2. ©1997 by M. Kostic Ch.4 (Ch.6 in current Text): Probability and Statistics Variations due to: Measurement System: Resolution and RepeatabilityMeasurement System: Resolution and Repeatability Meas. Procedure: RepeatabilityMeas. Procedure: Repeatability Measured Variable: Temporal & Spatial Var.Measured Variable: Temporal & Spatial Var.

3. ©1997 by M. Kostic Statistical Measurement Theory Sample - a set of measured dataSample - a set of measured data Measurand - measured variableMeasurand - measured variable (True) mean value: (x’) x mean(True) mean value: (x’) x mean

4. ©1997 by M. Kostic Mean Value and Uncertainty x’= x mean ± u x @ P% x mean is a P% probable estimate of x’ with uncertainty u x

5. ©1997 by M. Kostic Probability-Density Function More dense Less dense Range

6. ©1997 by M. Kostic Histogram-Frequency distribution K=7 intervals n j =7>5 1 3 4 2

7. ©1997 by M. Kostic Mean value and Variance

8. ©1997 by M. Kostic Infinite Statistics Probability-density function p(x) and Probability P%  =(x-x’)/  dim’less deviationFor x=x’,  =0 p(x)=dP/dx

9. ©1997 by M. Kostic 68. 27% 95. 45% Normal-Gaussian distribution  =(x-x’)/  99. 73%

10. ©1997 by M. Kostic Normal-Gaussian distribution Z 1 =1.02 ½P( z 1 =1.02 )=34.61% ½P( z 1 =1.02 )= ? Also, z 1 ( ½P =0.3461 ) =1.02 MathCAD file

11. ©1997 by M. Kostic Finite Statistics Student-t distributionStudent-t distribution t( =9,P=50% ) =? Also, P( =9, t =0.703 )=50% and ( P =50%, t =0.703 )=9, P, t are related t =N-1 MathCAD file 50=P%

12. ©1997 by M. Kostic Standard Deviation of the Means

13. ©1997 by M. Kostic Standard Deviation of the Means (2) x i y x      N color i x xx N S i=1 2 __ 1 1

14. ©1997 by M. Kostic Pooled Statistics M replicates of N repeated measurements

15. ©1997 by M. Kostic niyx ii,...2,1},,{:pointsdataGiven  Least-Square Regression aaaaxfy y mjiic c )...,,...,,( :function)choice(our Arbitrary 10,  a j foundbetocoefficientsarewhere :minimumbe should squareddeviationsofsumThe y(ydD i ici i i min) 2, 2   ni,...2,1  x i y i y ic, d i y(y ici ),   y x

16. ©1997 by M. Kostic Least-Square Regression (2) Click for Polynomial Curve-Fit Click for Arbitrary Curve-Fit

17. ©1997 by M. Kostic Correlation Coefficient Click for Polynomial Curve-Fit Click for Arbitrary Curve-Fit If S xy =S y and S xy =0, respectively For the simplest, zeroth order polynomial fit. Coeff. of determination

18. ©1997 by M. Kostic Data Outlier  limit  z OL  z OL (%P in or %P out ) %P out (z OL ) %P in (z OL ) Usually z OL = 3 or z OL = z OL (P out = 0.5-P in =0.1/N) if number of data N is large. (For P out =1%, z OL =2.33) Keep data if within ± z OL otherwise REJECT DATA as Outliers

19. ©1997 by M. Kostic Required #of Measurements