©1997 by M. Kostic
Ch.4: Probability and Statistics
Variations due to:
• Measurement System:
Resolution and Repeatability
• Meas. Procedure:
Repeatability
• Measured Variable:
Temporal & Spatial Var.
©1997 by M. Kostic
Statistical Measurement Theory
• Sample - a set of measured data
• Measurand - measured variable
• (True) mean value: (x’)
xmean
©1997 by M. Kostic
Mean Value and Uncertainty
x’= xmean ± ux @ P%
xmean is a P% probable estimate of x’
with uncertainty ux
Probability-Density Function
©1997 by M. Kostic
More dense
Range
Less dense
©1997 by M. Kostic
Histogram-Frequency distribution
K=7 intervals
nj=7>5
4
3
2
1
©1997 by M. Kostic
Mean value and Variance
©1997 by M. Kostic
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
©1997 by M. Kostic
Normal-Gaussian distribution
b=(x-x’)/s
68.27%
95.45%
99.73%
Normal-Gaussian distribution
©1997 by M. Kostic
½P(z1=1.02)=?
Z1=1.02
MathCAD
file
½P(z1=1.02)=34.61%
Also, z1( ½P=0.3461) =1.02
Finite Statistics
©1997 by M. Kostic
• Student-t distribution
50=P%
n=N-1
t
t(n=9,P=50%)=?
MathCAD
file
Also, P(n=9, t =0.703)=50%
and n(P =50%, t =0.703)=9
n, P, t are related
©1997 by M. Kostic
Standard Deviation of the Means
©1997 by M. Kostic
Pooled Statistics
M replicates of N repeated measurements
Pooled mean or average :
M
x =
1
MN
M
N
 x
j =1 i =1
or if N j  const x =
i, j
N
j =1
M
j
N
j =1
xj
; n j = N j 1
j
Pooled standard deviation :
M
Sx =
M
N
1
( xi , j  x j ) 2 =

M ( N  1) j =1 i =1
1
M
M
 Sxj
j =1
2
or S x =
n
j =1
M
Sx =
Sx
MN
or S x =
Sx
M
N
j =1
j
Sx j
n
j =1
Pooled standard deviation of the mean :
j
j
©1997 by M. Kostic
Least-Square Regression
Arbitrary (our choice function): yc
yc ,i = f ( xi , a0 , a1 ,...a j , ...am )
y
where aj are coefficients to be found
The sum of deviations squared
should be minimum :
2
D =  d i = (yi  yc , i ) 2  min
yi
yc,i
i
i = 1,2,...n
i
(yi  yc , i ) = d i
x
xi
Given data points: { xi , yi }, i = 1,2,...n
©1997 by M. Kostic
Least-Square Regression (2)
Given data points : {xi , yi }, i = 1,2,...n
to curve  fit with an arbitrary (our choice function) : yc
yc ,i = f ( xi , a0 , a1 ,...a j , ...am ) where a j are coefficien ts to be found
the sum of deviations squared should be minimum :
D =  d i = (yi  yc , i ) 2  min ; i = 1,2,...n; then ...
2
i
i
D
= 0  a j (for j = 0,1,2,...m) could be solved from (m  1) eqs.
a j
Click for Polynomial Curve-Fit
Click for Arbitrary Curve-Fit
©1997 by M. Kostic
Correlation Coefficient
Given data points : {xi , yi }, i = 1,2,...n and curve  fit function
yc ,i = f ( xi , a0 , a1 ,...a j , ...am ) with a j coefficien ts,
the correlatio n coefficien t, r , is :
r = 1
where :
1
2
S xy =  (yi  yc , i ) 2
n
i
S xy2
S
2
y
If Sxy=Sy and Sxy=0, respectively
, 0  r 1
and
For the simplest,
zeroth order polynomial fit.
1
2
Sy =
(y

y
)
 i
N 1 i
Click for Polynomial Curve-Fit
2
Click for Arbitrary Curve-Fit
©1997 by M. Kostic
Data Outlier
%Pin (zOL)
%Pout(zOL)
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
blimit= zOL = zOL(%Pin or %Pout)
©1997 by M. Kostic
Required #of Measurements
Mean precision interval : CI = 2d = u =  d = tn ,% P
2
Sx
; then...
N
 tn ,% P S x 
 ; since n = N  (m  1) m =0 = N  1
N = 
 d 
the calculatio n procedure is iterative (unless N  , too large)