Experimental
Experimental Errors
Errors and
and Error
Error
Analysis
Analysis
Rajeev Prabhakar
University of Texas at Austin, Austin, TX
Freeman Research Group Meeting
March 10, 2004
Topics covered
• Types of experimental errors
• Reducing errors
• Describing errors quantitatively
– Measured variables
– Quantities calculated from measured variables (error
propagation)
• Least-squares fit
– straight line
– polynomial
– arbitrary function
Ref: Bevington and Robinson, Data Reduction and Error Analysis for the
Physical Sciences, 2nd edition, McGraw Hill, NY 1992
Quality of Experiments and Experimental Error
• Accuracy
– Measure of correctness of result
• Precision
– Measure of reproducibility of result
• Types of errors
– Mistakes
• Wrong membrane used
• Number recorded incorrectly, 1.92 written as 1.29
– Systematic error
• Change in membrane casting conditions
• Incorrect calibration
– Random error
• Fluctuations in temperature, pressure etc.
Reducing Errors
• Planning experiments carefully
– Objective is to understand and reduce sources of
systematic error
– Don’t give in to the “it’s always been done this
way” mentality
• Use more accurate instruments
• Repeat the measurement, if possible
Uncertainties in Measured Variables
• Fluctuations
– Often independent of actual value
• Reading an analog instrument
– ½ of smallest scale division/least count
• Uncertainty estimate – standard deviation, σ
– ½ of least count
– Repeat measurement if possible and take
standard deviation of the data
Parameters calculated from measured variables
• Propagation of Errors
– e.g. Volume measurement
Vo = LoWo H o
If the measured values are L, W and H ,
 ∂V 
 ∂V 
 ∂V 
V ( L,W , H ) Vo + ∆L 
+ ∆W 

 + ∆H 

 ∂L WoHo
 ∂W  LoHo
 ∂H  LoWo
Error , ∆V = V − Vo
2
2
N
N




1
1
2
Variance, σ V =  ∑ (Vi − Vo )  = lim
Vi − V 
∑
N →∞ 
 N i =1

 N i =1

(
)
Propagation of errors – general formula
y = f (u , v)
 ∂y 
 ∂y 
yi − y = (ui − u )   + (vi − v)  
 ∂u 
 ∂v 
2
N


1
σ y2 = lim
y
−
y
∑
i

N →∞ 
N
1
i
=


(
)
2
2


1
y
y
∂
∂

 ∂y   ∂y  




2
2
2
σ y = lim
 ∑ (ui − u )   + ∑ (vi − v)   + ∑ 2(ui − u )(vi − v)     
N →∞
 ∂u 
 ∂v 
 ∂u   ∂v  
 N 
2
2
 ∂y 
 ∂y 
 ∂y   ∂y 
σ y2 = σ u2   + σ v2   + 2σ uv2    
 ∂u 
 ∂v 
 ∂u   ∂v 
(1)
co-variance
Neglecting the co - variance,
2
 ∂y 
 ∂y 
σ y2 σ u2   + σ v2  
 ∂u 
 ∂v 
2
(2)
Volume example – simplified equation
2
 ∂y 
2  ∂y 
σ σ   +σv  
 ∂u 
 ∂v 
2
y
2
2
u
2
2
2
2  ∂V 
2  ∂V 
2  ∂V 
σV σ L 
 + σW 
 +σH 

L
W
H
∂
∂
∂






Since V = LWH ,
σ σ (WH ) + σ
2
V
2
2
L
2
2
W
( HL ) + σ ( LW )
2
2
H
2
V 
2  V 
2  V 
σ σ   + σW   + σ H  
L
W 
H
2
V
2
2
2
2
L
2
2
2
 σV   σ L   σW   σ H 
 V   L  + W  + H 
    
 

2
Simplified form of equation valid when the dependent variable is a
product of independent variables to the 1 or -1 power only
Example - Limitations of simplified equation
If V = π R 2 H ,
2
 ∂V 
2  ∂V 
σ σ 
 +σH 

 ∂R 
 ∂H 
2
V
2
R
σ σ ( 2π RH ) + σ
2
V
2
2
2
R
2
H
(π R )
2
2
 2V 
2  V 
σ σ   +σH  
 R 
H
2
V
2
2
R
2
2
 σ V   2σ R   σ H 
 V   R  + H 
  
 

2
2
Least-squares fit to a straight line
• y = a + b*x
• Objective is to find the values of a and b that provide the
best prediction of y for a given x
2
χ
• Goodness-of-fit parameter,
1
2
2
χ = ∑  2 ( yi − a − bxi ) 
σ i

(3)
To minimize the error,
∂χ 2
∂χ 2
=0=
∂a
∂b
• Resulting equations for a and b – pgs. 104 and 113 of
Bevington
• Errors in a and b - Equations on pgs. 109 & 114 of Bevington
Least-squares fit to an equation linear in the
coefficients
• Example,
Polynomial: y = a + b*x + c*x2 + …..
Other functions: y = a + b*ex
• Similar mathematical treatment as for straight line. However,
determinants get bigger and more numerous. See pg 117
(last 3 lines) and 118 for example of quadratic equation.
• Alternate method – Matrix solution (pg. 121)
Matrix Method for equation linear in coefficients
m
y ( xi ) = ∑ ak f k ( xi )
k =1
The coefficient matrix a is given by,
a = βα −1
where
1

β k = ∑  2 yi f k ( xi )  ; row matrix
σ i

1

α lk = ∑  2 f l ( xi ) f k ( xi )  ; symmetric matrix
σ i

Error in the coefficients are elements of the inverse α matrix
- Diagonal elements are variances
σ a2k = α kk−1
- Off-diagonal elements are co-variances
σ a2l ak = α lk−1
Example – Quadratic fit to Permeability data
P = b + c ∆p + d ( ∆ p ) 2
Compare with,
m
y ( xi ) = ∑ ak f k ( xi ) = a1 f1 ( xi ) + a2 f 2 ( xi ) + a3 f 3 ( xi )
k =1
Therefore,
a1 = b; a2 = c; a3 = d ;
f1 = 1;
f 2 = ∆p;
f 3 = ( ∆p ) 2
Solve for a, b and c using matrix method
You will also obtain variances and covariances (previous slide)
Then, calculate error in P using eqs. (1) or (2).
Example in Excel file: Error analysis demo.xls; sheet: P
For more details, including an example, Bevington: pgs. 121-125
Least-squares fit to any function
• Best-fit for functions that are not linear in the coefficients
• Trial-and-error methods (Ch. 8, Bevington)
• Find coefficient values by using SOLVER function in Excel
•
2
χ
To get errors, vary one coefficient at a time to increase
by 1
• The change in the coefficient is it’s standard deviation
Example – S fit to dual mode equation
f *g
S =e+
1+ g * p
Determine coefficients by minimizing χ 2 ,
2
N


S
S
−


1
χ 2 =  ∑  calc exp t  
σi
 N i =1 
 
by varying the coefficients e, f and g.
Change initial guess values of the coefficients to
confirm best - fit values.
Then determine σ for each coefficient, by the
method described on the previous slide.
Example in Excel file: Error analysis demo.xls; sheet: S
Calculating Deff from best fit equations of P and S

dP   dp 
Deff (C2 ) =  P + ∆p
  dC 
∆
d
p
 p2

 p2 
b + 2c(∆p ) + 3d (∆p) 2 NUMR
=
=
fg
DENR
e+
(1 + g p ) 2
Calculate NUMR & DENR and their errors.
Calculate Deff .
 σ Deff
Error in Deff , 
D
 eff
2
  σ NUMR  2  σ DENR  2
 = 
 +

NUMR
DENR
 

 
References
Bevington
•Pgs. 1-6: Accuracy, precision and uncertainties
•Pgs. 41-48: Propagation of errors with specific examples
•Pgs. 101-114: Least-squares fit to a straight line
•Pgs. 115-125: Least-squares fit to a function linear in the
coefficients, including an example for a
quadratic fit
Microsoft Excel Help
•MINVERSE worksheet function
•MMULT worksheet function
Questions