Ch.4: Probability and Statistics Variations due to: Measurement System: Resolution and Repeatability Meas. Procedure: Repeatability Measured Variable: Temporal & Spatial Var.

Statistical Measurement Theory Sample - a set of measured data Measurand - measured variable (True) mean value: (x’) xmean

Mean Value and Uncertainty x’= xmean ± ux @ P% xmean is a P% probable estimate of x’ with uncertainty ux

Probability-Density Function Less dense More dense Range

Histogram-Frequency distribution K=7 intervals nj=7>5 1 3 4 2

Mean value and Variance

Probability-density function p(x) and Probability P% Infinite Statistics Probability-density function p(x) and Probability P% p(x)=dP/dx b=(x-x’)/s = dim’less deviation For x=x’, b=0

Normal-Gaussian distribution b=(x-x’)/s 99.73% 95.45% 68.27%

Normal-Gaussian distribution ½P(z1=1.02)=? Z1=1.02 MathCAD file ½P(z1=1.02)=34.61% Also, z1( ½P=0.3461) =1.02

t Finite Statistics Student-t distribution t(n=9,P=50%)=? n=N-1 MathCAD file Also, P(n=9, t =0.703)=50% and n(P =50%, t =0.703)=9 n, P, t are related

Standard Deviation of the Means

Pooled Statistics M replicates of N repeated measurements

Least-Square Regression y x ,... 2 , 1 }, { : points data Given = a x f y m j i c ) ... , ,... ( : function) choice (our Arbitrary 1 = found be to coefficients are where y : minimum be should squared deviations of sum The y (y d D i c min ) 2 , ® - = å n ,... 1 x i y c , d (y ) - = x

Least-Square Regression (2) Click for Polynomial Curve-Fit Click for Arbitrary Curve-Fit

Correlation Coefficient If Sxy=Sy and Sxy=0, respectively For the simplest, zeroth order polynomial fit. Click for Polynomial Curve-Fit Click for Arbitrary Curve-Fit

Data Outlier Usually zOL= 3 or zOL= zOL(Pout= 0.5-Pin=0.1/N) if number of data N is large. (For Pout=1%, zOL=2.33) Keep data if within ± zOL otherwise REJECT DATA as Outliers %Pin (zOL) %Pout(zOL) blimit= zOL = zOL(%Pin or %Pout)

Required #of Measurements