Ch.4: Probability and Statistics

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 _at_ Pxmean is a P probable
  estimate of xwith uncertainty ux

4
Probability-Density Function
Less dense
More dense
5
Histogram-Frequency distribution
K7 intervals
nj7gt5
6
Mean value and Variance
7
Infinite Statistics

• Probability-density function p(x) and Probability
  P

p(x)dP/dx
b(x-x)/s dimless deviation
For xx, b0
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Normal-Gaussian distribution
b(x-x)/s
9
Normal-Gaussian distribution
½P(z11.02)?
MathCAD file
Also, z1( ½P0.3461) 1.02
10
Finite Statistics

• Student-t distribution

t(n9,P50)?
MathCAD file
Also, P(n9, 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 StatisticsM replicates of N repeated
measurements
13
Least-Square Regression
y
x
14
Least-Square Regression (2)
Click for Polynomial Curve-Fit
Click for Arbitrary Curve-Fit
15
Correlation Coefficient
If SxySy and Sxy0, 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-Pin0.1/N) if number of data N is large. (For
Pout1, zOL2.33) Keep data if within
zOL otherwise REJECT DATA as Outliers
Pin (zOL)
Pout(zOL)
blimit zOL zOL(Pin or Pout)
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Required of Measurements