Measurement Process Characterization_8 docx

Block sizes Angle blocks normally come in sets of 1, 3, 5, 20, and 30 seconds 1, 3, 5, 20, 30 minutes 1, 3, 5, 15, 30, 45 degrees and blocks of the same nominal size from 4, 5 or 6 different sets can be calibrated simultaneously using one of the designs shown in this catalog. Design for 4 angle blocks● Design for 5 angle blocks● Design for 6 angle blocks● Restraint The solution to the calibration design depends on the known value of a reference block, which is compared with the test blocks. The reference block is designated as block 1 for the purpose of this discussion. Check standard It is suggested that block 2 be reserved for a check standard that is maintained in the laboratory for quality control purposes. Calibration scheme A calibration scheme developed by Charles Reeve (Reeve) at the National Institute of Standards and Technology for calibrating customer angle blocks is explained on this page. The reader is encouraged to obtain a copy of the publication for details on the calibration setup and quality control checks for angle block calibrations. Series of measurements for calibrating 4, 5, and 6 angle blocks simultaneously For all of the designs, the measurements are made in groups of seven starting with the measurements of blocks in the following order: 2-3-2-1-2-4-2. Schematically, the calibration design is completed by counter-clockwise rotation of the test blocks about the reference block, one-at-a-time, with 7 readings for each series reduced to 3 difference measurements. For n angle blocks (including the reference block), this amounts to n - 1 series of 7 readings. The series for 4, 5, and 6 angle blocks are shown below. Measurements for 4 angle blocks Series 1: 2-3-2-1-2-4-2 Series 2: 4-2-4-1-4-3-4 Series 3: 3-4-3-1-3-2-3 2.3.4.5. Designs for angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc345.htm (2 of 6) [5/1/2006 10:12:18 AM] Measurements for 5 angle blocks (see diagram) Series 1: 2-3-2-1-2-4-2 Series 2: 5-2-5-1-5-3-5 Series 3: 4-5-4-1-4-2-4 Series 4: 3-4-3-1-3-5-3 Measurements for 6 angle blocks Series 1: 2-3-2-1-2-4-2 Series 2: 6-2-6-1-6-3-6 Series 3: 5-6-5-1-5-2-5 Series 4: 4-5-4-1-4-6-4 Series 5: 3-4-3-1-3-5-3 Equations for the measurements in the first series showing error sources The equations explaining the seven measurements for the first series in terms of the errors in the measurement system are: Z 11 = B + X 1 + error 11 Z 12 = B + X 2 + d + error 12 Z 13 = B + X 3 + 2d + error 13 Z 14 = B + X 4 + 3d + error 14 Z 15 = B + X 5 + 4d + error 15 Z 16 = B + X 6 + 5d + error 16 Z 17 = B + X 7 + 6d + error 17 with B a bias associated with the instrument, d is a linear drift factor, X is the value of the angle block to be determined; and the error terms relate to random errors of measurement. 2.3.4.5. Designs for angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc345.htm (3 of 6) [5/1/2006 10:12:18 AM] Calibration procedure depends on difference measurements The check block, C, is measured before and after each test block, and the difference measurements (which are not the same as the difference measurements for calibrations of mass weights, gage blocks, etc.) are constructed to take advantage of this situation. Thus, the 7 readings are reduced to 3 difference measurements for the first series as follows: For all series, there are 3(n - 1) difference measurements, with the first subscript in the equations above referring to the series number. The difference measurements are free of drift and instrument bias. Design matrix As an example, the design matrix for n = 4 angle blocks is shown below. 1 1 1 1 0 1 -1 0 -1 1 0 0 0 1 0 -1 0 -1 0 1 -1 0 0 1 0 0 -1 1 0 0 1 -1 -1 0 1 0 0 -1 1 0 The design matrix is shown with the solution matrix for identification purposes only because the least-squares solution is weighted (Reeve) to account for the fact that test blocks are measured twice as many times as the reference block. The weight matrix is not shown. 2.3.4.5. Designs for angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc345.htm (4 of 6) [5/1/2006 10:12:18 AM] Solutions to the calibration designs measurements Solutions to the angle block designs are shown on the following pages. The solution matrix and factors for the repeatability standard deviation are to be interpreted as explained in solutions to calibration designs . As an example, the solution for the design for n=4 angle blocks is as follows: The solution for the reference standard is shown under the first column of the solution matrix; for the check standard under the second column; for the first test block under the third column; and for the second test block under the fourth column. Notice that the estimate for the reference block is guaranteed to be R*, regardless of the measurement results, because of the restraint that is imposed on the design. Specifically, Solutions are correct only for the restraint as shown. Calibrations can be run for top and bottom faces of blocks The calibration series is run with the blocks all face "up" and is then repeated with the blocks all face "down", and the results averaged. The difference between the two series can be large compared to the repeatability standard deviation, in which case a between-series component of variability must be included in the calculation of the standard deviation of the reported average. 2.3.4.5. Designs for angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc345.htm (5 of 6) [5/1/2006 10:12:18 AM] Calculation of standard deviations when the blocks are measured in two orientations For n blocks, the differences between the values for the blocks measured in the top ( denoted by "t") and bottom (denoted by "b") positions are denoted by: The standard deviation of the average (for each block) is calculated from these differences to be: Standard deviations when the blocks are measured in only one orientation If the blocks are measured in only one orientation, there is no way to estimate the between-series component of variability and the standard deviation for the value of each block is computed as s test = K 1 s 1 where K 1 is shown under "Factors for computing repeatability standard deviations" for each design and is the repeatability standard deviation as estimated from the design. Because this standard deviation may seriously underestimate the uncertainty, a better approach is to estimate the standard deviation from the data on the check standard over time. An expanded uncertainty is computed according to the ISO guidelines. 2.3.4.5. Designs for angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc345.htm (6 of 6) [5/1/2006 10:12:18 AM] 2. Measurement Process Characterization 2.3. Calibration 2.3.4. Catalog of calibration designs 2.3.4.5. Designs for angle blocks 2.3.4.5.1.Design for 4 angle blocks DESIGN MATRIX 1 1 1 1 Y(1) 0 1 -1 0 Y(2) -1 1 0 0 Y(3) 0 1 0 -1 Y(4) 0 -1 0 1 Y(5) -1 0 0 1 Y(6) 0 0 -1 1 Y(7) 0 0 1 -1 Y(8) -1 0 1 0 Y(9) 0 -1 1 0 REFERENCE + CHECK STANDARD + DEGREES OF FREEDOM = 6 SOLUTION MATRIX DIVISOR = 24 OBSERVATIONS 1 1 1 1 Y(11) 0 2.2723000 -5.0516438 -1.2206578 Y(12) 0 9.3521166 7.3239479 7.3239479 Y(13) 0 2.2723000 -1.2206578 -5.0516438 2.3.4.5.1. Design for 4 angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3451.htm (1 of 2) [5/1/2006 10:12:18 AM] Y(21) 0 -5.0516438 -1.2206578 2.2723000 Y(22) 0 7.3239479 7.3239479 9.3521166 Y(23) 0 -1.2206578 -5.0516438 2.2723000 Y(31) 0 -1.2206578 2.2723000 -5.0516438 Y(32) 0 7.3239479 9.3521166 7.3239479 Y(33) 0 -5.0516438 2.2723000 -1.2206578 R* 1 1. 1. 1. R* = VALUE OF REFERENCE ANGLE BLOCK FACTORS FOR REPEATABILITY STANDARD DEVIATIONS SIZE K1 1 1 1 1 1 0.0000 + 1 0.9749 + 1 0.9749 + 1 0.9749 + 1 0.9749 + Explanation of notation and interpretation of tables 2.3.4.5.1. Design for 4 angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3451.htm (2 of 2) [5/1/2006 10:12:18 AM] 2. Measurement Process Characterization 2.3. Calibration 2.3.4. Catalog of calibration designs 2.3.4.5. Designs for angle blocks 2.3.4.5.2.Design for 5 angle blocks DESIGN MATRIX 1 1 1 1 1 0 1 -1 0 0 -1 1 0 0 0 0 1 0 -1 0 0 -1 0 0 1 -1 0 0 0 1 0 0 -1 0 1 0 0 0 1 -1 -1 0 0 1 0 0 -1 0 1 0 0 0 1 -1 0 -1 0 1 0 0 0 0 1 0 -1 REFERENCE + CHECK STANDARD + DEGREES OF FREEDOM = 8 SOLUTION MATRIX DIVISOR = 24 OBSERVATIONS 1 1 1 1 1 Y(11) 0.00000 3.26463 -5.48893 -0.21200 -1.56370 Y(12) 0.00000 7.95672 5.38908 5.93802 4.71618 2.3.4.5.2. Design for 5 angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3452.htm (1 of 2) [5/1/2006 10:12:19 AM] Y(13) 0.00000 2.48697 -0.89818 -4.80276 -0.78603 Y(21) 0.00000 -5.48893 -0.21200 -1.56370 3.26463 Y(22) 0.00000 5.38908 5.93802 4.71618 7.95672 Y(23) 0.00000 -0.89818 -4.80276 -0.78603 2.48697 Y(31) 0.00000 -0.21200 -1.56370 3.26463 -5.48893 Y(32) 0.00000 5.93802 4.71618 7.95672 5.38908 Y(33) 0.00000 -4.80276 -0.78603 2.48697 -0.89818 Y(41) 0.00000 -1.56370 3.26463 -5.48893 -0.21200 Y(42) 0.00000 4.71618 7.95672 5.38908 5.93802 Y(43) 0.00000 -0.78603 2.48697 -0.89818 -4.80276 R* 1. 1. 1. 1. 1. R* = VALUE OF REFERENCE ANGLE BLOCK FACTORS FOR REPEATABILITY STANDARD DEVIATIONS SIZE K1 1 1 1 1 1 1 0.0000 + 1 0.7465 + 1 0.7465 + 1 0.7456 + 1 0.7456 + 1 0.7465 + Explanation of notation and interpretation of tables 2.3.4.5.2. Design for 5 angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3452.htm (2 of 2) [5/1/2006 10:12:19 AM] 2. Measurement Process Characterization 2.3. Calibration 2.3.4. Catalog of calibration designs 2.3.4.5. Designs for angle blocks 2.3.4.5.3.Design for 6 angle blocks DESIGN MATRIX 1 1 1 1 1 1 0 1 -1 0 0 0 -1 1 0 0 0 0 0 1 0 -1 0 0 0 -1 0 0 0 1 -1 0 0 0 0 1 0 0 -1 0 0 1 0 0 0 0 1 -1 -1 0 0 0 1 0 0 -1 0 0 1 0 0 0 0 1 -1 0 -1 0 0 1 0 0 0 0 0 1 0 -1 0 0 1 -1 0 0 -1 0 1 0 0 0 0 0 1 0 -1 0 REFERENCE + CHECK STANDARD + DEGREES OF FREEDOM = 10 SOLUTION MATRIX DIVISOR = 24 OBSERVATIONS 1 1 1 1 1 2.3.4.5.3. Design for 6 angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3453.htm (1 of 3) [5/1/2006 10:12:19 AM] [...]... of artifact calibration 2 Measurement Process Characterization 2.3 Calibration 2.3.5 Control of artifact calibration Purpose The purpose of statistical control in the calibration process is to guarantee the 'goodness' of calibration results within predictable limits and to validate the statement of uncertainty of the result Two types of control can be imposed on a calibration process that makes use of... regular intervals, the check standard values be plotted against time to check for drift or anomalies in the measurement process http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc352.htm (3 of 3) [5/1/2006 10:12:22 AM] 2.3.5.2.1 Example of Shewhart control chart for mass calibrations 2 Measurement Process Characterization 2.3 Calibration 2.3.5 Control of artifact calibration 2.3.5.2 Control of bias... of tables http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3453.htm (3 of 3) [5/1/2006 10:12:19 AM] 2.3.4.6 Thermometers in a bath 2 Measurement Process Characterization 2.3 Calibration 2.3.4 Catalog of calibration designs 2.3.4.6 Thermometers in a bath Measurement sequence Calibration of liquid in glass thermometers is usually carried out in a controlled bath where the temperature in the bath... instrument precision Because the measurements for a single design are completed in a short time span, this standard deviation estimates the basic precision of the instrument Designs should be chosen to have enough measurements so that the standard deviation from the design has at least 3 degrees of freedom where the degrees of freedom are (n - m + 1) with q n = number of difference measurements q m = number... chart for precision 2 Measurement Process Characterization 2.3 Calibration 2.3.5 Control of artifact calibration 2.3.5.1 Control of precision 2.3.5.1.1 Example of control chart for precision Example of a control chart for precision of a mass balance Mass calibrations usually start with the comparison of kilograms standards using a high precision balance as a comparator Many of the measurements at the... procedure that is sensitive to small changes in the process is discussed on another page For a Shewhart control procedure, the average and standard deviation of historical check standard values are the parameters of interest The check standard values are denoted by The baseline is the process average which is computed from the check standard values as The process standard deviation is with (K - 1) degrees... the control limits, the process is judged to be out of control and the current calibration run is rejected The best strategy in this situation is to repeat the calibration to see if the failure was a chance occurrence Check standard values that remain in control, especially over a period of time, provide confidence that no new biases have been introduced into the measurement process and that the long-term... in the following time sequence: where R1, R2, R3 represent the measurements on the standard resistance thermometer and T1, T2, , TK and T'1, T'2, , T'K represent the pair of measurements on the K test thermometers Assumptions regarding temperature The assumptions for the analysis are that: q Equal time intervals are maintained between measurements on the test items q Temperature increases by with... distribution of data points on the two sides of the control chart -indicating a change in either: q process average which may be related to a change in the reference standards or q variability which may be caused by a change in the instrument precision or may be the result of other factors on the measurement process Small changes only become obvious over time Unfortunately, it takes time for the patterns... subset t < 85 and plot y cc2 ul2 ll2 vs t subset t > 85 Revised control chart based on check standard measurements after 1985 http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3521.htm (3 of 3) [5/1/2006 10:12:22 AM] 2.3.5.2.2 Example of EWMA control chart for mass calibrations 2 Measurement Process Characterization 2.3 Calibration 2.3.5 Control of artifact calibration 2.3.5.2 Control of bias . 0.00000 -4 .80 276 -0. 786 03 2. 486 97 -0 .89 8 18 Y(41) 0.00000 -1.56370 3.26463 -5. 488 93 -0.21200 Y(42) 0.00000 4.716 18 7.95672 5. 389 08 5.9 380 2 Y(43) 0.00000 -0. 786 03 2. 486 97 -0 .89 8 18 -4 .80 276 R*. 0.00000 5. 389 08 5.9 380 2 4.716 18 7.95672 Y(23) 0.00000 -0 .89 8 18 -4 .80 276 -0. 786 03 2. 486 97 Y(31) 0.00000 -0.21200 -1.56370 3.26463 -5. 488 93 Y(32) 0.00000 5.9 380 2 4.716 18 7.95672 5. 389 08 Y(33) 0.00000. blocks http://www.itl.nist.gov/div8 98/ handbook/mpc/section3/mpc3452.htm (1 of 2) [5/1/2006 10:12:19 AM] Y(13) 0.00000 2. 486 97 -0 .89 8 18 -4 .80 276 -0. 786 03 Y(21) 0.00000 -5. 488 93 -0.21200 -1.56370 3.26463 Y(22) 0.00000 5. 389 08