Engineering Statistics Handbook Episode 3 Part 13 pdf

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] 1 Y(11) 0.0000 3.2929 -5.2312 -0.7507 -0.6445 -0.6666 Y(12) 0.0000 6.9974 4.6324 4.6495 3.8668 3.8540 Y(13) 0.0000 3.2687 -0.7721 -5.2098 -0.6202 -0.6666 Y(21) 0.0000 -5.2312 -0.7507 -0.6445 -0.6666 3.2929 Y(22) 0.0000 4.6324 4.6495 3.8668 3.8540 6.9974 Y(23) 0.0000 -0.7721 -5.2098 -0.6202 -0.6666 3.2687 Y(31) 0.0000 -0.7507 -0.6445 -0.6666 3.2929 -5.2312 Y(32) 0.0000 4.6495 3.8668 3.8540 6.9974 4.6324 Y(33) 0.0000 -5.2098 -0.6202 -0.6666 3.2687 -0.7721 Y(41) 0.0000 -0.6445 -0.6666 3.2929 -5.2312 -0.7507 Y(42) 0.0000 3.8668 3.8540 6.9974 4.6324 4.6495 Y(43) 0.0000 -0.6202 -0.6666 3.2687 -0.7721 -5.2098 Y(51) 0.0000 -0.6666 3.2929 -5.2312 -0.7507 -0.6445 Y(52) 0.0000 3.8540 6.9974 4.6324 4.6495 3.8668 Y(53) 0.0000 -0.6666 3.2687 -0.7721 -5.2098 -0.6202 R* 1. 1. 1. 1. 1. 1. R* = VALUE OF REFERENCE ANGLE BLOCK FACTORS FOR REPEATABILITY STANDARD DEVIATIONS SIZE K1 1 1 1 1 1 1 1 0.0000 + 1 0.7111 + 1 0.7111 + 1 0.7111 + 1 0.7111 + 2.3.4.5.3. Design for 6 angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3453.htm (2 of 3) [5/1/2006 10:12:19 AM] 1 0.7111 + 1 0.7111 + Explanation of notation and interpretation of tables 2.3.4.5.3. Design for 6 angle blocks http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3453.htm (3 of 3) [5/1/2006 10:12:19 AM] Estimates of drift The estimates of the shift due to the resistance thermometer and temperature drift are given by: Standard deviations The residual variance is given by . The standard deviation of the indication assigned to the ith test thermometer is and the standard deviation for the estimates of shift and drift are respectively. 2.3.4.6. Thermometers in a bath http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc346.htm (2 of 2) [5/1/2006 10:12:20 AM] 2. Measurement Process Characterization 2.3. Calibration 2.3.4. Catalog of calibration designs 2.3.4.7. Humidity standards 2.3.4.7.1.Drift-elimination design for 2 reference weights and 3 cylinders OBSERVATIONS 1 1 1 1 1 Y(1) + - Y(2) + - Y(3) + - Y(4) + - Y(5) - + Y(6) - + Y(7) + - Y(8) + - Y(9) - + Y(10) + - RESTRAINT + + CHECK STANDARD + - DEGREES OF FREEDOM = 6 SOLUTION MATRIX DIVISOR = 10 OBSERVATIONS 1 1 1 1 1 2.3.4.7.1. Drift-elimination design for 2 reference weights and 3 cylinders http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3471.htm (1 of 2) [5/1/2006 10:12:20 AM] Y(1) 2 -2 0 0 0 Y(2) 0 0 0 2 -2 Y(3) 0 0 2 -2 0 Y(4) -1 1 -3 -1 -1 Y(5) -1 1 1 1 3 Y(6) -1 1 1 3 1 Y(7) 0 0 2 0 -2 Y(8) -1 1 -1 -3 -1 Y(9) 1 -1 1 1 3 Y(10) 1 -1 -3 -1 -1 R* 5 5 5 5 5 R* = average value of the two reference weights FACTORS FOR REPEATABILITY STANDARD DEVIATIONS WT K1 1 1 1 1 1 1 0.5477 + 1 0.5477 + 1 0.5477 + 2 0.8944 + + 3 1.2247 + + + 0 0.6325 + - Explanation of notation and interpretation of tables 2.3.4.7.1. Drift-elimination design for 2 reference weights and 3 cylinders http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3471.htm (2 of 2) [5/1/2006 10:12:20 AM] standards at all nominal lengths. A check standard can also be a mathematical construction, such as the computed difference between the calibrated values of two reference standards in a design. Database of check standard values The creation and maintenance of the database of check standard values is an important aspect of the control process. The results from each calibration run are recorded in the database. The best way to record this information is in one file with one line (row in a spreadsheet) of information in fixed fields for each calibration run. A list of typical entries follows: Date1. Identification for check standard2. Identification for the calibration design3. Identification for the instrument4. Check standard value5. Repeatability standard deviation from design6. Degrees of freedom7. Operator identification8. Flag for out-of-control signal9. Environmental readings (if pertinent)10. 2.3.5. Control of artifact calibration http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc35.htm (2 of 2) [5/1/2006 10:12:20 AM] Control procedure is invoked in real-time for each calibration run The control procedure compares each new repeatability standard deviation that is recorded for the instrument with an upper control limit, UCL. Usually, only the upper control limit is of interest because we are primarily interested in detecting degradation in the instrument's precision. A possible complication is that the control limit is dependent on the degrees of freedom in the new standard deviation and is computed as follows: . The quantity under the radical is the upper percentage point from the F table where is chosen small to be, say, 05. The other two terms refer to the degrees of freedom in the new standard deviation and the degrees of freedom in the process standard deviation. Limitation of graphical method The graphical method of plotting every new estimate of repeatability on a control chart does not work well when the UCL can change with each calibration design, depending on the degrees of freedom. The algebraic equivalent is to test if the new standard deviation exceeds its control limit, in which case the short-term precision is judged to be out of control and the current calibration run is rejected. For more guidance, see Remedies and strategies for dealing with out-of-control signals. As long as the repeatability standard deviations are in control, there is reason for confidence that the precision of the instrument has not degraded. Case study: Mass balance precision It is recommended that the repeatability standard deviations be plotted against time on a regular basis to check for gradual degradation in the instrument. Individual failures may not trigger a suspicion that the instrument is in need of adjustment or tuning. 2.3.5.1. Control of precision http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc351.htm (2 of 2) [5/1/2006 10:12:21 AM] let f=sqrt(f) let sul=f*scc plot s scc sul vs t Control chart for precision TIME IN YEARS Interpretation of the control chart The control chart shows that the precision of the balance remained in control through 1990 with only two violations of the control limits. For those occasions, the calibrations were discarded and repeated. Clearly, for the second violation, something significant occurred that invalidated the calibration results. Further interpretation of the control chart However, it is also clear from the pattern of standard deviations over time that the precision of the balance was gradually degrading and more and more points were approaching the control limits. This finding led to a decision to replace this balance for high accuracy calibrations. 2.3.5.1.1. Example of control chart for precision http://www.itl.nist.gov/div898/handbook/mpc/section3/mpc3511.htm (2 of 2) [5/1/2006 10:12:21 AM] [...]... subset t > 85 let n = size y let cc2=ybar2 for i = 1 1 n let ul2=cc2 +3* sd2 let ll2=cc2 -3* sd2 plot y cc ul ll vs t 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 Example... 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 let ll=cc -3* sd characters * blank blank blank * blank blank blank lines blank solid dotted dotted blank solid dotted dotted plot y cc ul ll... dimension 500 30 skip 4 read mass.dat x id y bal s ds let n = number y let cutoff = 85.0 let tag = 2 for i = 1 1 n let tag = 1 subset x < cutoff xlimits 75 90 let m = mean y subset tag 1 let s = sd y subset tag 1 let lambda = 2 let fudge = sqrt(lambda/(2-lambda)) let mean = m for i = 1 1 n http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc3522.htm (2 of 3) [5/1/2006 10:12:22 AM] 2 .3. 5.2.2 Example... value can be compared algebraically with the control limits The calibration run is judged to be out-of-control if either: C > UCL or C < LCL http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc352.htm (2 of 3) [5/1/2006 10:12:22 AM] 2 .3. 5.2 Control of bias and long-term variability Actions to be taken If the check standard value exceeds one of the control limits, the process is judged to be out of control... level of change might seem insignificant, but the calculation of uncertainties for the calibration process depends on the control limits http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc3521.htm (2 of 3) [5/1/2006 10:12:22 AM] 2 .3. 5.2.1 Example of Shewhart control chart for mass calibrations Re-establishing the limits based on recent data and EWMA option If the limits for the control chart are... pattern emerges because the process average has actually shifted about one standard deviation, and the EWMA control chart is sensitive to small changes http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc3522.htm (3 of 3) [5/1/2006 10:12:22 AM] ... and the EWMA statistics are shown as black dots superimposed on the raw data The control limits are calculated according to the equation above where the process standard deviation, s = 0. 030 65 mg and k = 3 The EWMA statistics, and not the raw data, are of interest in looking for out-of-control signals Because the EWMA statistic is a weighted average, it has a smaller standard deviation than a single... multiple violations of the control limits for the EWMA statistics The target (average) and process standard deviation are computed from the check standard data taken prior to 1985 The computation of the EWMA statistic begins with the data taken at the start of 1985 In the control chart below, the control data after 1985 are shown in green, and the EWMA statistics are shown as black dots superimposed on... = 1 1 n http://www.itl.nist.gov/div898 /handbook/ mpc/section3/mpc3522.htm (2 of 3) [5/1/2006 10:12:22 AM] 2 .3. 5.2.2 Example of EWMA control chart for mass calibrations let upper = mean + 3* fudge*s let lower = mean - 3* fudge*s let nm1 = n-1 let start = 106 let pred2 = mean loop for i = start 1 nm1 let ip1 = i+1 let yi = y(i) let predi = pred2(i) let predip1 = lambda*yi + (1-lambda)*predi let pred2(ip1)... freedom let let let let alphau = 1 - 0.05/2 k = 6 v1 = k-1 t = tppf(alphau, v1) return the following value: THE COMPUTED VALUE OF THE CONSTANT T = 0.2570583E+01 Simplification for large degrees of freedom It is standard practice to use a value of 3 instead of a critical value from the t-table, given the process standard deviation has large degrees of freedom, say, v > 15 The control procedure is invoked . 4. 632 4 4.6495 3. 8668 3. 8540 Y( 13) 0.0000 3. 2687 -0.7721 -5.2098 -0.6202 -0.6666 Y(21) 0.0000 -5. 231 2 -0.7507 -0.6445 -0.6666 3. 2929 Y(22) 0.0000 4. 632 4 4.6495 3. 8668 3. 8540 6.9974 Y( 23) . -5.2098 -0.6202 -0.6666 3. 2687 Y (31 ) 0.0000 -0.7507 -0.6445 -0.6666 3. 2929 -5. 231 2 Y (32 ) 0.0000 4.6495 3. 8668 3. 8540 6.9974 4. 632 4 Y (33 ) 0.0000 -5.2098 -0.6202 -0.6666 3. 2687 -0.7721 Y(41). 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/mpc34 53. htm (1 of 3) [5/1/2006 10:12:19 AM] 1 Y(11) 0.0000 3. 2929 -5. 231 2 -0.7507 -0.6445