MIL-HDBK-17-1F Volume 1,Chapter 8 Statistical Methods 8.2.5.1 Example Table 8.2.5.1 presents tensile strength data(in ksi)for a unidirectional composite material,tested un- der room temperature dry conditions. TABLE 8.2.5.1 Room temperature dry tensile strength for a unidirectional composite material. 226 227 226 232 252 The mean and standard deviation for these data are x 232.6 and s=11.13.Using the normal model (Section 8.3.4.3),a B-basis value for these data is B=x-kBs=232.6-3.40711.13)=195 8.2.5.1 The first point to be made is that a B-basis value determined from as few as five specimens is not likely to be sufficiently reproducible for it to be regarded as a material constant for most applications.For the pre- sent discussion,the plausible assumption is made that the above data are a sample from a normal distri- bution with a mean of 230 and a standard deviation of 10. The theoretical population of B-basis values which corresponds to this assumed normal population of strength measurements can be calculated,and is displayed in Figure 8.2.5.1.Note that the observed ba- sis value is near the mean of this population of basis values.This is to be expected since the parameters of the hypothetical normal distribution have been based on the same set of data from which the basis value was determined.However,note also that values within +20 ksi of the basis value are also likely to be observed.Based on this analysis,one cannot rule out the possibility of the B-basis value of the next sample of five being as low as 180 ksi or as high as 220 ksi. 8.2.5.2 Mean and standard deviations of normal basis values Basis values calculated from small samples exhibit high variability.One way of quantifying this is to calculate the theoretical mean,standard deviation,and coefficient of variation of basis values from hypo- thetical populations as functions of the number of specimens.Of course,these calculations are going to depend on the statistical model chosen and the parameters selected for this model.However,the objec- tive of these calculations is not to provide rigid criteria,but rather to inform the user of the qualitative be- havior of basis values. A normal population with a mean of 100 and a standard deviation of 10 will be considered for the dis- cussion in this subsection.The 10%coefficient of variation is typical of what is observed for many mate- rial properties,and the mean of 100 is within an order of magnitude of most strength measurements (in ksi)for unidirectional composite materials.The choice of the normal population is made because the normal basis values procedures have broad appeal,and because the required calculations can be done in closed form.Sample sizes for basis values from Weibull populations should as a rule be larger than those for normal populations in order to achieve the same degree of reproducibility.Only basis values for a simple random sample are considered here;ANOVA basis values are discussed in the next subsection. The mean and one standard deviation limits for B-basis values from a normal population with a mean of 100 and a standard deviation of 10 is displayed in Figure 8.2.5.2(a)as a function of the number of specimens.Note the extremely high variability for sample sizes of ten or less. 8-11
MIL-HDBK-17-1F Volume 1, Chapter 8 Statistical Methods 8-11 8.2.5.1 Example Table 8.2.5.1 presents tensile strength data (in ksi) for a unidirectional composite material, tested under room temperature dry conditions. TABLE 8.2.5.1 Room temperature dry tensile strength for a unidirectional composite material. 226 227 226 232 252 The mean and standard deviation for these data are x = 232.6 and s = 11.13 . Using the normal model (Section 8.3.4.3), a B-basis value for these data is B = x − − kBs = 232.6 3.407(11.13) = 195 8.2.5.1 The first point to be made is that a B-basis value determined from as few as five specimens is not likely to be sufficiently reproducible for it to be regarded as a material constant for most applications. For the present discussion, the plausible assumption is made that the above data are a sample from a normal distribution with a mean of 230 and a standard deviation of 10. The theoretical population of B-basis values which corresponds to this assumed normal population of strength measurements can be calculated, and is displayed in Figure 8.2.5.1. Note that the observed basis value is near the mean of this population of basis values. This is to be expected since the parameters of the hypothetical normal distribution have been based on the same set of data from which the basis value was determined. However, note also that values within ±20 ksi of the basis value are also likely to be observed. Based on this analysis, one cannot rule out the possibility of the B-basis value of the next sample of five being as low as 180 ksi or as high as 220 ksi. 8.2.5.2 Mean and standard deviations of normal basis values Basis values calculated from small samples exhibit high variability. One way of quantifying this is to calculate the theoretical mean, standard deviation, and coefficient of variation of basis values from hypothetical populations as functions of the number of specimens. Of course, these calculations are going to depend on the statistical model chosen and the parameters selected for this model. However, the objective of these calculations is not to provide rigid criteria, but rather to inform the user of the qualitative behavior of basis values. A normal population with a mean of 100 and a standard deviation of 10 will be considered for the discussion in this subsection. The 10% coefficient of variation is typical of what is observed for many material properties, and the mean of 100 is within an order of magnitude of most strength measurements (in ksi) for unidirectional composite materials. The choice of the normal population is made because the normal basis values procedures have broad appeal, and because the required calculations can be done in closed form. Sample sizes for basis values from Weibull populations should as a rule be larger than those for normal populations in order to achieve the same degree of reproducibility. Only basis values for a simple random sample are considered here; ANOVA basis values are discussed in the next subsection. The mean and one standard deviation limits for B-basis values from a normal population with a mean of 100 and a standard deviation of 10 is displayed in Figure 8.2.5.2(a) as a function of the number of specimens. Note the extremely high variability for sample sizes of ten or less
MIL-HDBK-17-1F Volume 1,Chapter 8 Statistical Methods 0.30 0.25 0.20- 0.15- 0.10 Observed bosis volue 0.05 0.00- 10 160 180 200 220 240 B-basis Value (ksi) FIGURE 8.2.5.1 B-basis value population for a sample of size five. 100 80 enIDA 60 40 Mean=100.S.D.=10 20 0 5 10 15 20 25 Number Of Specimens FIGURE 8.2.5.2(a)Normal B-basis values with one-sigma limits. 8-12
MIL-HDBK-17-1F Volume 1, Chapter 8 Statistical Methods 8-12 FIGURE 8.2.5.1 B-basis value population for a sample of size five. FIGURE 8.2.5.2(a) Normal B-basis values with one-sigma limits
MIL-HDBK-17-1F Volume 1,Chapter 8 Statistical Methods 0.40 0.35- 0.30- Population CV =10 0.25 可 0.20 6 苦 0.15 0.10 0.05 5 10 15 20 25 30 35 Number Of Specimens FIGURE 8.2.5.2(b)C.V.of B-basis values:normal model. The coefficient of variation(CV)is the ratio of the standard deviation to the mean.It is,therefore, easy to obtain the CV as a function of sample size from the information in Figure 8.2.5.2(a).Figure 8.2.5.2(b)displays these CV values,with a horizontal line at 10%provided for reference. Since an A-basis value is a 95%lower confidence limit on the first population percentile,while a B-basis value is a 95%lower confidence limit on the tenth percentile,it is obvious that,for a given amount of reproducibility in the basis values,substantially more data is required for A-basis than for B-basis.If one assumes that the measurements are a sample from a normal distribution,then it is reasonable to de- cide on the number of specimens as for B-basis and then multiply the resulting n by three to get an A-basis sample size.This is based on the assumption that the population coefficient of variation is less than 15%. 8.2.5.3 Basis values using the ANOVA method When the data come from several batches,and the between-batch variability is substantial,the flow- chart(Figure 8.3.1)might indicate that the ANOVA method of Section 8.3.5.2 should be used.To decide how many specimens are required when the data are to come from several batches,begin by acting as if the data were from a single batch,and selecting a sample size,say n,based on the discussion of the previous subsection.If J is the number of specimens per batch(assumed equal for all batches)and p is the correlation between any two measurements taken on specimens from the same batch,then the num- ber of specimens required for comparable reproducibility in the multi-batch case is approximately n=Jp+1-pn 8.2.5.3 8-13
MIL-HDBK-17-1F Volume 1, Chapter 8 Statistical Methods 8-13 FIGURE 8.2.5.2(b) C.V. of B-basis values: normal model. The coefficient of variation ( CV ) is the ratio of the standard deviation to the mean. It is, therefore, easy to obtain the CV as a function of sample size from the information in Figure 8.2.5.2(a). Figure 8.2.5.2(b) displays these CV values, with a horizontal line at 10% provided for reference. Since an A-basis value is a 95% lower confidence limit on the first population percentile, while a B-basis value is a 95% lower confidence limit on the tenth percentile, it is obvious that, for a given amount of reproducibility in the basis values, substantially more data is required for A-basis than for B-basis. If one assumes that the measurements are a sample from a normal distribution, then it is reasonable to decide on the number of specimens as for B-basis and then multiply the resulting n by three to get an A-basis sample size. This is based on the assumption that the population coefficient of variation is less than 15%. 8.2.5.3 Basis values using the ANOVA method When the data come from several batches, and the between-batch variability is substantial, the flowchart (Figure 8.3.1) might indicate that the ANOVA method of Section 8.3.5.2 should be used. To decide how many specimens are required when the data are to come from several batches, begin by acting as if the data were from a single batch, and selecting a sample size, say n , based on the discussion of the previous subsection. If J is the number of specimens per batch (assumed equal for all batches) and ρ is the correlation between any two measurements taken on specimens from the same batch, then the number of specimens required for comparable reproducibility in the multi-batch case is approximately ~n = J +1 n ρ ρ − 8.2.5.3
MIL-HDBK-17-1F Volume 1,Chapter 8 Statistical Methods If p=0,there is no between-batch variability;hence=n.At the other extreme,if p=1,there is perfect correlation within each batch(that is,each batch consists of J copies of a single value),and=Jn,one needs n batches to have the same degree of reproducibility as n specimens in the uncorrelated (p=0) case.In practice,p is unknown.For sample size guidelines,letting p=1/2 in Equation 8.2.5.3 is ade- quate for most applications.This suggests that (n(J+1)/(2J))batches of size J are necessary for the same degree of reproducibility as a single sample of size n.It is usually preferable to divide a fixed num- ber of specimens among as many batches as is possible.However,testing a new batch is much more expensive than testing several more specimens within a single batch.It is sometimes the case that the variability between two panels from the same batch,processed and tested separately,is comparable to the variability between two panels from different batches.When this is the case,it is reasonable to sub- stitute multiple panels within a batch for multiple batches. Suppose that an A-basis ANOVA value is desired which has the same degree of reproducibility as a B-basis value would have for a single sample of size n=5.First,make the adjustment to an A-basis sam- ple size:n=3.5=15,as described in Section 8.2.5.2.Next,assuming moderate between-batch vari- ability and a batch size of(say)J=3,calculate that nA[(J+1)/(2J)]=10 batches are required for the de- sired degree of reproducibility,for a total of 30 specimens. 8.3 CALCULATION OF STATISTICALLY-BASED MATERIAL PROPERTIES Section 8.3 contains computational methods for obtaining B-and A-basis values from composite ma- terial test data. 8.3.1 Guide to computational procedures The procedure used to determine a basis value depends on the characteristics of the data.The step- by-step procedure for selecting the appropriate computational method is illustrated by the flowchart in Figure 8.3.1.Details for the specific computational methods are provided in later sections. Two approaches are used,with the selection dependent on whether the data are structured or not. The k-sample Anderson-Darling test in Section 8.3.2 examines the differences among groups of data to determine if they are significant or negligible,which also determines whether the data should be treated as structured or unstructured.The difference between structured and unstructured data is considered in Section 8.3.2.Briefly,data sets which either cannot be grouped,or for which there are negligible differ- ences among such groups,are called unstructured.Otherwise,the data are said to be structured.All data should be examined for outliers,using the test in Section 8.3.3.From this point,different ap- proaches are used for analysis depending on whether the data are unstructured or structured. The approach for unstructured data is described first.If unstructured data were grouped and the dif- ferences among the groups found to be negligible,the groups are combined.The test for outliers should be performed again on the combined data.Tests for goodness-of-fit(Section 8.3.4.1)are performed for the Weibull,normal,and lognormal distributions in succession.If the observed significance level (OSL) for the Weibull distribution is greater than 0.05,indicating an adequate fit for the data to the Weibull distri- bution,then a Weibull basis value is recommended(Section 8.3.4.2).If the OSL for the Weibull distribu- tion is less than 0.05 and the OSL for the normal distribution is greater than 0.05,then the normal basis value should be used (Section 8.3.4.3).If the OSL's from both the Weibull and normal goodness-of-fit tests are less than 0.05,and the OSL for the lognormal distribution is greater than 0.05,then a lognormal basis value is recommended(Section 8.3.4.4).If none of the three OSL's are greater than 0.05,then the nonparametric basis value procedures are recommended (Section 8.3.4.5).Section 8.3.4 provides the rationale for the order of the distribution selection.An alternative approach is to use the basis values cor- responding to the best-fitting model.Exploratory data analysis (EDA)techniques,described in Section 8.3.6,can provide graphical illustrations of the data distribution in support of the goodness-of-fit tests. 8-14
MIL-HDBK-17-1F Volume 1, Chapter 8 Statistical Methods 8-14 If ρ = 0 , there is no between-batch variability; hence ~n = n . At the other extreme, if ρ = 1, there is perfect correlation within each batch (that is, each batch consists of J copies of a single value), and ~n = Jn , one needs n batches to have the same degree of reproducibility as n specimens in the uncorrelated ( ρ = 0 ) case. In practice, ρ is unknown. For sample size guidelines, letting ρ =1/2 in Equation 8.2.5.3 is adequate for most applications. This suggests that (n(J+1) / (2J) ) batches of size J are necessary for the same degree of reproducibility as a single sample of size n . It is usually preferable to divide a fixed number of specimens among as many batches as is possible. However, testing a new batch is much more expensive than testing several more specimens within a single batch. It is sometimes the case that the variability between two panels from the same batch, processed and tested separately, is comparable to the variability between two panels from different batches. When this is the case, it is reasonable to substitute multiple panels within a batch for multiple batches. Suppose that an A-basis ANOVA value is desired which has the same degree of reproducibility as a B-basis value would have for a single sample of size n = 5. First, make the adjustment to an A-basis sample size: nA = 3• 5 = 15, as described in Section 8.2.5.2. Next, assuming moderate between-batch variability and a batch size of (say) J = 3, calculate that nA (J+1) / (2 J) = 10 batches are required for the desired degree of reproducibility, for a total of 30 specimens. 8.3 CALCULATION OF STATISTICALLY-BASED MATERIAL PROPERTIES Section 8.3 contains computational methods for obtaining B- and A-basis values from composite material test data. 8.3.1 Guide to computational procedures The procedure used to determine a basis value depends on the characteristics of the data. The stepby-step procedure for selecting the appropriate computational method is illustrated by the flowchart in Figure 8.3.1. Details for the specific computational methods are provided in later sections. Two approaches are used, with the selection dependent on whether the data are structured or not. The k-sample Anderson-Darling test in Section 8.3.2 examines the differences among groups of data to determine if they are significant or negligible, which also determines whether the data should be treated as structured or unstructured. The difference between structured and unstructured data is considered in Section 8.3.2. Briefly, data sets which either cannot be grouped, or for which there are negligible differences among such groups, are called unstructured. Otherwise, the data are said to be structured. All data should be examined for outliers, using the test in Section 8.3.3. From this point, different approaches are used for analysis depending on whether the data are unstructured or structured. The approach for unstructured data is described first. If unstructured data were grouped and the differences among the groups found to be negligible, the groups are combined. The test for outliers should be performed again on the combined data. Tests for goodness-of-fit (Section 8.3.4.1) are performed for the Weibull, normal, and lognormal distributions in succession. If the observed significance level (OSL) for the Weibull distribution is greater than 0.05, indicating an adequate fit for the data to the Weibull distribution, then a Weibull basis value is recommended (Section 8.3.4.2). If the OSL for the Weibull distribution is less than 0.05 and the OSL for the normal distribution is greater than 0.05, then the normal basis value should be used (Section 8.3.4.3). If the OSL's from both the Weibull and normal goodness-of-fit tests are less than 0.05, and the OSL for the lognormal distribution is greater than 0.05, then a lognormal basis value is recommended (Section 8.3.4.4). If none of the three OSL's are greater than 0.05, then the nonparametric basis value procedures are recommended (Section 8.3.4.5). Section 8.3.4 provides the rationale for the order of the distribution selection. An alternative approach is to use the basis values corresponding to the best-fitting model. Exploratory data analysis (EDA) techniques, described in Section 8.3.6, can provide graphical illustrations of the data distribution in support of the goodness-of-fit tests
MIL-HDBK-17-1F Volume 1,Chapter 8 Statistical Methods START REMOVE REMOVE OUTLIER OUTLIER ARE THE DATA FROM A SINGLE BATCH/ GROUPT SECTION 8.3.2 No TEST BATCH/GROUP SAMPLES FOR OUTLIERS SECTION8.3.3.1 INVESTIGATE SOURCE OF VARIABILITY No CAUSE FOR Yes- OUTLIER BETWEEN-BATCH/GROUP VARIATION? Yes SECT0N8.3.2.2 EQUALITY OF VARIANCE? No Yes (DIAGNOSTIC TEST) SECT10N83.5.2.1 TEST SINGLE SAMPLE FOR OUTLIE No SECTION 8.3.3.1 INVESTIGATE DEPARTURE FROM INVESTIGATE STANDARD MODELS AND/OR SOURCES OF SOURCES OF VARIABILITY VARIABILITY ANOVA METHOD SECT10N8,3.5,2 CAUSE FOR OUTLIER TEST FOR WEIBULLNESS SECTION 8.3.4.2 Yes. WEIBULL METHOD SECT1oN8,3.4.2.3 No TEST FOR NORMALITY NORMAL METHOD SECTION 8.3.4.3 SECTION 8.3.4.3.3 No TEST FOR LOGNORMALITY LOGNORMAL SECTION 8.3.4.4 METHOD SECTION 8.3.4.4 No NONPARAMETRIC METHOD SECTION 8.3.4.5 FIGURE 8.3.1 Flowchart illustrating computational procedures for B-basis material property values. The ANOVA method applies to the simple multiple-batch case.Other scenarios may be addressed by linear regression(RECIPE). The acceptance of data analyzed by linear regression for inclusion in MIL-HDBK-17 is under consideration. 8-15
MIL-HDBK-17-1F Volume 1, Chapter 8 Statistical Methods 8-15 FIGURE 8.3.1 Flowchart illustrating computational procedures for B-basis material property values.1 1 The ANOVA method applies to the simple multiple-batch case. Other scenarios may be addressed by linear regression (RECIPE). The acceptance of data analyzed by linear regression for inclusion in MIL-HDBK-17 is under consideration