
[µm]

Data from all devices

Bootstrap results

A

B

C

D

F

G

H

I

J

K

N

O

Q

S

T

U

V

W

X

Mean

Low

High

1

0.5

17.2

10.7

6.9

15.1

6.8

7.1

15.7

5.2

0.0

1.3

8.3

14.4

13.6

3.9

2.2

4.5

7.9

5.4

10.51

1.5

2.0

22.2

14.3

10.3

18.2

11.7

12.6

18.3

0.1

8.3

0.1

3.6

14.2

 

19.9

18.6

7.4

3.9

6.4

10.7

7.2

14.06

2

4.4

25.2

17.0

12.9

20.6

15.2

16.1

20.7

2.2

11.1

3.9

6.3

18.5

9.0

23.3

22.4

10.2

6.7

8.3

13.4

10.1

16.58

3

10.3

29.1

21.4

17.6

23.7

20.5

20.8

25.2

8.8

16.2

23.0

11.5

25.6

13.6

27.8

27.4

15.1

8.8

12.3

18.9

16.2

21.75

4

16.5

32.1

25.2

21.5

27.6

24.8

24.4

29.3

12.9

20.6

32.0

16.3

29.1

18.1

31.2

31.2

19.5

12.3

16.4

23.2

20.5

26.17

6

27.6

37.7

31.9

28.2

35.0

31.8

30.6

36.3

18.9

28.0

38.1

24.2

36.2

26.1

37.2

38.5

27.0

15.4

23.9

30.2

27.4

32.98

8

38.2

43.4

37.5

34.0

40.8

37.6

36.5

42.1

24.1

34.1

46.7

30.5

42.3

33.1

43.1

45.4

33.1

21.3

30.4

36.6

33.2

39.41

12

54.8

55.6

47.3

44.1

47.6

47.3

48.0

51.7

34.7

44.1

63.4

41.0

52.4

45.2

53.8

59.2

42.8

27.4

41.5

47.5

43.7

51.21

16

68.2

67.2

55.6

53.0

55.3

55.4

58.6

59.9

45.8

52.5

78.7

50.0

60.8

55.2

62.7

71.1

50.9

41.1

51.0

57.4

53.8

61.37

24

78.8

84.1

69.0

68.7

69.1

69.2

76.6

73.2

63.5

66.1

93.8

65.0

74.3

70.0

76.1

86.4

64.3

56.0

66.6

72.1

68.5

76.26

32

84.1

92.8

79.3

78.9

79.4

79.8

88.4

82.9

75.9

76.2

98.0

76.6

83.9

80.3

85.2

93.8

74.9

79.7

78.3

82.6

79.9

85.63

48

92.7

98.6

91.4

91.8

89.0

92.5

97.8

93.7

90.2

88.2

99.8

90.3

94.7

93.6

94.7

98.7

88.3

91.4

91.7

93.1

91.6

94.62

64

96.3

99.9

96.3

96.7

95.4

97.8

99.6

98.0

95.5

93.7

100.0

96.3

98.7

97.6

98.0

99.7

94.5

99.2

97.3

97.4

96.6

98.18

96

99.4

100.0

98.9

99.2

99.2

99.6

100.0

100.0

98.5

97.4

100.0

99.4

100.0

99.3

99.7

100.0

98.7

100.0

100.0

99.4

99.11

99.72

128

99.9

100.0

99.5

99.7

99.5

99.8

100.0

100.0

99.3

98.1

100.0

99.6

100.0

99.8

100.0

100.0

99.7

100.0

100.0

99.73

99.49

99.89

In Table 2, all data points that are outside the confidence limits are in bold and data points of more than absolute value 5% outside the confidence limits are in yellow cells. So, if any column contains more than 4 data in a yellow cell, than that data set is an outlier. Using this criterion, the sets A, B, J, N, T, U and W were considered outliers.

The next step is to recalculate mean distribution of SRM 114p and its 95 % confidence limits without the outliers. The reason of this recalculation is to have a better-defined distribution. This was done again using the bootstrap method and the results are shown in Table 3. Figure 2 shows the data with the bootstrap mean and 95 % confidence limits, after the outliers were eliminated. It is clear that the distribution obtained is narrower than shown in Figure 1. The data in Table 3 could be used to correct the data obtained by all methods as shown in Appendix D-1.

Figure 2: Data with the Bootstrap mean after the outliers are eliminated.

Table 3: Bootstrap mean and 95 % confidence limits calculated without the outliers.

Size
[µm]

1

1.5

2

3

4

6

8

12

16

24

32

48

64

96

128

Mean

7.8

11.4

13.8

18.6

22.8

29.9

36.0

46.1

54.8

69.4

79.9

91.9

96.8

99.3

99.6

Low

5.5

8.7

11.1

16.0

20.2

27.5

33.7

44.2

53.0

67.4

78.0

90.5

95.9

98.8

99.3

High

10.4

14.1

16.4

21.2

25.3

32.1

38.3

48.1

56.6

71.6

82.2

93.5

97.8

99.7

99.9

Therefore, the methodology that uses the mean curve to correct the data measured by various instruments would be:

This procedure was applied to the four cements used in this study using all the data sets available with the exception of sets R and L because they were clear outliers. Appendix D-1 shows graphs of all data.

This method could then be used to correct data obtained with any method and could be considered as a calibration of the instrument. Unfortunately, it is not that simple. If the results obtained would have being considered outliers, (i.e., sets of data that are more than 5 % absolute value outside the confidence limits obtained with the Bootstrap method), the correction is not enough. This can be seen in Figure 3A where sets W, J or N are still clearly outside the confidence limits. On the other hand if the data set is within the confidence limits, the correction factor will reduce the spread of the data (Figure 3B). Therefore, the reference SRM 114p could be used in two ways:

Figure 3: Example of correction obtained using the correction factor. The cement used was CCRL 131. (A) only the outliers R and L were not plotted; (B) the outliers identified by the bootstrap method are not included


Next: Determination Up: Reference distribution Previous: Reference distribution