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B-1 APPENDIX B: Example of Analysis Technique Example for Determining Invalid Data DATA SOURCE: AASHTO T314 Direct Tension Failure Strain (%) SAMPLES: Performance Graded Binder Samples 195 and 196 Sample Sample (Y-X) - 195, (X) 196, (Y) (Ymed-Xmed) Count = Number of Laboratories 60 60 60 Median 1.355 1.31 0.05 0.875 Percentile 1.85 1.91625 0.315 0.125 Percentile 1.00625 0.9525 -0.2375 Range of Inner 75% = (87.5th Percentile Value) - (12.5th Percentile Value) 0.84375 0.96375 0.5525 (1.555) x (Range of Inner 75%) =Dist Beyond Inner 75% for 4.725 Std Dev 1.312031 1.498631 0.8591375 Invalid Upper Limit = (87.5 th Percentile) + [(1.555) x (Range of Inner 75%)] 3.162031 3.414881 1.1741375 Invalid Lower Limit = (12.5 th Percentile) - [(1.555) x (Range of Inner 75%)] -0.30578 -0.54613 -1.0966375 Table of Statistics and Limits Table 16 â Table of Statistics and Limits The data at the right is in descending order for Sample 195, (X). The laboratory numbers were assigned in ascending order to make them easier to locate in the column. The data for Sample 195 appears in the Column 2. Data for Sample 196 appears in the Column 3. The fourth column, labeled (X-Y) - (Ymed -Xmed), is the difference between the Sample 196 result and the Sample 195 result for each laboratory minus the difference between the median value for Sample 196 and the median value for Sample 195. The values in this fourth column provide an indication of the variation that can be expected between two test results determined by an individual laboratory. This column is ultimately used to estimate the repeatability. Column 2 in the table at the right, containing data for sample X, and the Table of Statistics and Limits above can be used to demonstrate how Invalid Data was determined. The 87.5th percentile was determined using a function available in Microsoft EXCEL software. The value corresponding to the 87.5th percentile is 1.85, as shown in the table above. Similarly, the value corresponding to the 12.5th percentile was determined to be 1.00625. The range of the Inner 75% of the data extends from the 87.5th percentile down to the 12.5th percentile, providing a range of 1.85 - 1.00625 = 0.84375. The limits for determining Invalid Data are located at 1.555 times the Range of the Inner 75% beyond the 87.5th percentile and below the 12.5th percentile. (For normally distributed data, these upper and lower limits are equivalent to 4.725 standard deviations from the center of the data. Since Invalid Data having extreme values can greatly affect the average value of the data, the median is used to estimate the center of the data rather than the average value.) In this case, (1.555) x (Range of the Inner 75%) = 1.555 x 0.84375 = 1.312031, as shown in the table above. The upper limit for determining Invalid Data is then equal to the value of the (87.5th percentile) + (1.312031) = (1.85) + (1.312031) = 3.162031. There are two data points for sample X having values greater than 3.162031. Those values for Sample 195 were reported for laboratories #1 and #2 and are shown as gray shaded in Column 2 of the table at the right. Laboratories #1 and #2 are then eliminated from any further analysis. The lower limit for determining Invalid Data is equal to the value of the (12.5th percentile) - (1.312031) = (1.00625) - (1.312031) = -0.30578. For Sample 195, there are no results reported below -0.305781, so no other data is determined to be invalid for Sample 195. Similarly using Column 3 in the table at the right and the table above, Invalid Data is determined for Sample 196. Any data above 3.414881 or below -0.54613 are considered to be invalid. Again the results for laboratories #1 and #2 are above the upper limit and are shown as gray shaded in the data at the right in Column 3. LAB Sample Sample (Y-X)- 195, (X) 196, (Y) (Ymed-Xmed) 1 4.89 5.28 0.39 2 3.82 3.82 0 3 2.57 2.41 -0.16 4 2.3 2.32 0.02 5 2.034 2.211 0.177 6 2 1.46 -0.54 7 1.97 2.24 0.27 8 1.85 1.91 0.06 9 1.85 1.78 -0.07 10 1.85 1.63 -0.22 11 1.84 1.81 -0.03 12 1.82 1.92 0.1 13 1.82 1.2 -0.62 14 1.77 1.67 -0.1 15 1.76 1.28 -0.48 16 1.67 1.59 -0.08 17 1.66 1.45 -0.21 18 1.63 2.06 0.43 19 1.62 1.91 0.29 20 1.62 1.19 -0.43 21 1.55 1.26 -0.29 22 1.54 1.79 0.25 23 1.54 1.39 -0.15 24 1.53 1.48 -0.05 25 1.53 0.72 -0.81 26 1.44 1.29 -0.15 27 1.428 1.517 0.089 28 1.42 1.71 0.29 29 1.39 1.12 -0.27 30 1.36 1.38 0.02 31 1.35 0.93 -0.42 32 1.31 1.36 0.05 33 1.28 1.2 -0.08 34 1.24 1.23 -0.01 35 1.24 0.71 -0.53 36 1.23 1.29 0.06 37 1.22 1.26 0.04 38 1.21 1.48 0.27 39 1.19 1.26 0.07 40 1.18 1.33 0.15 41 1.18 1.21 0.03 42 1.18 1.04 -0.14 43 1.17 1.57 0.4 44 1.16 1.42 0.26 45 1.13 1.08 -0.05 46 1.13 1.04 -0.09 47 1.099 1.33 0.231 48 1.09 1.33 0.24 49 1.09 1.2 0.11 50 1.08 1.05 -0.03 51 1.07 1.24 0.17 52 1.05 0.91 -0.14 53 0.98 0.99 0.01 54 0.97 1.06 0.09 55 0.84 1.27 0.43 56 0.808 0.702 -0.106 57 0.69 0.77 0.08 58 0.63 0.58 -0.05 59 0.6 1 0.4 60 0.5 0.38 -0.12 DATA Table 17 â Example Data
B-2 The same criteria is applied to Column 4 of the table at the right, marked (Y-X) - (Ymed - Xmed). Any values above 1.1741375 or below -1.0966375 would be considered as Invalid Data. In this case, there are no values that are considered invalid. However, the results from laboratories #1 and #2 are shown as gray shaded and are not included in further analysis because the results for those two laboratories were invalid for Samples 195 and 196. Any laboratory having any invalid results in any of the columns at the right is totally removed from any further analysis of this data for reproducibility or repeatability. The diagram below identifies the data points for laboratories #1 and #2 that are eliminated from further analysis. Using a similar process, the data remaining after eliminating results for laboratories #1 and #2 are then analyzed for Outliers. Determination of Invalid Data Using the (1.555) x (Inner 75%) Rule Alternating Dot and Dashed Lines - Median Values Dashed Line - Diagonal Thru Center of Data Dotted Lines - Limits for Invalid Data -3 -2 -1 0 1 2 3 4 5 6 -3 -2 -1 0 1 2 3 4 5 6 Sample 195 Sa m pl e 19 6 Invalid Data: Lab #1 Invalid Data: Lab #2 Figure 6 â Determination of Invalid Data
B-3 Example for Determining Outliers TEST DATA: AASHTO T314 Direct Tension Failure Strain (%) SAMPLES: AMRL Performance Graded Binder Samples 195 and 196 Sample Sample (Y-X)- 195, (X) 196, (Y) (Ymed-Xmed) Count = Number of Laboratories 58 58 58 Median 1.33 1.29 0.04 0.875 Percentile 1.84875 1.8975 0.30875 0.125 Percentile 0.98875 0.9375 -0.2475 Range of Inner 75% = (87.5th Percentile Value) - (12.5th Percentile Value) 0.86 0.96 0.55625 (0.674) x (Range of Inner 75%) = Dist Beyond 75% for 2.7 Std Dev 0.57964 0.64704 0.3749125 Outlier Upper Limit = (87.5 th Percentile) + [(0.674) x (Range of Inner 75%)] 2.42839 2.54454 0.6836625 Outlier Lower Limit = (12.5 th Percentile) - [(0.674) x (Range of Inner 75%)] 0.40911 0.29046 -0.6224125 Table of Statistics and Limits Table 18 â Table of Statistics and Limits Laboratories #1 and #2, whose results were determined to be Invalid Data, have been eliminated from the data at the right. The data remaining is arranged in descending order for Sample 195 and will be analyzed for Outliers in a manner similar to that previously applied to determine Invalid Data. Once again, the data for Sample 195 appears in Column 2. Data for Sample 196 appears in Column 3. The fourth column, marked (Y-X)-(Ymed-Xmed), is the difference between the Sample 196 result and the Sample 195 result for each laboratory minus the difference between the median value for Sample 196 and the median value for Sample 195. (New median values were calculated after laboratories # 1 and # 2 were removed.) The values in this fourth column provide an indication of the variation that can be expected between two test results determined by an individual laboratory. This column will ultimately be used to determine an estimate of repeatability. Column 2 and the above Table of Statistics and Limits can be used to demonstrate how Outliers were determined. The 87.5th percentile, for the data remaining after the elimination of Invalid Data, was determined using a function available in Microsoft EXCEL software. The value corresponding to the 87.5th percentile is 1.84875, as shown in the table above. Similarly, the value corresponding to the 12.5th percentile was determined to be 0.98875. The range of the Inner 75% of the data extends from the 87.5th percentile down to the 12.5th percentile, providing a range of 1.84875 - 0.98875 = 0.86. The limits for determining Outliers are located at 0.674 times the Range of the Inner 75% beyond the 87.5th percentile and below the 12.5th percentile. (For normally distributed data, these limits are equivalent to 2.7 standard deviations from the center of the data. Since Outliers having extreme values can greatly affect the average value of the data, the median is used to estimate the center of the data rather than the average value.) In this case, (0.674) x (Range of the Inner 75%) = 0.674 x 0.86 = 0.57964, as shown in the table above. The upper limit for determining Invalid Data is then equal to the value of the (87.5th percentile) + (0.57964) = (1.84875) + (0.57964) = 2.42839. There is one point in Column 2 having a value greater than 2.42839. That value was reported by laboratory #3 and is shown as gray shaded at the top of Column 2. Laboratory #3 is then eliminated from any further analysis. The lower limit for determining Outliers is equal to the value of the (12.5th percentile) - (0.57964) = (0.98875) - (0.57964) = 0.40911. For Sample 195, there are no results reported below 0.40911, so no other point is determined to be an Outlier for Sample 195. LAB Sample Sample (Y-X)- 195, (X) 196, (Y) (Ymed-Xmed) 3 2.57 2.41 -0.12 4 2.3 2.32 0.06 5 2.034 2.211 0.217 6 2 1.46 -0.5 7 1.97 2.24 0.31 8 1.85 1.91 0.1 9 1.85 1.78 -0.03 10 1.85 1.63 -0.18 11 1.84 1.81 0.01 12 1.82 1.92 0.14 13 1.82 1.2 -0.58 14 1.77 1.67 -0.06 15 1.76 1.28 -0.44 16 1.67 1.59 -0.04 17 1.66 1.45 -0.17 18 1.63 2.06 0.47 19 1.62 1.91 0.33 20 1.62 1.19 -0.39 21 1.55 1.26 -0.25 22 1.54 1.79 0.29 23 1.54 1.39 -0.11 24 1.53 1.48 -0.01 25 1.53 0.72 -0.77 26 1.44 1.29 -0.11 27 1.428 1.517 0.129 28 1.42 1.71 0.33 29 1.39 1.12 -0.23 30 1.36 1.38 0.06 31 1.35 0.93 -0.38 32 1.31 1.36 0.09 33 1.28 1.2 -0.04 34 1.24 1.23 0.03 35 1.24 0.71 -0.49 36 1.23 1.29 0.1 37 1.22 1.26 0.08 38 1.21 1.48 0.31 39 1.19 1.26 0.11 40 1.18 1.33 0.19 41 1.18 1.21 0.07 42 1.18 1.04 -0.1 43 1.17 1.57 0.44 44 1.16 1.42 0.3 45 1.13 1.08 -0.01 46 1.13 1.04 -0.05 47 1.099 1.33 0.271 48 1.09 1.33 0.28 49 1.09 1.2 0.15 50 1.08 1.05 0.01 51 1.07 1.24 0.21 52 1.05 0.91 -0.1 53 0.98 0.99 0.05 54 0.97 1.06 0.13 55 0.84 1.27 0.47 56 0.808 0.702 -0.066 57 0.69 0.77 0.12 58 0.63 0.58 -0.01 59 0.6 1 0.44 60 0.5 0.38 -0.08 DATA Table 19 â Example Data
B-4 Similarly using Table 17 and Column 3 of Table 18, Outliers are determined for Sample 196. Any point above 2.54454 or below 0.29046 would be considered to be an Outlier. There are no points that exceed the Outlier limits for Sample 196, however, laboratory #3 appears as gray shaded in Column 3 of Table 18 since laboratory #3 was previously eliminated based on results for Sample 195. From Table 17, the upper and lower Outlier limits for the fourth column of Table 18, marked (Y-X)-(Ymed-Xmed), are 0.6836625 and -0.6224125, respectively. In the fourth column, the value for laboratory #25, -0.77, is beyond the lower Outlier limit. Therefore, -0.77 is considered to be an Outlier and laboratory #25 is eliminated from any further analysis. The results for laboratory #25 are shown as gray shaded in Table 18. The diagram below identifies the points that were eliminated as Outliers. The core data points remaining after eliminating results from laboratories #1, #2, #3, and #25 (i.e. those points contained in the hexagon) were used in the final analysis to estimate repeatability and reproducibility. Determination of Outliers Using the (0.674) x Inner 75% Rule Alternating Dot and Dashed Lines - Median Values Dashed Line - Diagonal Thru Center of Data Dotted Lines - Outlier Limits -0.5 0.5 1.5 2.5 3.5 -0.5 0.5 1.5 2.5 3.5 Sample 195 Sa m pl e 19 6 Core data used for the final analysis are contained in the hexagon. Outlier: Lab #3 Outlier: Lab #25 Figure 7 â Determination of Outliers