Distribution of repeatability parameters

The ASTM C 805 International Standard contains precision statements for the rebound index of the rebound hammers (ASTM, 2008). It is given for the precision that the within-test standard deviation of the rebound index is 2.5 units, as “single-specimen, single-operator, machine, day standard deviation”.

Therefore, the range of ten readings should not exceed 12 units (taking into account a k = 4.5 multiplier given in ASTM C 670). The multiplier is actually the one digit round value of the p=0.95 probability level critical value (k = 4.474124) for the standardized range statistic of a N(  ,1) normal distribution population at n = 10 according to Harter, 1960. Dependence of the within-test standard deviation on the average rebound index is not indicated. Particular literature data support the ASTM C 805 suggestions (e.g. Mommens, 1977).

Based on the ASTM C 805 implications it can be summarized that the probability distribution for the range

(r

R

) of ten rebound index readings is supposed to follow a normal probability distribution, where r

R

= 12 at a

p = 0.95 probability level if n = 10; and the within-test standard deviation of the rebound index can be supposed to be mean value of an undetermined probability distribution and s

R

= 2.5 if n = 10.

There are two underlying assumptions in the precision statements of the rebound index given in the ASTM C 805 International Standard: (1) the within-test standard deviation of the rebound index has a constant value independently of the properties of the actual concrete and of the actual operator error, and (2) the percentage points of the standardized ranges of N (  ,1) normal probability distribution populations can be applied for the determination of the acceptable range of rebound index readings at test areas. No indication is given in the ASTM C 805 either about the probability distribution of the within-test standard deviation of the rebound index or its percentile level for which the value is given in the standard. In the absence of the above information one may assume – as a first estimate – that the within-test standard deviation of the rebound index has a normal probability distribution and the value s

R

= 2.5 is its mean value.

An extended statistical analysis has been made on the previously detailed 8955 data-pairs of corresponding average rebound indices and standard deviations of rebound indices that were collected from 48 different sources (in which the number of in-situ test areas was 4785 and the number of laboratory test areas was 4170). It can be realized in Fig. 4.13a that the distribution of the within-test standard deviation of the rebound index has a strong positive skewness (  = 1.7064), therefore, the assumption of the normal probability distribution should be rejected. Fit of distributions resulted that a three-parameter Dagum distribution (also referred in the literature as generalized logistic-Burr or inverse Burr distribution) gives the best goodness of fit out of more than 60 different types of distributions. Goodness of fit analysis was performed by running the Kolmogorov-Smirnov test, the Anderson-Darling test and the χ

²

-test.

The parameters of the distribution function are as follows:

f(s

R

; a, b, c)

_{a 1}

b R

1 b R

c 1 s c

c ab s















 



 







 







(D

f

: s

R

= 0.23 to 7.80) Eq. (4.1)

where: a = 1.7958, b = 3.7311, c = 1.2171

x

7,2 6,4 5,6 4,8 4 3,2 2,4 1,6 0,8

f(x)

0,18

0,16

0,14

0,12

0,1

0,08

0,06

0,04

0,02

0

- - - - - 0.16

0.12

0.08

0.04

0 - - - -1 2 3 4 5 6 7 f(sR), -

sR,

-ƒ ( s

R

;a,b,c)

_{a 1}

b R

1 b R

c 1 s c

c ab s















 



 







 







a)

x

7,2 6,4 5,6 4,8 4 3,2 2,4 1,6 0,8

f(x)

0,18

0,16

0,14

0,12

0,1

0,08

0,06

0,04

0,02

0

- - - - - 0.16

0.12

0.08

0.04

0 - - - -1 2 3 4 5 6 7 f(sR),

-sR, -b)

sR= 2.5 (p=0.885)

Fig. 4.13 a) Relative frequency histogram of the standard deviation of the rebound index together with the best

goodness of fit probability density function (PDF), b) with the indication of s

R

= 2.5.

It can be realized that the s

R

= 2.5 value does not coincide either with the modus (= mode), or the median (= 50th percentile), or the mean value, but rather corresponds to a p = 88.5% probability level (Fig.

13b). If one would estimate the probability distribution with a N (1.667, 0.75) normal distribution (for which the goodness of fit is considerably weaker than that of the Dagum distribution) then the s

R

= 2.5 value would correspond to a p = 86.7% probability level. The mean value is E[s

R

] = 1.667; the median value is m[s

R

] = 1.5; the mode value is Mo[s

R

] = 1.45; the 95% percentile value is v

95

[s

R

] = 3.1526; for the analysed range of s

R

= 0.23 to 7.80. Value of s

R

= 2.5 exceeds the experimental values in 88.5% of the cases.

Next check is the analysis of the rebound index ranges (r

R

= R

max

– R

min

) at 8342 test areas in the case of real measurements (in which the number of in-situ test areas was 4785 and the number of laboratory test areas was 3557). (Note that the analysis of the standard deviation of the rebound index in the previous paragraphs is based on more test areas (8955) than that of the range of the rebound index (for that only 8342 test areas were available). In the technical literature several references include only the average rebound index and the standard deviation of the rebound index, without the publication of the individual rebound index readings. That is the reason of the difference between the sizes of the examined database.) Fig. 4.14 indicates the empirical probability histogram together with the best goodness of fit four-parameter Burr distribution corresponding to the 8342 test areas. The parameters of the distribution function are as follows:

f(r

R

; a, b, c, d)

_a1

b R

1 ab R

c d 1 r

c

c d ab r



















 



 



 



 



(D

f

: r

R

= 1 to 24) Eq. (4.2)

where: a = 0.89001, b = 4.0809, c = 3.755, d = 0.41591

One can again realize a strong positive skewness ( = 1.9432), and the median (= 50th percentile) for the rebound index ranges at the test areas is found to be m[r

R

] = 4. The mean value is E[r

R

] = 4.8068 and the mode value is Mo[r

R

] = 3.75. Considering the value of r

R

= 12 as of the ASTM C 805 proposal, a p = 98.7%

probability level can be determined. The rebound index range at a test area corresponding to the p = 95%

probability level as of the ASTM C 805 target is found to be v

95

[r

R

] = 9. The mean value is E[r

R

] = 4.8068; the mode value is Mo[r

R

] = 3.75 for the analysed range of r

R

= 1 to 24.

x ^{14 16 18 20 22 24}

12 10 8 6 4 2

f(x)

0,3 0,28 0,26 0,24 0,22 0,2 0,18 0,16 0,14 0,12 0,1 0,08 0,06 0,04 0,02 0

rR, --

- - - - - - - 0.28 0.24 0.20 0.16 0.12 0.08 0.04 0

f(rR), -

- - - - - - - - -

2 4 6 8 10 12 14 16 18 20 22 24

ƒ (r

R

;a,b,c,d)

_{a 1}

b R

1 ab R

c d 1 r

c c

d ab r















 



 



 



 



 



b)

Fig. 4.14 Relative frequency histogram of the range of rebound index readings together with the best goodness of

fit probability density function (PDF).

It can be concluded that the supposition of having normal probability distribution for both r

R

and s

R

should be rejected; implications given in ASTM C 805 do not fit to empirical findings.

After the above statistical analyses that are only partly confirming the assumptions of ASTM C 805, the next check can be the analysis of the assumption of ASTM C 670 that actually suggests the application of the theory of standardized ranges (  = r/  ) for N(  ,1) normal probability distribution populations for the determination of the multiplier applied to the maximum acceptable range (ASTM, 2003). One may realize for the rebound hardness method (if 10 replicate readings are considered at each test area) that the suggested value of the multiplier is k = 4.5 according to ASTM C 670, which is the one-digit round value of the percentage point of the standardized range (  ) for a sample of n = 10 from a N(  ,1) normal probability distribution population corresponding to a cumulative probability of p = 95% (  = 4.474124;

see e.g Harter, 1960). The standardized ranges usually can not be applied for actual measurements as the real standard deviation (  ) is not known. Therefore, the studentized ranges (  = r/s) can be introduced for N(  , 

²

) normal probability distribution populations for the selection of the multiplier applied to the maximum acceptable range. Based on the number of the measured results an appropriate degree of freedom (  ) for the independent estimate s

²

of 

²

should be selected. For large samples (  → ∞ ) the percentage point of the studentized range (  ) approaches to the percentage point of the standardized range (  ). Fig. 4.15 indicates the cumulative distribution function of the calculated studentized ranges ( 

R

= r

R

/s

R

) corresponding to the 8342 test areas together with the percentage points of the standardized ranges for n = 10 of N(  ,1) for cumulative probabilities of p = 0.01 % to 99.99% (based on Harter, 1960).

It is assumed for the present analysis that the comparison of the empirical studentized ranges ( 

R

) with the standardized ranges (  ) is acceptable due to the unusually large number of measured data. It can be realized that the median (= 50th percentile) values are almost equal; for the empirical values of the studentized ranges m[ 

R

] = 2.991 and for the standardized ranges by Harter (1960) m[ ] = 3.024202. It is demonstrated in the technical literature that the probability distribution of the standardized ranges (  ) has a positive skewness (  = 0.3975), therefore the mean value E[  ] does not equal to the median value, but E[  ] = 3.077505 (Harter, 1960). The probability distribution of the empirical studentized ranges ( 

R

) corresponding to the 8342 test areas, however, has a negative skewness (  = –0.26501), and the mean value is E[ 

R

] = 2.9794. Fit of distributions resulted that a four-parameter Pearson VI distribution (also referred in the literature as beta prime or inverse beta distribution) gives the best goodness of fit out of more than 60 different types of distributions.

The parameters of the distribution function are as follows:

f(

R

; a, b, c, d) =

_ab

R 1 a R

c 1 d ) b , a ( c

c d





 



 

 



 



  







 

 



  

 B

(D

f

: 

R

= 0.555 to 4.786) Eq. (4.3)

where

) b a (

) b ( ) a ) ( b , a

(  



 

B is the Euler Beta function,

and a = 41399.0, b = 27867.0, c = 35.186, d = –49.297

Fig. 4.15 clearly indicates the difference in the probability distributions of the percentage points of the

standardized ranges (  ) by Harter (1960) and that of the empirical studentized ranges (

R

) corresponding

to the 8342 test areas. One can realize that at the cumulative probability level of p = 95% the difference is

considerable; v

95

[ ] = 4.474124 and v

95

[ 

R

] = 3.635.

As the selection of the analysed 8342 test areas was free of any filtering, it is assumed that a further increase in the number of the data points would not result a better fit between the probability distributions of the percentage points of the standardized ranges (  ) and that of the empirical studentized ranges ( 

R

).

Based on the present comprehensive statistical analysis, the application of Table 1 of ASTM C 670 for the rebound hardness method is suggested to be reconsidered.

0,0 0,2 0,4 0,6 0,8 1,0

0 1 2 3 4 5 6 7

F(θ

R

), F(ω)

θ

R

, ω ,

-1.0 0.8 0.6 0.4

0.2 0

0

1 2 3 4 5 6 7

ω  θ

R

 

Fig. 4.15 Cumulative probability distribution function (CDF) of the calculated studentized ranges ( 

R

= r

R

/s

R

) corresponding to the 8342 test areas together with the standardized ranges for n = 10 of N (  ,1) for cumulative

probabilities of p = 0.01% to 99.99%.

The relative frequency histograms are constructed for the coefficient of variation of rebound index readings based on the analysis of 8955 test areas (from which 4170 are laboratory and 4785 are in-situ test areas, with total number of individual rebound index readings exceeding eighty thousand), as well. A strong positive skewness is realized over the analyzed range (  = 2.2472 for the coefficient of variation) (Fig. 4.16).

The findings confirm that experimental data are available for the repeatability parameters of concrete strength (Soroka, 1971; Shimizu et al, 2000). It was demonstrated in the literature – based on an extensive analysis of 10788 drilled core samples taken from 1130 existing reinforced concrete buildings – that the coefficient of variation of concrete strength had a lognormal probability distribution with strong positive skewness, while normal probability distribution was found for the compressive strength itself (conventional concretes were studied with compressive strength lower than 50 MPa; Shimizu et al, 2000).

Similar observation can be made considering the distributions of the standard deviation and the coefficient of variation of concrete strength indicated earlier in Fig. 4.11.

Surface hardness and compressive strength of concrete are interrelated material properties, therefore, it is expected that the probability distribution of the coefficient of variation of rebound index readings has a positive skewness.

Goodness of fit analysis of sixty different probability distributions has demonstrated that the probability distribution of the coefficient of variation (V

R

) of rebound index readings follows a three parameter Dagum distribution (a = 2.2255; b = 3.1919; c = 2.7573), of which mean value is E[V

R

] = 4.4021%; the median value is m[V

R

] = 3.8%; the mode value is Mo[V

R

] = 3.125%; the 95% percentile value is v

95

[V

R

] = 9.2132%;

for the analysed range of V

R

= 0.43% to 31.12% (Fig. 4.16). The parameters of the distribution function

are as follows:

f(V

R

; a, b, c)

_a1

b R

1 ab R

c 1 V c

c ab V





 





 





 

 



 

 

 





 (D

f

: V

R

= 0.43% to 31.12%) Eq. (4.4)

where: a = 2.2255, b = 3.1919, c = 2.7573

x

30 25 20 15 10 5

f(x)

0,28 0,26 0,24 0,22 0,2 0,18 0,16 0,14 0,12 0,1 0,08 0,06 0,04 0,02 0

- - - - - - - -0.28 0.24 0.20 0.16 0.12 0.08 0.04 0

f(VR),

-VR,

-- - - - - -

5 10 15 20 25 30

ƒ ( V

R

;a,b,c)

_{a 1}

b R

1 b R

c 1 s c

c ab s





 





 



 



 



 

 

 







Fig. 4.16 Relative frequency histogram of the coefficient variation of rebound index readings together with the best

goodness of fit probability density function (PDF).

In document Rebound surface hardness and related properties of concrete (Pldal 64-69)