Statistics and Measurement. (Section 2.2 of Vardeman and Jobe plus some)

Transcription

1 Statistics and Measurement (Section 2.2 of Vardeman and Jobe plus some) 1

2 A Measurement Sstem is One s Looking Glass Process Measurement Sstem 2

3 Basic Issues in Metrolog Validit (Am I reall tracking what I want to track?) Precision (Consistenc of measurement) Accurac (Getting the right answer on average) 3

4 Basic Measurement Model (that allows for the realit of error ) For a single item: where: x and ε = = = the "true" value = x+ ε the measured value the (random) measurement error (a r.v.) with mean β and std. dev. σ measurement 4

5 Measurement Model σ measurement x β x + β 5

6 Measurement Model Multiple items = x+ ε where now x is random with mean and standard deviation σ x µ x (this measures real process or item-to-item variation) Now µ = µ + β x and = > x measurement x σ σ σ σ 6

7 Estimating Measurement Variation For a sample of m measurements on the same item producing and s the mean of is x+ β the mean of s is σ the interval estimates measurement ( m 1) s ( m 1) s, χ χ m 1,upper m 1,lower σ measurement 7

8 Estimating Part/Process Variation For a sample of measurements on n different items producing and s the mean of is µ x + β the mean of s is σ + σ (not σ ) the interval estimates x measurement x ( n 1) s ( n 1) s χ, n 1,upper χn 1,lower σ = σ + σ x measurement 8

9 Estimating Part/Process Variation An estimate of unit-to-unit variation free of measurement variation is (for m measurements on a single unit and n on different units) is ( max 0, ) x s s σ = and this estimate has standard error s s + 2 s s n 1 m 1 9

10 Example n=30 different widget diameters with = inch and s =.05 inch m=10 measurements made on the same widget diameter produce =.997 inch and s =.008 inch Estimate part-to-part standard deviation as ( ) σ x = max 0,(.05) (.008) =.049 inch with standard error (.05) (.008) 2 (.05) (.008) =.007 inch 10

11 Improving Accurac Through Calibration and Curve Fitting Calibration experiments get true /goldstandard-measurement values x and local measurements and seek a conversion method The relevant statistical methodolog is curve-fitting/regression analsis Regression analsis can provide both point conversions and measures of uncertaint (the latter through inversion of prediction limits ) 11

12 Simple Linear Regression SLR Model is = β + β x+ ε 0 1 Pictoriall: 12

13 Mandel (NBS/NIST) Example Gold-standard and local measurements on n = 14 specimens (units not given) x x

14 Mandel Example 95% Prediction Limits (x=1653) 14

15 Evaluating Measurement Precision in an R&R Stud Often there are operator effects that should be considered part of measurement imprecision Repeatabilit variation is variation characteristic of one operator remeasuring one part Reproducibilit variation is variation characteristic of man operators measuring a single part once (exclusive of repeatabilit variation) 15

16 Tpical Gauge R&R Data Laout E.g. I=2 parts, J=3 operators and m=2 repeats per cell Part Operator

17 Estimates of R&R Sigmas Repeatabilit variation is within cells σ repeatabilit = d 2 R ( m) (based on ranges) or σ repeatabilit = MSE (based on ANOVA) Describing reproducibilit variation requires more subtlet (common rangebased methods are wrong) 17

18 Estimates of Reproducibilit Sigma (Using Ranges) One must correct/allow for repeatabilit variation in considering differences between operators (as earlier in σ x ) i Let be the range of the part i cell means Further, let be the mean of these ranges σ reproducibilit 1 R = max 0, d2( J) m d2( m) ANOVA-based estimator is on page 27 of V&J 18

19 BTW Standard Errors For estimates of repeatabilit sigma: d 3( m) σ repeatabilit based on ranges d ( m) IJ and 2 σ repeatabilit based on ANOVA 2 IJ( m 1) For ANOVA estimate of reproducibilit sigma use Chiang s program (take the Stat531 link from Vardeman s Web page) usuall these are HUGE 19

20 I=3, J=3 and m=2 Example 20

21 Workshop Exercises See problem 5.8, page 249 and the n=8 measured hardnesses there. Suppose that previous measurement of a single part m=5 times gave s=.044 mm. Find σ x and a standard error for this estimate. For the Mandel example, about what corrected value would ou associate with a local measurement of 1000? What 95% limits would ou associate with this? 21