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

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

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

4. Probability-Density Function
Less dense
More dense
Range

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

6. Mean value and Variance

7. 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

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

9. 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

10. 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

11. Standard Deviation of the Means

12. Pooled Statistics M replicates of N repeated measurements

13. Least-Square Regressiony 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

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

15. 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

16. 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)

17. Required #of Measurements